**On the Efficiency in Electrical Networks with AC and DC Operation Technologies: A Comparative Study at the Distribution Stage**

#### **Oscar Danilo Montoya 1,2,\*, Federico Martin Serra <sup>3</sup> and Cristian Hernan De Angelo <sup>4</sup>**


Received: 23 July 2020; Accepted: 18 August 2020; Published: 20 August 2020

**Abstract:** This research deals with the efficiency comparison between AC and DC distribution networks that can provide electricity to rural and urban areas from the point of view of grid energy losses and greenhouse gas emissions impact. Configurations for medium- and low-voltage networks are analyzed via optimal power flow analysis by adding voltage regulation and devices capabilities sources in the mathematical formulation. Renewable energy resources such as wind and photovoltaic are considered using typical daily generation curves. Batteries are formulated with a linear representation taking into account operative bounds suggested by manufacturers. Numerical results in two electrical networks with 0.24 kV and 12.66 kV (with radial and meshed configurations) are performed with constant power loads at all the nodes. These simulations confirm that power distribution with DC technology is more efficient regarding energy losses, voltage profiles and greenhouse emissions than its AC counterpart. All the numerical results are tested in the General Algebraic Modeling System widely known as GAMS.

**Keywords:** alternating current networks; direct current networks; optimal power flow; non-linear optimization; control of power electronic converters

#### **1. Introduction**

#### *1.1. General Context*

Presently electrical distribution networks are essential systems in economic development around the word [1,2]; these grids are also responsible for distributing energy from large-scale power systems to all end users at medium and low voltage levels [3], which implies that in terms of size, the distribution networks are the lengthiest infrastructure inside of the power system [4,5]. This is important since higher losses can be presented at distribution networks in comparison to power systems (transmission and sub-transmission networks), e.g., in the Colombian context, energy losses at distribution networks can be between 6% and 18% while losses at transmission networks can be between 1% and 2% [6]. Recent advancements in power electronics, renewable energy, and energy storage technologies have made distribution networks be focused on the current modernization of power systems. In this sense, three main tendencies can be identified as follows: (i) expanding the existing distribution

networks using conventional AC technologies considering AC–DC inverters to interface the distributed energy resources [7]; (ii) Use of DC feeders to expand distribution networks taking the advantages of renewables and batteries that can work directly under the DC paradigm [8]; (iii) design hybrid distribution networks using AC and DC feeders taking the advantages of these technologies regarding reliability and security in the network operation, particularly in the new microgrids environment [3]. To analyze these possible distribution network configurations, power flow and optimal power flow models (steady-state analysis) appear to be essential tools in the literature [9]. These methods (convex and heuristics) determine the state variable (voltage magnitudes and angles) for a particular load condition being applicable to AC and DC networks with minimal changes [3].

#### *1.2. Motivation*

The analysis of electrical distribution networks from the point of view of power flow and optimal power flow is a fundamental step to validate the efficiency of these grids regarding energy losses, voltage profiles and conductor chargeability. In this sense, this research is motivated in the analysis of electrical distribution networks using AC or DC technologies in order to identify their performance regarding efficiency in terms of energy losses and greenhouse gas emissions, when it is selected one of both technologies for distributed electricity at medium-voltage level [2].

#### *1.3. Brief Literature Survey*

Electrical distribution networks had been designed under the AC paradigm for decades [10,11]; however, presently multiple reports can be found where distribution networks are analyzed under the DC paradigm, some of them are recompiled below.

The authors of [3] have presented an optimal power flow model for multi-terminal DC networks in medium-voltage levels where the energy losses in power converters have been added with quadratic constraints. These constraints allow the obtaining of an equivalent convex optimization model easily solved with semidefinite programming. In Reference [12] an optimization model for optimal phase-balancing in DC low-voltage distribution circuits with a bipolar configuration, which is represented through a Mixed-Integer Non-Linear Programming (MINLP) multi-objective model, has been presented. Numerical results demonstrated that phase-balancing reduces energy losses significantly when compared to the benchmark case. The authors of [13] have proposed the optimal location of photovoltaic sources in DC networks to minimize the total greenhouse gas emissions of CO2 in rural networks. The proposed optimization model has an MINLP structure and it was solved through the General Algebraic Modeling System (GAMS) optimization software. In Reference [8,14] three approaches for optimal operation of battery energy storage systems in DC networks using day-ahead economic formulations, have been presented. The main idea of those works is the minimization of the daily energy purchasing costs in slack nodes by using metaheuristic and convex optimization methods with excellent results when these are compared to the benchmark cases. In Reference [15] a convex optimization model added to a branch and bound approach to solve the problem of optimal reconfiguration of DC networks, has been proposed. The main advantage of this approach is that the global optimal solution is guaranteed via second-order cone optimization, applied to a study case using real-time simulations. The authors in [16] have proposed a MINLP model for optimal location and reallocation of battery energy storage systems in DC grids to reduce the daily energy losses and the total energy purchasing costs in the conventional sources. Numerical results demonstrated that the location of the batteries is dependent on the performance index used as the objective function, i.e., energy losses or energy purchasing costs; in this sense, authors have demonstrated a multi-objective compromise between both objectives. All the simulations were carried-out in the GAMS optimization package.

Regarding AC distribution networks multiple works related with power system planning and operation have been proposed in scientific literature. Some of these works are: optimal reconfiguration of distribution networks [17–19], optimal location of shunt capacitor banks and distributed generators [20–22], optimal selection of wire gauges in radial distribution networks [23–25], optimal location and coordination of protective devices [26–28], optimal location and operation of battery energy storage systems [7,14,29], and optimal planning of AC distribution networks including new substations [5,30].

It is important to mention that for all aforementioned approaches regardless the operation technology, i.e., AC or DC paradigms, the concepts of power flow and optimal power flow analyses are essential to determine their operative conditions [31]. This clearly implies that these concepts can be used to address both technologies and compare them regarding greenhouse gas emissions, energy purchasing costs and grid energy losses as will be addressed in this contribution.

#### *1.4. Contribution and Scope*

The main contributions of this research can be summarized as follows:


Additionally, the main considerations taken into account in the development of this work are: (i) to make distribution AC and DC technologies it is supposed that all the loads in the AC grid operate with unity power factor (only applicable for AC loads), (ii) in the case of loads connected in the DC grid that require reactive power support (e.g., motors), these are interfaced via power inverters that can provide this support without affecting the operative condition of the DC grid, and (iii) a low-voltage grid operating with 240 V and 60 kW of load is considered to present the effect of the AC or DC distribution network in residential applications, while a medium-voltage grid (12.66 kV) allows to compare AC and DC technologies when considerable reactive power appear in loads.

Observe that this research is focused on the efficiency comparison between AC and DC grids from the point of view of the distribution stage, i.e., when AC or DC technologies are used to transfer power from conventional and renewable generators to loads and batteries; for this reason, we assume that power losses in all the conversion stage are similar when these devices are interfaced in AC and DC grids, which allows us to consider them as equals in both scenarios for comparison purposes in the distribution layer.

Please note that we include ahead in this paper a section dedicated to the analysis of voltage source converters since these devices are essential in the DC distribution paradigm [3]. We introduced these devices with a classical passivity-based controller that operates these devices in the inversion mode, i.e., these are used to provide AC power to three-phase loads assuming a constant voltage source in the DC side [32]. However, these devices can also be used as the main sources when AC conventional networks interface with DC distribution feeders [33].

#### *1.5. Organization of the Document*

The remainder of this document is rearranged as follows: Section 2 presents the complete mathematical formulation of the multiperiod optimal power flow problem for AC grids as well as the necessary simplifications to derive the equivalent DC model. Section 3 presents the main characteristics of the GAMS software to solve non-linear non-convex optimization problems with a small test feeder composed by six nodes and five lines that operates with 240 V to to meet a total load about 60 kW. Section 4 presents the integration of three-phase loads in DC networks using voltage source converter interface. In addition, a general control design to guarantee sinusoidal voltage profile in the AC load regardless the active and reactive power consumption via passivity-based control methods is presented. Section 5 presents all the numerical information regarding the medium-voltage distribution network analyzed, which is composed by 33 nodes, 32 lines and operates with a nominal voltage of 12.66 kV. This system has four renewable generators (two photovoltaic-based generators and two wind turbines), and three battery energy storage systems (note that these renewable sources and battery energy storage systems are indeed composed by DC sub-networks interfaced with power electronic converters to manage the power transferred (absorbed) to (from) the DC or AC distribution networks). Section 6 presents all the numerical simulations on the 33-nodes test feeder considering multiple simulation cases. Section 7 shows the main conclusions derived from this work as well as some possible future works.

#### **2. Mathematical Formulation**

To compare electrical AC and DC distribution to provide electrical service in rural areas we assume that all the power consumptions have unity power factor and the voltage profile for both technologies is the same. The main characteristic of the proposed formulation is that the distribution network lacks of a voltage controlled node, since the operation is governed by the best coordination between renewable, batteries, and fossil fuels that guarantees the power supply to all the loads during a daily operative scenario. Regarding possible objective function in rural isolated areas two operative scenarios are considered: (i) minimization of the total grid energy losses, and (ii) minimization of the greenhouse emissions by diesel generators. Both objective functions are formulated under an optimal power flow environment.

In the case of the mathematical model of the battery energy storage systems we assume a linear representation to facilitate the implementation in GAMS environment based on the simplified model proposed in [34] which considers that in the conversion stage all the energy losses are neglected, i.e., 100% of efficiency in all the power electronic interface [7,14]; nevertheless, if more accurate battery models are required, then, references [35,36] can be consulted.

#### *2.1. Optimal Power Flow Model in AC Grids*

The Optimal power flow (OPF) problem in AC networks is a classical non-linear non-convex optimization problem due to the presence of the active and reactive power balance equations. Here, we consider OPF formulation presented in [7] to analyze distribution networks without constant voltage suppliers. The complete formulation of the OPF problem for AC grids is presented below:

Objective Functions

$$\min z\_{\mathbb{B}^{\mathbf{e}}}^{\mathsf{a}\mathbf{c}} = \sum\_{t \in \mathcal{T}} \sum\_{i \in \mathcal{N}} T\_i^{\mathbb{B}^{\mathbf{e}}} p\_{i,t}^{d\mathbf{g}} \Delta t\_{\prime} \tag{1a}$$

$$\min z\_{\text{loss}}^{\text{ac}} = \sum\_{t \in \mathcal{T}} \sum\_{i \in \mathcal{N}} \sum\_{ij \in \mathcal{N}} Y\_{ij} v\_{i,t} v\_{j,t} \cos \left(\delta\_{j,t} - \delta\_{i,t} + \theta\_{ij} \right) \Delta t\_{\text{s}} \tag{1b}$$

where *z*ac ge and *z*ac loss are the objective function values related to the amount of greenhouse emissions and energy losses per day, respectively. *Tge <sup>i</sup>* represents the quantity of CO2 emitted to the atmosphere in *kg kWh* by a diesel generator connected at node *i*, *p dg <sup>i</sup>*,*<sup>t</sup>* is the active power delivered by the diesel generator connected at node *i* in the period of time *t*; Δ*t* is the length of the period of time considered (typically 1 h). *Yij* is the value of the admittance that relates nodes *i* and *j*, which have voltages *vi*,*<sup>t</sup>* and *vj*,*<sup>t</sup>* at each period of time *t*. *δi*,*<sup>t</sup>* (*δj*,*t*) is the angle of the voltage at node *i* (*j*) in the interval of time *t*, and *θij* is the angle of the admittance between nodes *i* and *j*. Please note that N and T are the sets that contains all the nodes in the grid and the total of periods of time of the operation horizon.

#### Set of Constraints

The power balance equations in the AC power flow are related to the amount of active and reactive power injected at each node *i* in each period of time *t*. These take the following form:

$$p\_{i,t}^{d\text{g}} + p\_{i,t}^{rs} + p\_{i,t}^b - p\_{i,t}^d = v\_{i,t} \sum\_{j \in \mathcal{N}} \mathbf{Y}\_{ij} v\_{j,t} \cos \left(\delta\_{i,t} - \delta\_{j,t} - \theta\_{i\bar{j}}\right),\tag{2a}$$

$$q\_{i,t}^{d\mathcal{S}} + q\_{i,t}^{rs} + q\_{i,t}^b - q\_{i,t}^d = \upsilon\_{i,t} \sum\_{j \in \mathcal{N}} \mathcal{Y}\_{ij} \upsilon\_{j,t} \sin \left(\delta\_{i,t} - \delta\_{j,t} - \theta\_{ij} \right), \tag{2b}$$

where *prs <sup>i</sup>*,*<sup>t</sup>* and *<sup>q</sup>rs <sup>i</sup>*,*<sup>t</sup>* are the active and reactive power generation by renewable sources connected at node *i* in the period of time *t*; *p<sup>b</sup> <sup>i</sup>*,*<sup>t</sup>* and *<sup>q</sup><sup>b</sup> <sup>i</sup>*,*<sup>t</sup>* are the active and reactive power capabilities in batteries and *p<sup>d</sup> <sup>i</sup>*,*<sup>t</sup>* and *<sup>q</sup><sup>d</sup> <sup>i</sup>*,*<sup>t</sup>* represent the active and reactive power demands, respectively. Please note that in the literature it is recommended that batteries can operate with unity power factor which implies that *qb <sup>i</sup>*,*<sup>t</sup>* = 0 [14,37].

Constraints related with batteries are listed below:

$$So\mathbb{C}\_{i,t+1}^{b} = SoC\_{i,t}^{b} - \boldsymbol{\varrho}\_{i}^{b}\boldsymbol{p}\_{i,t}^{b}\Delta t,\tag{3a}$$

$$\text{SoC}\_{i}^{b,\text{min}} \le \text{SoC}\_{i,t}^{b} \le \text{SoC}\_{i}^{b,\text{max}},\tag{3b}$$

$$p\_i^{b, \text{min}} \le p\_{i,t}^b \le p\_i^{b, \text{max}},\tag{3c}$$

$$\text{SoC}^{b}\_{i,t\_0} = \text{SoC}^{b,\text{initial}}\_i,\tag{3d}$$

$$\text{So} \mathsf{C}\_{i,t\_f}^b = \text{So} \mathsf{C}\_i^{b,\text{final}},\tag{3e}$$

where *SoC<sup>b</sup> <sup>i</sup>*,*<sup>t</sup>* is the state-of-charge of the battery *b* connected at node *i* in the period of time *t*, which is bounded by *SoCb*,min *<sup>i</sup>* and *SoCb*,max *<sup>i</sup>* ; note that the state-of-charge can be interpreted as the quantity of energy stored in the battery in percentage. *ϕ<sup>b</sup> <sup>i</sup>* is the charging/discharging coefficient of the battery *b*. *pb*,min *<sup>i</sup>* and *<sup>p</sup>b*,max *<sup>i</sup>* represent the minimum and maximum power allowable draws/injections at node *i* for secure operation of the battery at each period of time. *SoCb*,initial *<sup>i</sup>* and *SoCb*,final *<sup>i</sup>* represent the initial and final state-of-charges defined by the utility to operate the battery, i.e., the initial condition of the economic dispatch problem at *t*<sup>0</sup> and the final operative consign at the end of the operation period *tf* .

The complete interpretation of the mathematical models (1)–(3) is as follows: Expression (1a) determines the value of the objective function regarding greenhouse gas emissions to the atmosphere by diesel generation. Equation (1b) defines the total energy losses in all the conductors of the network during the operation horizon (i.e., typically 24 h). Expressions (2a) and (2b) define the power balance constraints regarding active and reactive components of the power at each node. Equation (3a) calculates future the state-of-charge in the battery for each period of time as function of the current charge and the power injection/absorption to/from the grid. Expressions (3b) and (3c) determine the maximum and minimum values allowed to the state-of-charge (energy stored) in the battery as well as its maximum power injection (discharging state) or absorption (charging state), respectively. Finally, Equations (3d) and (3e) determine the initial state-of-charge of the battery and the final operative consign defined by the utility. These are defined as function of the operational requirements of the network. Nevertheless, in specialized literature it is recommended for Ion-Lithium batteries to start and end the day with 50% of state of charge [14].

**Remark 1.** *The mathematical model for the optimal operation of AC networks with renewables and batteries defined from (1) to (3) is non-linear and non-convex due to the power balance constraints which makes difficult to reach the global optimum [8]. For this reason, here we recurred to the GAMS optimization software to solve this problem to make our results comparable to those obtained from the mathematical model regarding DC grids reported in next subsection.*

**Remark 2.** *The studied optimization model (1)–(3) corresponds to a single-phase representation of AC distribution network that can be used if: (i) the AC network is indeed a single-phase network which is the most typical scenario in low-voltage distribution environments, or (ii) it is a three-phase balanced distribution network that can be represented through a single-phase equivalent model [7].*

#### *2.2. Optimal Power Flow Model in DC Grids*

The mathematical formulation of the optimal power flow problem for DC networks can be obtained directly from the AC formulation by simplifying the objective function regarding energy losses minimization. Also, power balance constraints can be simplified as follows:


With these assumptions, the objective function of the DC optimal power flow and the power balance constraint takes the following forms:

Objective Function Regarding Energy Losses

$$\min z\_{\text{loss}}^{\text{dc}} = \sum\_{t \in \mathcal{T}} \sum\_{i \in \mathcal{N}} \sum\_{\text{i} \dot{\jmath} \in \mathcal{N}} G\_{\text{i}\dot{\jmath}} \upsilon\_{i,t} \upsilon\_{\text{j},t} \Delta t\_{\text{s}} \tag{4}$$

where *z*dc loss is the amount of energy losses in all the branches of the DC network.

Power Balance Constraint

$$p\_{i,t}^{d\p} + p\_{i,t}^{rs} + p\_{i,t}^b - p\_{i,t}^d = \upsilon\_{i,t} \sum\_{j \in \mathcal{N}} G\_{ij} \upsilon\_{j,t\_{\prime}} \tag{5}$$

**Remark 3.** *The complete mathematical model of the optimal power flow problem in DC grids is composed by the greenhouse gas emission objective function (1a), the objective function regarding energy losses minimization defined by (4), the power balance constraint (5) and the remainder of constraints defined in (3).*

It is worthy to mention that the model of the optimal power flow in DC grids is also non-linear and non-convex due to the product between voltage variables in the power balance constraint, which makes necessary to use specialized software (i.e., GAMS) to solve it efficiently [16].

#### **3. Solution Methodology**

To solve the optimal power flow problems in AC and DC grids in this paper it is selected the GAMS software to implement their mathematical structures with a non-linear programming (NLP) solver that typically works with interior point methods to reach the optimal solution [36,38].

GAMS software has been largely used in specialized literature to address non-linear non-convex optimization problems in many areas of engineering, some of these works are: optimal planning and operation of power systems with batteries in AC and DC networks [36,37,39,40]; optimal location of distributed generators [13,22,41,42]; optimal design of osmotic power plants [43]; optimal design of water distribution networks [44]; stability analysis in DC networks [45]; optimal design of thermoacoustic engines [46]; optimal location of protective devices [47], and optimal planning of distribution networks [30], among others.

In general terms, GAMS software is a powerful optimization package that solves complex optimization problems focused on the mathematical formulation of the problem rather than the solution methodology [38]. This represents an ideal situation to introduce students and researchers with mathematical optimization [36]. The main characteristics of the GAMS software can be summarized as follows:


A numerical example is presented to understand the use of GAMS software to solve optimization problems with non-linear and non-convex structure. To do so, below it is presented the solution of the power flow problem for a small test feeder that can be operated with AC or DC technology. In the case of the AC technology, it is important to mention that this grid corresponds to a low-voltage distribution network with a single-phase structure, which is the most typical operation case in low-voltage applications [48]. The configuration of this test feeder is presented in Figure 1. This test feeder is composed by six nodes and five distribution lines (radial topology). The information of the branches and loads is reported in Table 1. Please note that to make both configurations comparable, we assume unity power factor in all the loads. In addition, this system operates with 240 V typically found in Colombian AC grids.

**Figure 1.** Electrical configuration for the 6-nodes test system used in the GAMS example.

**Table 1.** Branches and load information.


Please note that this example is a typical low-voltage distribution network where distribution transformers have nominal power of 75 kVA. The implementation of the optimal power flow problem for this test system considers:


The simplified mathematical model for the optimal power flow in AC grids is presented below:

$$\min \min \; p\_{\text{loss}} = \sum\_{i \in \mathcal{N}} v\_i \sum\_{j \in \mathcal{N}} \mathbf{Y}\_{ij} v\_j \cos \left(\delta\_i - \delta\_j - \theta\_{ij} \right), \tag{6a}$$

$$p\_i^{d\xi} - p\_i^d = \upsilon\_i \sum\_{j \in \mathcal{N}} \Upsilon\_{ij} \upsilon\_j \cos \left(\delta\_i - \delta\_j - \theta\_{ij}\right),\tag{6b}$$

$$q\_i^{d\mathcal{g}} - q\_i^d = v\_i \sum\_{j \in \mathcal{N}} Y\_{ij} v\_j \sin \left(\delta\_i - \delta\_j - \theta\_{ij} \right). \tag{6c}$$

The implementation of the simplified mathematical model (6) for the optimal power flow analysis in AC grids is presented in Listing 1.

In the case of the DC model the set of equations reported in (6) can be simplified as presented in Equation (7)

$$\text{minim } p\_{\text{loss}} = \sum\_{i \in \mathcal{N}} \upsilon\_i \sum\_{j \in \mathcal{N}} G\_{ij} \upsilon\_{j\prime} \tag{7a}$$

$$p\_i^{d\chi} - p\_i^d = v\_i \sum\_{j \in \mathcal{N}} G\_{ij} v\_{j\prime} \tag{7b}$$

The implementation of this simplified mathematical model (6) for the optimal power flow analysis in DC grids is presented in Listing 2.

Once both models (i.e., Equations (6) and (7)) are solved using the CONOPT solver in GAMS by using algorithms presented in Listings 1 and 2, we reach the solution of the voltage profiles reported in Table 2. Please note that for both cases the lowest voltage occurs at node 6 being 207.86 V for the AC grid and 208.24 V for the DC network. These results imply a difference of about 0.38 V between both networks.


**Table 2.** Voltage profile for the AC and DC grids.

Additionally, power losses in the AC grid are 2.40 kW, and 2.39 kW in the DC case (i.e., a difference about 10 W). These results demonstrate that even considering unity power factor at all the loads, the distribution using DC technology is more efficient than the AC counterpart. The voltage drop in lines is lower due to the irrelevance of inductive reactance of the lines.

An important fact when comparing AC and DC configurations is the amount of reactive power required by the AC grid to operate adequately. In this sense, in this small example the grid needs to generate about 3.21 kVAr, which corresponds to reactive power losses through all the lines. This obviously does not occurs in the DC paradigm since reactive effect is not presented as previously mentioned. An additional simulation case in this small test feeder is made in the case of the AC distribution network by considering that all the loads are operated with a lagging power factor of 0.95. In these conditions, the total grid losses exhibit by this network are about 2.77 kW and the minimum voltage profile is about 221.04 V at node 6. These results imply that in comparison to the unity power factor operation case, the reactive power consumption at all the loads makes that the power losses to

be increased about 0.37 kW, requiring at the substation node 23.40 kVAr to supply the reactive power requirements at all the loads; and that the voltage at the node has worsened about 5.76 V. Please note that when reactive power is considered for loads in the AC grid, its behavior affects and worsens the results of the comparison with respect to the unity Please note that these results are particularly important when voltage increase to medium level, since these differences become in tens of volts and hundreds of watts. In Section 6 these impacts will be widely discussed under an economic dispatch environment in the 33-node test feeder taking into account higher penetration of renewables and battery energy storage systems.

**Listing 1.** Algorithm implemented in GAMS for OPF model (6).

```
1 SETS
2 i set of nodes /N1*N6/
3 g set of generators /G1/
4 map (g,i) Associates node with gen /G1.N1 /;
5 alias (i,j);
6 SCALARS
7 vmax Maximum voltage bound /1.10/
8 vmin Minimum voltage bound /0.90/
9 v0 Slack voltage /1.00/
10 d0 Slack angle /0.00/;
11 PARAMETER Pd(i)
12 /N1 0,N2 0.16, N3 0.11, N4 0.15, N5 0.10, N6 0.08/;
13 TABLE Ybus (i,j ,*)
14 Yij Thij
15 N1.N1 21.9587119756815 -0.977131269610956
16 N2.N1 21.9587119756815 2.164461383978837
17 N1.N2 21.9587119756815 2.164461383978837
18 N2.N2 34.2649004085076 -0.946099794913962
19 N3.N2 12.3355889037129 2.250751567181446
20 N2.N3 12.3355889037129 2.250751567181446
21 N3.N3 39.8699085005043 -0.930090535155209
22 N4.N3 16.4707942748332 2.094290056213377
23 N5.N3 11.2786162705081 2.339716087328560
24 N3.N4 16.4707942748332 2.094290056213377
25 N4.N4 16.4707942748332 -1.047302597376417
26 N3.N5 11.2786162705081 2.339716087328560
27 N5.N5 31.9116612264763 -0.909021422256346
28 N6.N5 20.7328356619391 2.174363262762511
29 N5.N6 20.7328356619391 2.174363262762511
30 N6.N6 20.7328356619391 -0.967229390827282;
31 VARIABLES
32 ploss Power losses variable
33 v(i) Magnitude of the voltage at node i.
34 d(i) Angle of the voltage at node i.
35 p(g) Active power generation at node i.
36 q(g) Reactive power generation at node i.;
37 v.lo(i) = vmin ; v.up(i) = vmax ;
38 d.fx('N1') = d0;
39 v.fx('N1') = v0;
40 EQUATIONS
41 ObjFun Objective function
42 PowerA (i) Active power balance per node .
43 PowerR (i) Reactive power balance per node .;
44 * Mathematical structure
45 ObjFun .. ploss =E= SUM (i,v(i) *SUM (j,v(j)* Ybus (i,j,'Yij')*
46 cos (d(i) -d(j) -Ybus (i,j,'Thij'))));
47 PowerA (i).. sum ( g$map (g,i) ,p(g)) - Pd(i)=E= v(i)* SUM(j,v(j)*
48 Ybus (i,j,'Yij')* cos (d(i) -d(j) -Ybus (i,j,'Thij')));
49 PowerR (i).. sum ( g$map (g,i) ,q(g)) =E= v(i) *SUM (j,v(j)*
50 Ybus (i,j,'Yij')* sin (d(i) -d(j) -Ybus (i,j,'Thij')));
51 MODEL OPF / ALL /;
52 OPTIONS decimals = 4;
53 SOLVE OPF us NLP min ploss ;
54 DISPLAY ploss .l, v.l, p.l;
```

```
1 SETS
2 i set of nodes /N1*N6/
3 g set of generators /G1/
4 map (g,i) Associates node with gen /G1.N1 /;
5 alias (i,j);
6 SCALARS
7 vmax Maximum voltage bound /1.10/
8 vmin Minimum voltage bound /0.90/
9 v0 Slack voltage /1.00/;
10 PARAMETER Pd(i)
11 /N1 0, N2 0.16, N3 0.11, N4 0.15, N5 0.10, N6 0.08/;
12 TABLE Ybus (i,j ,*)
13 Gij
14 N1.N1 39.2538523925385
15 N2.N1 -39.2538523925385
16 N1.N2 -39.2538523925385
17 N2.N2 58.8728228019427
18 N3.N2 -19.6189704094041
19 N2.N3 -19.6189704094041
20 N3.N3 68.7863929415566
21 N4.N3 -32.9475833900613
22 N5.N3 -16.2198391420912
23 N3.N4 -32.9475833900613
24 N4.N4 32.9475833900613
25 N3.N5 -16.2198391420912
26 N5.N5 52.7481410288836
27 N6.N5 -36.5283018867925
28 N5.N6 -36.5283018867925
29 N6.N6 36.5283018867925
30 VARIABLES
31 ploss Power losses variable
32 v(i) Magnitude of the voltage at node i.
33 p(g) Active power generation at node i.;
34 v.lo(i) = vmin ; v.up(i) = vmax ;
35 v.fx('N1') = v0;
36 EQUATIONS
37 ObjFun Objective function
38 PowerA (i) Active power balance per node .;
39 * Mathematical structure
40 ObjFun .. ploss =E= SUM (i,v(i) *SUM (j,v(j)* Ybus (i,j,'Gij')));
41 PowerA (i).. sum ( g$map (g,i) ,p(g)) - Pd(i) =E= v(i)* SUM (j,v(j) * Ybus (i,j,'Gij'));
42 MODEL OPF / ALL /;
43 OPTIONS decimals = 4;
44 SOLVE OPF us NLP min ploss ;
45 DISPLAY ploss .l, v.l, p.l;
```
**Listing 2.** Algorithm implemented in GAMS for the OPF model (7).

It is worth mentioning that as described in Section 5, the comparisons made in this research regarding the efficiency comparison between AC and DC paradigms are focused on the distribution stage, which implies that we will not consider power losses in the power electronic interfaces used for interfacing renewable sources and battery energy storage systems. Nevertheless, these will be able to be explored in future research regarding energy distribution technologies and power electronic interfaces for batteries and renewables.

**Remark 4.** *Regarding optimal implementation in GAMS environment of the OPF problem for AC and DC grids depicted in Listings 1 and 2, we can observe that the DC model is pretty simple with less variables (no angles), which makes this easiest to be solved since its nonlinearities are soft when compared to the AC model that contains trigonometric functions.*

#### **4. Generation Reactive Power in DC Grids with Voltage Source Converter Interfaces**

To provide apparent power to three-phase loads in medium- and low-voltage levels using DC distribution feeders, it is required a power electronic interface, i.e., voltage source converter (VSC) (A voltage source converter interface corresponds to a power electronic converter that is fed by a DC source that provides the required active power in the AC side, which is managed by controlling the switches (on or off) states via pulse-width modulation techniques to provide a sinusoidal voltage signal to AC loads [3,32]) that manages the active power interchange between the DC grid and the load at the same time that the reactive power is correctly provided to this load by the converter. In Figure 2 it is depicted the interconnection of a three-phase load to a DC distribution network with a voltage source converter and RLC filter.

**Figure 2.** Interconnection of three-phase loads to DC distribution networks via VSCs.

**Remark 5.** *The VSC interface is a power electronic device that can provide active and reactive power to AC loads when at the DC side it is interconnected a constant voltage source (inversion mode of operation), as the case of the DC distribution paradigm [32]; however, the same device can also be employed to transform AC energy into DC energy when is operated in the conversion mode, as the case of high-voltage direct power transmission systems [33], i.e., the case of DC distribution is used to interface the conventional AC grid to the DC working as the main transformer [49,50].*

To demonstrate that it is possible to manage the active and reactive power consumption in the three-phase load we consider the following facts:


Please note that the control objective in the power electronic interface presented in Figure 2 is to maintain the voltage across the capacitor *Cf* with purely sinusoidal form as presented below:

$$\begin{aligned} v\_a^\star &= \sqrt{2} V\_{\text{rms}} \sin \left(\omega t\right), \\ v\_b^\star &= \sqrt{2} V\_{\text{rms}} \sin \left(\omega t - 2 \frac{\pi}{3}\right), \\ v\_c^\star &= \sqrt{2} V\_{\text{rms}} \sin \left(\omega t + 2 \frac{\pi}{3}\right). \end{aligned} \tag{8}$$

where *V*rms is the root-mean-square value of the voltage in the point of load connection, *ω* is the angular frequency of the three-phase voltage which is defined as 2*π f* , being *f* the electrical frequency en hertz.

Since the desired voltages are three sinusoidal signals defined with positive sequence, then, the control design for the VSC presented in Figure 2 can be designed using the Park's reference frame

as demonstrated in [32]. The complete dynamical model of the three-phase VSC to support active and reactive power to a three-phase load as presented in Figure 2 takes the following form in the *dq*−reference frame:

$$\mathbf{L}\_f \frac{d}{dt} \mathbf{i}\_d = m\_d \boldsymbol{\upsilon}\_{d\mathbf{c}} - \mathbf{R}\_f \mathbf{i}\_d + \omega \mathbf{L}\_f \mathbf{i}\_q - \boldsymbol{\upsilon}\_{d\mathbf{v}} \tag{9a}$$

$$L\_f \frac{d}{dt} \dot{\mathbf{i}}\_d = m\_q \boldsymbol{\upsilon}\_{dc} - \mathcal{R}\_f \dot{\mathbf{i}}\_q - \omega L\_f \dot{\mathbf{i}}\_d - \boldsymbol{\upsilon}\_{q\_f} \tag{9b}$$

$$\mathbb{C}\_f \frac{d}{dt} v\_d = i\_d - \omega \mathbb{C}\_f v\_q - j\_{d\prime} \tag{9c}$$

$$\mathbb{C}\_f \frac{d}{dt} v\_q = i\_q + \omega \mathbb{C}\_f v\_d - j\_{q'} \tag{9d}$$

where *md* and *mq* are the modulation indexes in the *dq*−reference frame; *idq* and *jdq* are the current that flow in the inductor of the filter and the current absorbed by the three-phase load, *vdq* are the *dq* components of the voltage across the capacitor *Cf* .

The main characteristic of the dynamical model (9) is that it corresponds to a Hamiltonian system which can be easily controllable via passivity-based control theory as presented in [32]. The Hamiltonian model of (9) takes the following form:

$$\mathcal{D}\dot{\mathbf{x}} = \left[\mathcal{J} - \mathcal{R}\right]\mathbf{x} + \mathbf{g}u + \mathcal{J}\_{\prime} \tag{10}$$

where D is known as the inertia matrix based on its similarities with mechanical systems, J corresponds to the interconnection matrix which is skew-symmetric, R is the damping matrix which is positive semidefinite, *g* is the control input matrix, and *z* is a vector that contains external inputs. Please note that *x* is the vector of state variables and *u* is a vector with control inputs, respectively. Each of the aforementioned parameters and variables can be easily defined by comparing (9) and (10).

**Remark 6.** *The dynamical system (10) can be asymptotically stabilized by using an incremental representation as presented in [32] with a proportional-integral strategy over the passive output y as follows:* ˜

$$
\hbar \ddot{\mathbf{u}} = -\mathcal{K}\_p \ddot{\mathbf{y}} - \mathcal{K}\_i \int\_0^{t\_f} \ddot{\mathbf{y}} dt\_\prime \tag{11}
$$

*where Kp and Ki are defined as diagonal positive definite matrices and y*˜ *is gTx*˜*. In addition, the complete control is defined from the incremental model as u* = *u*˜ + *u*-*, where u is obtained by evaluating the equilibrium point xin (10).*

To show that the power electronic interface presented in Figure 2 is able to control active and reactive power in a three-phase load, let us consider the following parameters: *f* = 50 Hz, *Lf* = 1.25 mH, *Rf* = 0.20 Ω, *Cf* = 45 μF, *V*rms = 100 V and *vdc* = 311 V. The three-phase load is modeled as a combination between a resistance of 2 Ω and an inductance of 7.958 mH connected in parallel. During the period of time between 0 s and 150 ms only the resistive load is connected, then, when time simulation is greater than 150 ms, the inductive load is also connected. It is important to mention that these parameters imply an equivalent active power consumption of about 5 kW added with 4 kVAr at each phase.

In Figure 3 the voltage and current profile provided to the *a*-phase by the VSC interface to the three-phase load are presented.

Please note that the behavior of the *a*−phase current depicted in Figure 3 demonstrates that if the voltage profile is supported at the load terminals via passivity-based control as defined in (11), then, the active and reactive power required by the load is guaranteed. This implies that the power electronic interface presented in Figure 2 can be used to interface three-phase loads to DC distribution networks with local reactive power support (i.e., unity power factor operation). The main advantage

of this interface is that the reactive power is provided locally to the load and no power losses are observed by the DC grid caused by reactive currents, which clearly shows that DC grids are more efficient than AC grids when loads have power factors lower than unity, as it will be presented in section of Results.

**Figure 3.** Behavior of the voltage and current in the *a*-phase of the three-phase load.

#### **5. Test System**

The comparison of the AC and DC technologies for energy distribution in medium-voltage levels is made by using the 33-nodes test feeder reported in [7]. This test feeder is designed to be operated at 12.66 kV with the connections depicted in Figure 4. The information regarding branches and loads for this test feeder are reported in Table 3. Please note that this test feeder corresponds to a three-phase distribution network typically used to study the problem of the optimal location of distributed generators in power distribution networks as reported in [51].

To evaluate the effect of the renewable generation in the daily operation of this test feeder, we consider four renewable generators previously located in this test system with the information reported in Table 4.

The connection of the generators for each test system is described as follows: at the node 13 it is connected the photovoltaic generator PV1 and the wind turbine WT1 with nominal rates of 450 kW and 825 kW, respectively. At the node 25, it is connected a second PV2 with a nominal power rate of 1500 kW while at the node 30 it is connected the second wind generator WT2 with the nominal capability of 1200 kW. The information regarding battery energy storage systems considered in this test feeder is reported in Table 5. We assume that the utility has located three batteries, which are distributed as follows: at node 6, a C-type battery is located; at node 14 an A-type battery is used, and at the node 31, a B-type battery is considered.

**Figure 4.** Electrical configuration for the 33-nodes test system.





It is worth mentioning that each one of the renewable source or battery energy storage system corresponds to a DC sub-network interfaced with a power electronic converter that manages the power transferred(absorbed)/to(from) the distribution network regardless whether this is operated under AC or DC paradigm [3].



To evaluate the daily variation of the active and reactive power consumption and emulating the hourly price behavior, we consider the load and cost curves reported in Figure 5. Furthermore, as the peak of the electricity, we assume the information reported by CODENSA utility from Colombia in May 2019, which is COP\$/kWh 479.3389.

**Figure 5.** Typical behavior of load consumption and electricity spot market cost.

The numerical information about the demand and cost curves presented in Figure 5 can be consulted in [7].

#### **6. Numerical Results and Discussion**

In this section, all the numerical results reached by GAMS after implementing the OPF models for AC and DC networks are described. To make a fair comparison between both distribution technologies the following simulation scenarios are proposed:


It is important to mention that as recommended in [14] all the batteries start and end the day with 50% of state-of-charge and during the day (for Ion-Lithium batteries) this variable can be moved from 10% to 90%.

#### *6.1. Operation of the AC Network*

In the operation of this grid, we consider three objective functions as follows: Case 1: minimization of the energy losses during the operation horizon, Case 2: the minimization of the energy purchase costs in the conventional generator (node 1) considering the cost curve reported in Figure 5, and Case 3: the minimization of the total CO2 gas emissions in the slack source considering that the 33-nodes test feeder is a rural grid fed by a diesel generator. Here, we consider as reported in [13] that CO2 emission coefficient is 1300 lb/MWh.

Table 6 presents all the numerical results regarding the three objective functions for the S1.


**Table 6.** Simulation results in the 33-nodes test feeder operated with unity power factor.

The results in the S1 reported in Table 6 allows to observe that:


In Table 7 the numerical results for the S2, i.e., the operation of the 33-nodes test feeder considering active and reactive power consumptions are presented. In general, numerical results presents the same behavior reported in the analysis of the S1. Nevertheless, we can notice that energy losses are drastically affected by the presence of the reactive power consumptions inside of the network. Note that in the first case, energy losses have been incremented at least 5 times in comparison to the unity power factor case. In addition, regarding the minimization of the energy purchase costs in the spot market and the greenhouse gas emissions, the increments are about 1.04 times for both cases, which allows concluding that reactive power practically does not produce effects in those objectives when compared to the operation with unitary power factor.


**Table 7.** Simulation results in the 33-nodes test feeder operated with active and reactive power loads.

#### *6.2. Operation of the DC Network*

To evaluate the performance of the 33-nodes test feeder, it was considered that the system can be represented by a DC equivalent as defined in the S1. In this sense, the reactive power loads and reactance of this model are removed. The implementation of optimal power flow for this DC medium-voltage distribution network is reported in Table 8.

**Table 8.** Simulation results in the 33-nodes test feeder operated with DC technology.


From the results reported in Table 8 we can note that:


#### *6.3. Efficiency Comparison for Different Power Factors*

To demonstrate that the DC distribution network is an attractive alternative to provide electrical service to industrial users connected at medium-voltage levels, let us compare the efficiency of this technology with conventional AC grids for different percentages of reactive power consumptions.

**Remark 7.** *Recall that DC distribution networks are able to provide active and reactive power support to AC loads by using a voltage source converter (VSC) that interfaces the DC grid to the AC load as can be demonstrated in [32], where an isolated (i.e., rural) AC grid receives voltage and power support by interconnecting linear and non-linear loads to DC distribution grids via a VSC.*

In Figure 6 it is presented the amount of energy losses (objective function minimize) in the AC grid when the reactive power load changes from 0 to 120% of the peak value reported in Table 3; in addition, the total energy losses are also presented in the DC equivalent network, assuming that the energy losses at all the VSCs are about 10% of the total energy losses of the network. Please note that in the case of the DC grid the reactive power is provided directly at the load side, which implies that this power is provided by the power electronic interface as explained in Section 4. For this reason, the active

power losses for DC grids remains constant for different percentage of reactive power demands at the load side, since no currents are associated with this power flow in DC distribution lines.

**Figure 6.** Amount of daily energy losses for different penetrations of reactive power consumptions at the load side.

The information regarding daily energy losses in the AC and DC equivalent networks make evident that the efficiency of the AC grid is deteriorated rapidly as a function of the amount of reactive power consumption at all the loads due to the exponential increment of the total energy losses. Nevertheless, in the case of the DC network the efficiency is always constant regardless the reactive power consumption. This is explained by the fact that the VSCs that interface the AC loads are able to locally generate reactive power, which implies that the DC distribution makes the node can sense their effects in its lines. Please note that when the loads in the 33-nodes test feeders are 100% reactive power consumption, the AC grid has 948.979 kWh/day of energy losses, while in the case of the DC equivalent these losses are about 205.588 kWh/day. These results imply that the AC grid has at least 4.6 times more energy losses than the DC equivalent, which confirms that the DC technology is a promissory alternative to provide the electricity service at medium- and low-voltage distribution levels with higher efficiency levels regarding energy losses, compared at the distribution stage, i.e., involving energy losses in conductors used in the electricity distribution. It is important to mention that this high relation (i.e., 4.6 times for the 33-nodes test feeder) is largely influenced by the relation between active and reactive power demand in the distribution network under study. This implies that this data can be considered to be an indicator, but more studies regarding energy efficiency at all the electronic interfaces (renewable generators, energy storage devices and controllable loads) are needed to determine the overall energy efficiency at distribution levels.

#### *6.4. Effect of Renewable Energy Variations in the Economic Dispatch*

In this subsection it is explored the possible operation scenario where renewable energy has important variations regarding weather conditions such as cloudy and rainy days, including very low-speed winds. In this case, we consider as objective function the total energy losses minimization during daily operation, i.e., the Case 1. To consider all the possible operation scenarios in a real network, we consider that the amount of renewable energy varies from 0% to 100% in steps of 20%. In addition, we consider that all the loads in the 33-nodes test feeder operate under normal conditions, i.e., 100% of active and reactive power consumption.

Table 9 presents the behavior of the AC and DC distribution networks when there are higher variations in renewable energy production.


**Table 9.** Daily energy losses as a function of the renewable energy variations.

From results reported in Table 9 we can observe that: (i) the difference regarding daily energy losses between AC and DC grids remains practically constant with an average value of 739.0792 kWh/day. This implies that at all the possible renewable energy penetration scenarios the DC grid has better behavior in terms of grid energy losses, which can be explained by the possibility to provide local reactive power with the VSC interface. This latter is not the case of the AC grid where the reactive energy flows from the substation to the loads; (ii) the division between the DC and AC energy losses presented in the last column of Table 9 shows that the efficiency of the DC grid in comparison to the AC case increases as a function of the renewable energy penetration in the grid; this behavior is mainly associated with the important reductions in the power flow through the lines caused by local injections of active power by renewable sources; and (iii) the total energy reduction in the AC grid when renewable energy penetration passes from 0 % to 100% is about 59.72%, while in the case of the DC network this reduction is about 84.46%, which entails that the same level of renewable energy penetration provides more positive impacts in a distribution network designed under the DC paradigm in contrast with the conventional AC grids.

#### **7. Conclusions and Future Works**

A comparative study regarding energy efficiency in AC and DC electrical networks for power distribution from the point of view of optimal power flow analysis was presented in this paper. This study allowed to confirm that AC and DC technologies have identical performances in residential applications, i.e., unity power factor, since the amount of energy losses, greenhouse gas emissions of CO2 or energy purchase costs are practically the same for both technologies. Nevertheless, in the case of high penetration of reactive power consumptions in AC networks (mainly in industrial applications), it was demonstrated that the performance of the AC grid is rapidly deteriorated compared with the DC equivalent, due to the need to transport this reactive power from the substation towards the loads. This increases the magnitude of the current through the lines, being translated into higher energy losses during the operation horizon. This situation does not happen in the case of the DC grids where reactive power is directly provided by the VSCs that interfaces all the AC loads, which implies that the efficiency of the DC distribution system remains constant regardless the reactive power requirements of the load.

To solve the optimal power flow models regarding the daily operation of AC and DC grids, we have introduced the GAMS software to efficiently solves both models with low computation effort, i.e., processing times about 5 s in all the simulation cases and scenarios. This low-computational time is important since multiple simulation cases can be analyzed before taking the final decision in regards with the day-ahead economic dispatch environment, which makes the GAMS software an attractive alternative for tertiary control in distribution networks. In addition, the GAMS package is a proper tool to solve complex optimization problems by focusing the attention on correctly developing the optimization models rather than the solution technique. This represents an ideal framework to easily introduce engineers and researchers in mathematical optimization; for this reason, this paper has been addressed in a tutorial form.

As future work it will be possible to analyze the following problems: (i) propose convex reformulations for optimal power flow analyses in AC and DC networks that will ensure reaching the global optimum of the problem under well-defined operative conditions, which are very attractive for real economic dispatch applications, (ii) make a comparative study between AC and DC grids considering transient operation scenarios such as suddenly load disconnections or short-circuit cases, which can be used in protective devices coordination studies for these grids, and extend the economic dispatch optimization model to three-phase distribution networks and bipolar DC configurations operated under unbalanced loads scenarios to analyze their efficiency in terms of power losses and voltage profiles.

**Author Contributions:** Conceptualization, methodology, software and writing—review and editing, O.D.M., F.M.S. and C.H.D.A. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by the Universidad Tecnológica de Bolívar under grant CP2019P011 associated with the project: "Operación eficiente de redes eléctricas con alta penetración de recursos energéticos distribuidos considerando variaciones en el recurso energético primario".

**Acknowledgments:** The authors want to express to thanks to Universidad Nacional de San Luis and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) in Argentina, and Universidad Distrital Francisco José de Caldas and Universidad Tecnológica de Bolivar in Colombia.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### *Article* **LC Impedance Source Bi-Directional Converter with Reduced Capacitor Voltages**

#### **Dogga Raveendhra \*, Rached Dhaouadi, Habibur Rehman and Shayok Mukhopadhyay**

College of Engineering, American University of Sharjah, Sharjah 26666, UAE; rdhaouadi@aus.edu (R.D.); rhabib@aus.edu (H.R.); smukhopadhyay@aus.edu (S.M.)

**\*** Correspondence: doggaravi12@gmail.com

Received: 3 June 2020; Accepted: 17 June 2020; Published: 28 June 2020

**Abstract:** This paper proposes an LC (Inductor and Capacitor) impedance source bi-directional DC–DC converter by redesigning after rearranging the reduced number of components of a switched boost bi-directional DC–DC converter. This new converter with a conventional modulation scheme offers several unique features, such as a) a lower number of components and b) reduced voltage stress on the capacitor compared to existing topologies. The reduction of capacitor voltage stress has the potential of improving the reliability and enhancing converter lifespan. An analysis of the proposed converter was completed with the help of a mathematical model and state-space averaging models. The converter performance under different test conditions is compared with the conventional bi-directional DC–DC converter, Z-source converter, discontinuous current quasi Z-source converter, continuous current quasi Z-source converter, improved Z-source converter, switched boost converter, current-fed switched boost converter, and quasi switched boost converter in the Matlab Simulink environment. MATLAB/Simulink results demonstrate that the proposed converter has lesser components count and reduced capacitors' voltage stresses when compared to the topologies mentioned above. A 24 V to 18 V LC-impedance source bi-directional converter and a conventional bidirectional converter are built to investigate the feasibility and benefits of the proposed topology. Experimental results reveal that capacitor voltage stresses, in the case of proposed topology are reduced by 75.00% and 35.80% in both boost and buck modes, respectively, compared to the conventional converter circuit.

**Keywords:** bi-directional converter; LC impedance source converter; DC–DC power converter; bi-directional power flow

#### **1. Introduction**

The study, development, and applications of bidirectional power converters are gaining a lot of attention due to their vital role in areas like renewable energy systems, DC microgrids, hybrid energy storage systems, smart mobility, etc. A bidirectional DC-DC converter (BDC) allows power flow in both directions. This functionality is not available in a traditional unidirectional DC-DC converter. Because of this flexibility, BDCs are widely used in several applications, such as battery-powered electric vehicles (BEVs) or hybrid electric vehicles (HEVs), power trains, uninterruptable power supplies (UPS), smart grids, charging stations for BEVs and plug-in hybrid electric vehicles (PHEV), aerospace, defense, aerospace, and non-conventional energy sources such as photovoltaic (PV) arrays, fuel cells (FCs), and wind turbines. Specifically, BDCs are widely adopted by the electric vehicle industry to achieve objectives, such as battery charging/discharging and energy recovery during regeneration modes of operation in electric vehicles. In case of the BEVs, electric energy needs to flow in both directions, i.e., from the motor to the battery and vice versa in regenerative mode. To avoid pollutant emissions, the electric vehicle must be powered only by batteries or other electrical sources (fuel

cells, solar panels, etc.) [1–4]. In all the above-mentioned applications, a BDC is preferred for saving space by eliminating a separate boost and buck converter. A BDC can offer some benefits, like cost reduction, improved power density, and effective utilization of the converter [4]. Figure 1 shows the typical structure of the bidirectional DC–DC converters. The BDC, shown in Figure 2, helps to enhance the system efficiency and performance by interfacing with power and energy storage devices [5]. It also avoids a couple of individual unidirectional converters for achieving bidirectional power flow. The BDC's mode of operation (buck or boost) is mainly decided by power flow direction and voltage levels of sources/energy storage elements. Accordingly, the controller must be designed to regulate the voltage/current of the system. While designing DC–DC converters, the main functional objectives are high power density and high efficiency. The high density can be achieved by increasing the switching frequency [6] due to the reduction in reactive components size. However, the problem is that increasing the switching frequency increases the switching losses, which leads to efficiency reduction. This problem can be addressed by adopting wide-bandgap power devices along with suitable gate drivers instead of conventional Si devices.

**Figure 1.** Structure of bi-directional DC–DC converters.

**Figure 2.** Conventional bidirectional converter (BDC).

In general, conventional step-up DC–DC converters are classified into isolated and non-isolated converters. Isolated converters like fly-back, push–pull, forward, half-bridge, and full-bridge converters have a high voltage gain by keeping a high enough transformer turns ratio. However, there is a problem with voltage spikes due to transformer leakage inductance, which leads to high power losses across the switch. On the other hand, in non-isolated converters, a high duty cycle is required to get a high voltage gain, which leads to decreasing efficiency due to reverse recovery problems [7]. In addition, non-isolated converters also have the problem of voltage stress nearly equal to the output voltage, causing a reduction of the device's reliability. Many DC–DC converter topologies are introduced to mitigate the problems mentioned above, such as interleaving topologies for the reduction of current ripple [8,9], soft-switching techniques to mitigate voltage spikes and efficiency improvement [10], and cascading boost converters [11] and incorporating a coupled inductor [12] in the conventional boost topology to get a high conversion gain. Input current ripples are reduced with the help of an interleaving concept, which leads to improving the source life. Additionally, it offers the flexibility of current sharing to enhance the power handling capacity [8,9].

On the other hand, several other converter topologies are suggested in the literature; most of these are designed to meet the various objectives, such as reliability, capacitor voltage reduction, and input current ripple reductions, by placing an impedance network between input DC source and

switching network in various fashions. An X-shaped LC impedance network, as shown in Figure 3a, is placed to get the voltage boosting capability by operating a switching network in the shoot-through mode [13]. As an alternative to the Z-source converter, the same authors proposed a quasi Z-source (qZS) converter in two variants based on input current, namely continuous input current q-ZS (qZS-CC) and discontinuous input current q-ZS (qZS-DC) [14,15] with a reduced current and capacitor voltage stresses, respectively. The main variation between these two topologies is the input side inductor connection with the supply. In case of qZS-CC, the inductor is placed directly in series with the source, and it tries to always maintain constant input current, whereas the source current is of discontinuous nature in the case of qZS-DC, which increases the stress on the source [15]. Later, Yu Tang et al. proposed an improved Z-source (IZS) converter [16] with reduced capacitor stresses. In this paper, the authors claim that the utilization of a low voltage capacitor reduces the inrush current, the resonance between the Z-source inductor and capacitors, and the cost and volume of the system compared to a conventional Z-source converter [17]. The switched boost converter is proposed with a reduced passive components count, achieved by replacing one pair of LCs with power semi-conductor devices to have the same kind of buck-boost conversion, as shown in Figure 3b [18]. However, this topology uses more power semiconductor devices compared to the topologies mentioned above.

**Figure 3.** Impedance DC–DC converters. (**a**) Z-source DC–DC converter, (**b**) switched boost DC–DC converter.

A SL-ZS converter is proposed with an enhanced gain by placing switched inductors instead of inductors in the impedance network [19]. However, this topology suffers from a large component count (six power diodes and two inductors higher than the ZS converter) in the switching network. Alternatively, the SL-qZS converter proposed in [20], consists of switched inductors in place of standard inductors in the qZS converter to reduce the capacitor voltage and startup inrush current compared to the SL-ZS converter. However, the downside of this topology is a higher component count. Hossein Fathi et al. [21] proposed an enhanced boost ZS converter (EB-ZSC), achieved by replacing the impedance network with switched impedance to enhance the conversion gain further. Although this topology increases gain, it suffers from a higher component count (four inductors, four capacitors, and five power diodes). Additionally, this topology suffers from the usage of sophisticated control platforms to achieve smoother voltage control in the case of adjustable speed-controlled drive applications. Moreover, with a similar concept of variations in the impedance network either in ZS or qZS as discussed above, there are several other impedance source topologies, such as a diode-assisted qZS (DA-qZS) converter [22], a capacitor-assisted qZS (CS-qZS) converter [22], and an enhanced boost quasi ZS (EB-qZS) converter [23], which are proposed in the literature. Though these topologies are mainly proposed for DC–AC power conversion applications due to high reliability (operation during shoot-through mode), they are equally applicable for bi-directional applications and are widely used in micro/nano-grid applications [18,23].

For most of these topologies, it has been suggested to incorporate switched-inductor, switched-capacitor, and hybrid switched-capacitor/switched-inductor structures resulting in high boosting factors. However, the effect of nonlinearity can be increased by increasing the energy storage

elements in the circuit, which leads to a higher output current and voltage distortion [24]. Additionally, introducing more energy storage elements in the circuit affects the control complexity, total cost, size, volume, losses, and weight of the converter [25,26]. Moreover, these topologies are suffering from the usage of more capacitors and higher capacitor voltage stresses. Additionally, the voltage across most of the capacitors is generally more than the supply voltage in the case of impedance source topologies in order to perform the voltage boost functionality. Hence, high-voltage Z-capacitors should be used, which may increase the volume and system cost. Capacitors are prone to failure in the field operation of power electronic converters [27]. Hence, due to the stricter reliability constraints brought by aerospace, automotive, defense, space, and energy industries, the stresses and usage of capacitors should be reduced to enhance the converter's reliability [28]. Therefore, to enhance the life and converter reliability, either reduction in capacitors usage or voltage stresses on the capacitor is highly recommended [29,30].

In this paper, the LC impedance bi-directional DC–DC converter (LC-BDC) is proposed by placing one inductor between source and half-bridge, and one capacitor between source and the load, as shown in Figure 4 [31]. These small passive components are arranged in such a way that the converter offers several features, such as lower capacitor voltages, which in turn reduces the cost, size, and volume of the converter and also increases the reliability while achieving the desired functionality. This topology reduces the voltage stresses on the device due to the usage of small passive components compared to existing converters in the case of SiC converters, which are less immune to parasitic components. The paper is organized as follows: the working principle, modes of operation, mathematical modeling, and state-space average models of the proposed topology are discussed in Section 2. The concept validation using simulation and experimentation, along with the respective results, are presented in Section 3. Additionally, to demonstrate the effectiveness of the proposed topology, a detailed comparative analysis of the proposed converter and conventional converter is carried out along with the results of the proposed converter. Moreover, a separate simulation-based comparative analysis of the proposed LC converter with eight similar boost/buck-boost converter topologies is presented in Section 4. Finally, conclusions are presented in Section 5 of this paper.

**Figure 4.** LC impedance bi-directional DC–DC converter (LC-BDC).

#### **2. Proposed System**

The LC bidirectional converter shown in Figure 4 is an advanced version of a conventional bidirectional converter and switched boost bidirectional converter, designed to reduce the voltage stresses on the capacitor. The primary function of the inductor is to store energy during the converter "on" period and release the stored energy during the "off" period of the primary device. The inductor is also used to eliminate the current ripple. Another energy storage element, the capacitor, is used to eliminate the ripple in the output voltage in both cases, namely the conventional BDC and the proposed BDC. Switches *M1* and *M2* are unidirectional switches used to realize the bidirectional power flow in the test setup which operate in a complementary fashion. For the forward direction of power flow, *M1* must be in the "on" state, and *S1* acts as the main switch operating at switching frequency, while *D2* acts as a freewheeling diode. Similarly, *M2* must be in the "on" position for the reverse direction of power flow, and *S2* acts as the main switch, which operates at switching frequency, while *D1* acts as a freewheeling diode. *VLV* and *RHV* are source and load in boost mode, and *VHV* and *RLV* are source and load in buck mode. The gating signals for boost switch (*G1*) and buck switch (*G2*) complement each other. The duty cycle of boost switch (*S1*) and buck switch (*S2*) is denoted as δ<sup>1</sup> and δ2, respectively. *Ts* represents the switching period of switches *S1* and *S2*.

#### *2.1. Boost Operation*

The equivalent circuit and idealized waveforms in boost mode of the LC-BDC converter are depicted in Figures 5 and 6, respectively. The converter operation is considered to be in boost mode, during which the switch (*S1*) is pulse-modulated and the diode *D2* freewheels. The boost mode operation is further categorized into two sub-modes of operation over a switching period, and the equivalent circuit of each sub-mode is depicted as shown in Figure 7.

**Figure 5.** LC bi-directional DC–DC converter in boost mode.

**Figure 6.** Characteristic waveforms during various boost modes of operation.

**Figure 7.** Equivalent circuit of boost operation (**a**) in mode 1 and (**b**) in mode 2.

#### 2.1.1. Mode 1 (*t0* < *t* < *t1*): (*S1* ON, *D2* OFF)

In this mode, switch *S1* is turned on by applying a gate signal. The inductor *L* starts charging linearly through switch *S1*, and the capacitor *C* will discharge through load *RHV*. Hence, the diode *D2* goes into the "off" state. The equivalent circuit during this mode of operation is shown in Figure 7a. The current through the inductor *L*(*iL*) and the voltage across the capacitor *C* are given by

$$V\_L = V\_{LV} = L\frac{di\_L}{dt} \tag{1}$$

$$i\_L(t) = \frac{V\_{LV}}{L}(t - t\_0) + i\_L(t\_0) \tag{2}$$

$$\dot{q}\_{\mathbb{C}} = -\frac{\upsilon\_{HV}}{R\_{HV}}\tag{3}$$

$$
\sigma\_{\mathbb{C}}(t) = i\_{HV}(t)\mathbb{R}\_{HV} - V\_{LV} \tag{4}
$$

This mode of operation ends when the gate pulses to switch S1 are withdrawn.

#### 2.1.2. Mode 2 (*t1* < *t* < *t2*): (*S1* OFF, *D2* ON)

At the instant when the gate pulses of switch *S1* are removed, the switch *S1* goes into the "off" state due to which the voltage across the inductor brings the diode *D2* into the forward-biased state. The equivalent circuit during this mode is shown in Figure 7b. In this mode, both inductor and source feed power to the load, and the inductor charges the capacitor. Hence, there is a formation of the *LC* tank in this mode, which can offer zero voltage switching to the upper switch with the proper selection of the snubber capacitor. In this mode, the current through *L*(*iL*) reaches its minimum value. The current flowing through the inductor *L*(*iL*) and the voltage across the capacitor *C* are given by

$$V\_L = V\_{LV} - V\_{HV} = L\frac{di\_L}{dt} \tag{5}$$

$$i\_L(t) = \frac{V\_{LV} - V\_{HV}}{L}(t - t\_1) + i\_L(t\_1) \tag{6}$$

$$v\_{\mathbb{C}}(t) = i\_{HV}(t)R\_{HV} - V\_{LV} \tag{7}$$

$$i\_{\mathbb{C}\mathcal{E}}(t) = i\_{\mathbb{L}}(t) - i\_{HV}(t) \tag{8}$$

This mode ends at *t* = *Ts* when the gate signal is provided to *S1* in the next switching cycle. Similar operation (Mode 1 and Mode 2) continues for several switching cycles until a power flow is required in the forward direction

#### *2.2. Buck Mode of Operation of LC-BDC Converter*

BDC operates in buck mode when there is a requirement of power flow in the reverse direction, and its equivalent circuit is shown in Figure 8. The converter operation is considered to be in reverse buck mode, during which the switch *S2* is pulse-modulated and the diode *D1* in a freewheeling mode. The buck mode of operation is further categorized into two sub-modes (i.e., mode 3 and mode 4) of operation over a switching period. The operating mode from mode 3 to mode 4 in buck mode is similar to the mode 2 to mode 1 of the boost mode of operation, respectively. Figure 9 illustrates the characteristic waveforms of the converter in buck mode, and its equivalent circuits in each sub-mode are depicted as shown in Figure 10.

**Figure 8.** Equivalent circuit in buck mode.

**Figure 9.** Characteristic waveforms during various modes in buck operation.

**Figure 10.** Equivalent circuit of buck operation (**a**) in mode 3 and (**b**) in mode 4.

#### 2.2.1. Mode 3 (*t3* < *t* < *t4*): (*S2* ON, *D1* OFF)

This mode starts at *t* = *t3* when the gate signal is given to *S2*. At this instant, the main switch S2 comes into conduction, and the diode *D1* goes into the "off" state. The supply *VHV* then directly energizes the inductor *L.* The capacitor is also discharged through the inductor. It leads to the formation of the *LC* tank, as shown in Figure 10a, similar to mode 2. This feature offers the resonating switching functionality to the upper switch. The current flowing through *L*(*iL*) and the voltage across capacitor *C* (*vC*) are given as

$$
\upsilon\_{Lb}(t) = V\_{HV} - \upsilon\_{LV} \tag{9}
$$

$$i\_{Lb}(t) = \frac{V\_{HV} - \upsilon\_{LV}}{L}(t - t\mathfrak{z}) + i\_{Lb}(t\mathfrak{z}) \tag{10}$$

$$
\sigma\_{\mathbb{C}}(t) = V\_{HV} - i\_{LV}(t)\mathbb{R}\_{LV} \tag{11}
$$

$$i\_{\mathbb{C}}(t) = i\_{\text{inb}}(t) - I\_{Lb}(t) \tag{12}$$

This mode continues until the gate pulse of *S2* is withdrawn at *t* = *t4*.

#### 2.2.2. Mode 4 (*t4* < *t* < *t5*): (*S2* OFF, *D1* ON)

This mode starts at *t* = *t4* when the gate pulses to the main switch are removed. Hence *S2* goes into the "off" state. The voltage across the inductor brings diode *D1* into "on" state, and it continues until *t* = *t5*. The energy stored in inductor L discharges through the load. The capacitor charges from the source. During this mode, the current flowing through the inductor *L* and voltage across the capacitor *C* can be expressed as

$$
\sigma\_{Lb}(t) = -i\_{LV}(t)\mathbb{R}\_{HV} \tag{13}
$$

$$i\_{Lb}(t) = \frac{-\upsilon\_{LV}}{L}(t - t\_4) + i\_{Lb}(t\_4) \tag{14}$$

$$
\sigma\_{\mathbb{C}}(t) = V\_{HV} - i\_{LV}(t)\mathcal{R}\_{LV} \tag{15}
$$

$$i\_{\mathbb{C}}(t) = i\_{\text{inb}}(t) \tag{16}$$

This mode ends at *t* = *Ts* when the gate signal is given to *S2* in the next switching cycle. Similar operation of mode 3 and mode 4 continues, for several switching cycles, until power flow is required in the reverse direction.

#### *2.3. State Space Analysis*

This section presents the development of a small-signal AC model followed by the derivation of the state-space model equations for one complete switching cycle. For this analysis, few assumptions are considered; (i) the converter is operating in continuous conduction mode, and (ii) there is no trace resistance. For the proposed converter, the state variables are the current through the inductor *iL* and the voltage across the coupling capacitor *VC*. A complete derivation of the state-space model and small-signal analysis for boost mode is presented. A similar derivation method can also be used for buck mode. With the inclusion of the parasitic components during both "on" and "off" states, the system can be represented with the help of the state-space model as follows.

$$
\begin{bmatrix}
\frac{d\dot{l}\_L}{dt} \\
\frac{d\upsilon\_C}{dt}
\end{bmatrix} = \begin{bmatrix}
0 & -\frac{1}{C(R + r\_C)}
\end{bmatrix} \begin{bmatrix}
\dot{i}\_L \\
\upsilon\_C
\end{bmatrix} + \begin{bmatrix}
\frac{1}{L} & 0 \\
0 & \frac{-R}{(R + r\_C)C}
\end{bmatrix} \begin{bmatrix}
V\_{LV} \\
\dot{i}\_{Load}
\end{bmatrix} \tag{17}
$$

during "off" state:

$$
\begin{bmatrix}
\frac{d\dot{l}\_{l}}{dt} \\
\frac{d\mathbf{v}\_{\mathcal{C}}}{dt}
\end{bmatrix} = \begin{bmatrix}
\left(\frac{-Rr\_{\mathcal{C}}}{(R+r\_{\mathcal{C}})} - r\_{L}\right)\frac{1}{L} & \frac{-R}{(R+r\_{\mathcal{C}})}\frac{1}{L} \\
\frac{R}{(R+r\_{\mathcal{C}})\mathcal{C}} & \frac{1}{(R+r\_{\mathcal{C}})\mathcal{C}}
\end{bmatrix} \begin{bmatrix}
\dot{i}\_{L} \\
\upsilon\_{\mathcal{C}}
\end{bmatrix} + \begin{bmatrix}
\frac{1}{L} & \frac{Rr\_{\mathcal{C}}}{(R+r\_{\mathcal{C}})} \\
0 & \frac{-R}{(R+r\_{\mathcal{C}})}
\end{bmatrix} \begin{bmatrix}
V\_{LV} \\
\dot{i}\_{Load}
\end{bmatrix} \tag{18}
$$

Here, *ron*—on-state resistance of switching device, *rL*—the equivalent series resistance of the inductor, and *rC*—equivalent series resistance of the capacitor.

The state-space average model of the converter can be written as follows.

$$\dot{\mathbf{x}} = [A\_1 \delta\_1 + A\_2(1 - \delta\_1)]\mathbf{x} + [B\_1 \delta\_1 + B\_2(1 - \delta\_1)]\mathbf{L}\tag{19}$$

$$\begin{array}{rcl} \text{Here,} & \mathbf{x} & = \begin{bmatrix} \dot{i}\_{L} \\ \mathbf{z}\_{\text{CC}} \end{bmatrix} \cdot A\_{1} & = & \begin{bmatrix} -\frac{(r\_{m} + r\_{\text{L}})}{L} & 0 \\ 0 & -\frac{1}{\mathbb{C}(R + r\_{\text{C}})} \end{bmatrix}, \quad B\_{1} & = & \begin{bmatrix} \frac{1}{L} & 0 \\ 0 & \frac{-R}{(R + r\_{\text{C}})\mathbb{C}} \end{bmatrix}, \quad A\_{2} & =\\ \begin{bmatrix} \frac{-Rr\_{\text{C}}}{(R + r\_{\text{C}})\mathbb{C}} - r\_{\text{L}} \end{bmatrix}, \quad B\_{2} & = & \begin{bmatrix} \frac{1}{L} & \frac{Rr\_{\text{C}}}{(R + r\_{\text{C}})\mathbb{C}} \\ 0 & \frac{-R}{(R + r\_{\text{C}})\mathbb{C}} \end{bmatrix}, \quad B\_{3} = & \begin{bmatrix} \frac{1}{L} & \frac{Rr\_{\text{C}}}{(R + r\_{\text{C}})\mathbb{C}} \\ 0 & \frac{-R}{(R + r\_{\text{C}})\mathbb{C}} \end{bmatrix}, \quad u = \begin{bmatrix} V\_{LV} \\ i\_{Lnd} \end{bmatrix} \end{array}$$

$$
\delta\_1 T\_s = t\_1 - t\_0 \text{€} \\
t\_2 - t\_1 = (1 - \delta\_1) T\_s \tag{20}
$$

The duty ratio of the main switch *S1* is defined as

$$
\delta\_1 = \frac{t\_1 - t\_0}{t\_2} \tag{21}
$$

The turn-off duty cycle of the main switch *S1* is

$$
\delta'\_1 = \frac{t\_2 - t\_1}{t\_2} \tag{22}
$$

Substituting the duty ratio values from (20)–(22) in Equations (17)–(19) and then incorporating the perturbation effect into the state variables and other variables around the steady-state values gives

$$\mathfrak{i}\_{\mathcal{L}} = \mathcal{I}\_{\mathcal{L}} + \hat{\mathfrak{i}}\_{\mathcal{L}\prime} \mathfrak{v}\_{\mathcal{L}\prime} = V\_{\mathcal{L}V} + \mathfrak{d}\_{\mathcal{L}V\prime} \mathfrak{v}\_{\mathcal{C}} = V\_{\mathcal{C}} + \mathfrak{f}\_{\mathcal{C}\prime} \mathfrak{v}\_{\mathcal{H}\prime} = V\_{\mathcal{H}V} + \mathfrak{f}\_{\mathcal{H}V\prime} \delta\_{1} = D\_{1} + d\_{1} \tag{23}$$

where *D1* is the duty ratio of the main switch under steady-state condition. After solving the above state-space equation, the steady-state gains of the converter can be obtained as

$$V\_{\mathbb{C}} = \frac{D\_1 V\_{LV}}{1 - D\_1} \& I\_L = \frac{I\_{HV}}{1 - D\_1} \tag{24}$$

Comparing small-signal AC parameters while ignoring the considerably very small second-order quantities, and then solving the equations gives the following two transfer functions.

#### 2.3.1. Control-to-Output Transfer Function

From the small-signal AC model, the control-to-output (output voltage to duty ratio) transfer function can be found under the condition of *v*ˆ*in* = 0&ˆ*iL* = 0, which is shown in Equation (25).

*v*ˆ*o* <sup>1</sup><sup>−</sup> <sup>ˆ</sup> *d*1 = [−*R*(*R*+*rc*)×(1+*CSrc*)× ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *RVLVrL* <sup>−</sup> *<sup>R</sup>*<sup>2</sup>*VLVD*<sup>1</sup> <sup>2</sup> <sup>−</sup> *<sup>R</sup>*<sup>2</sup>*VLV* + *RVLVron* + *VLVrLrc* + *VLVrcron* + <sup>2</sup>*R*<sup>2</sup>*VLVD*<sup>1</sup> + <sup>2</sup>*IloadR*<sup>2</sup>*rL* +*IloadR*<sup>2</sup>*rc* + *IloadR*<sup>2</sup>*ron* <sup>−</sup> <sup>2</sup>*IloadR*<sup>2</sup>*D*1(*rL* + *rc*) + *IloadR*<sup>2</sup>*D*<sup>1</sup> <sup>2</sup>*rc* <sup>−</sup> *IloadR*<sup>2</sup>*D*<sup>1</sup> <sup>2</sup>*ron* + *LRSVLV* +*LSVLVrc* + <sup>2</sup>*IloadRrLrc* + *IloadRrcron* + *IloadLR*<sup>2</sup>*<sup>S</sup>* + *IloadLRSrc* <sup>−</sup> <sup>2</sup>*IloadRD*1*rLrc* <sup>−</sup> *IloadLR*<sup>2</sup>*SD*<sup>1</sup> −*IloadRD*<sup>1</sup> <sup>2</sup>*rcron* <sup>−</sup> *IloadLRSD*1*rc* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎛ ⎜⎜⎜⎜⎜⎜⎝ *RrL* + *Rrc* + *rLrc* <sup>−</sup> <sup>2</sup>*R*2*D*<sup>1</sup> + *<sup>R</sup>*<sup>2</sup> +*R*2*D*<sup>1</sup> <sup>2</sup> <sup>−</sup> *RD*1*rc* + *RD*1*ron* + *<sup>D</sup>*1*rcron* ⎞ ⎟⎟⎟⎟⎟⎟⎠ × ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *CLR*2*S*<sup>2</sup> <sup>−</sup> *CR*2*SD*1*rc* + *CronR*2*SD*<sup>1</sup> + *CR*2*Src* + *CrLR*<sup>2</sup>*<sup>S</sup>* + *<sup>R</sup>*2*D*<sup>1</sup> <sup>2</sup> <sup>−</sup> <sup>2</sup>*R*2*D*<sup>1</sup> + *<sup>R</sup>*<sup>2</sup> +2*CLRS*2*rc* <sup>−</sup> *CRSD*1*r*<sup>2</sup> *<sup>c</sup>* + 2*CronRSD*1*rc* + *CRSr*<sup>2</sup> *<sup>c</sup>* + 2*CrLRSrc* + *LRS* − *RD*1*rc* +*ronRD*<sup>1</sup> + *Rrc* + *rLR* + *CLS*2*r*<sup>2</sup> *<sup>c</sup>* + *CronSD*1*r*<sup>2</sup> *<sup>c</sup>* + *CrLSrc* <sup>2</sup> + *LSrc* + *ronD*1*rc* + *rLrc* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (25)

By neglecting parasitic components, it can be simplified as

$$\begin{array}{c} \frac{\psi\_{HV}}{1 - d\_1} = \frac{RV\_{LV} - LSV\_{LV} - 2RV\_{LV}D\_1 + RV\_{LV}D\_1^{-2} - l\_{load}LRS + l\_{load}LRSD\_1}{(D\_1 - 1)^2 (CLRS^2 + LS + RD\_1^2 - 2RD\_1 + R)}\\ \implies \frac{\upsilon\_{HV}}{1 - d\_1} = \frac{RV\_{LV}(1 - D\_1)^2 - LSV\_{LV} - (1 - D\_1)l\_{load}LRS}{(D\_1 - 1)^2 (CLRS^2 + LS + R(1 - D\_1)^2)} \end{array} \tag{26}$$

#### 2.3.2. Control-to-Input Transfer Function

From the small-signal AC model, the inductor current-to-control (input current to duty ratio) transfer function can be found under the condition of *v*ˆ*LV* = 0&ˆ*iLoad* = 0, which is shown in Equation (27).

ˆ*iin* <sup>1</sup><sup>−</sup> <sup>ˆ</sup> *d*1 = −(*R*+*rc*) ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ <sup>2</sup>*R*<sup>2</sup>*VLV* <sup>−</sup> *IloadR*<sup>3</sup>*D*<sup>1</sup> <sup>2</sup> <sup>−</sup> *RVLVrc* <sup>−</sup> *IloadR*<sup>3</sup> + *VLVrcron* + <sup>2</sup>*IloadR*<sup>3</sup>*D*<sup>1</sup> + <sup>2</sup>*R*<sup>2</sup>*VLVD*<sup>1</sup> + *IlaodR*<sup>2</sup>*rL* + *IloadR*<sup>2</sup>*ron* + *IlaodRrLrc* +*IloadRrcron* <sup>−</sup> *CR*<sup>3</sup>*SVLVD*<sup>1</sup> <sup>−</sup> *CIloadR*<sup>3</sup>*SrL* <sup>−</sup> *CIlaodR*<sup>3</sup>*SrC* <sup>−</sup> *CR*<sup>2</sup>*SVLVrL* <sup>−</sup> *CRSVLVr*<sup>2</sup> *<sup>c</sup>* <sup>−</sup> <sup>2</sup>*CR*<sup>2</sup>*SVLVrc* + *CSVLVr*<sup>2</sup> *<sup>c</sup> ron* <sup>−</sup>*CIlaodLR*<sup>3</sup>*S*<sup>2</sup> <sup>−</sup> *CLR*2*S*<sup>2</sup>*VLV* + *CR*<sup>3</sup>*SVLVVD*<sup>1</sup> <sup>2</sup> <sup>−</sup> *CIloadLR*<sup>2</sup>*SrC* <sup>−</sup> *CIloadR*<sup>3</sup>*SD*<sup>1</sup> <sup>2</sup>*rC* + *CIloadR*<sup>3</sup>*SD*<sup>1</sup> <sup>2</sup>*ron* <sup>−</sup> *CRSVLVrLrC* +*CRSVLVrcron* <sup>−</sup> *CLRS*<sup>2</sup>*VLVrc* + <sup>2</sup>*CIloadR*<sup>3</sup>*SD*1*rL* + <sup>2</sup>*CIloadR*<sup>3</sup>*SD*1*rC* + *CR*<sup>2</sup>*SVLVD*1*rc* + *CIloadRSrLr*<sup>2</sup> *<sup>c</sup>* + *CIloadRSr*<sup>2</sup> *<sup>c</sup> ron* +*CIloadR*<sup>2</sup>*Srcron* + *CIloadLR*<sup>3</sup>*S*2*D*<sup>1</sup> + 2*CIloadR*<sup>2</sup>*SD*1*rLrc* + *CIloadLR*<sup>2</sup>*S*2*D*1*rc* + *CIloadR*<sup>2</sup>*SD*<sup>1</sup> <sup>2</sup>*ronr* + *RVLVron* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎡ ⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣ ⎛ ⎜⎜⎜⎜⎜⎜⎝ *RrL* + *Rrc* + *rLrc* <sup>−</sup> <sup>2</sup>*R*2*D*<sup>1</sup> + *<sup>R</sup>*<sup>2</sup> +*R*2*D*<sup>1</sup> <sup>2</sup> <sup>−</sup> *RD*1*rc* + *RD*1*ron* + *<sup>D</sup>*1*rcron* ⎞ ⎟⎟⎟⎟⎟⎟⎠ × ⎛ ⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝ *CLR*2*S*<sup>2</sup> <sup>−</sup> *CR*2*SD*1*rc* + *CronR*2*SD*<sup>1</sup> + *CR*2*Src* + *CrLR*<sup>2</sup>*<sup>S</sup>* + *<sup>R</sup>*2*D*<sup>1</sup> <sup>2</sup> <sup>−</sup> <sup>2</sup>*R*2*D*<sup>1</sup> + *<sup>R</sup>*<sup>2</sup> +2*CLRS*2*rc* <sup>−</sup> *CRSD*1*r*<sup>2</sup> *<sup>c</sup>* + 2*CronRSD*1*rc* + *CRSr*<sup>2</sup> *<sup>c</sup>* + 2*CrLRSrc* + *LRS* − *RD*1*rc* + *ronRD*<sup>1</sup> +*Rrc* + *rLR* + *CLS*2*r*<sup>2</sup> *<sup>c</sup>* + *CronSD*1*r*<sup>2</sup> *<sup>c</sup>* + *CrLSr*<sup>2</sup> *<sup>c</sup>* + *LSrc* + *ronD*1*rc* + *rLrc* ⎞ ⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠ ⎤ ⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (27)

By neglecting parasitic components, it can be simplified as

$$\begin{array}{l}\stackrel{\hat{i}}{\frac{\hat{i}\_{\text{in}}}{1-d\_{1}}} = \frac{2V\_{LV} - I\_{\text{load}}R(1-D\_{1}) + CRV\_{LV}S}{(1-D\_{1})(CLRS^{2} + LS + RD\_{1}^{2} - 2RD\_{1} + R)}\\ \implies \frac{\stackrel{\hat{i}\_{\text{in}}}{1-d\_{1}}}{1-d\_{1}} = \frac{RV\_{LV}(1-D\_{1})^{2} - LSV\_{LV} - I\_{\text{load}}(1-D\_{1})}{(1-D\_{1})(CLRS^{2} + LS + R(1-D\_{1})^{2})}\end{array} \tag{28}$$

#### 2.3.3. Step and Bode Responses of LC-BDC

From the derived transfer functions, the step responses of various variables are presented in Figure 11. From these results, it can be understood that the variations in input currents and capacitor voltages for both line and load disturbance are low in the case of the proposed converter compared to the existing converter. Moreover, the step responses reveal that the proposed LC-BDC converter response is the same for inductor current and capacitor current and capacitor voltage against the duty ratio, whereas input current transients against duty ratio variations are reduced in LC-BDC. It can also be noted that in the case of supply variations, the transient responses of the inductor current, output voltage, load current, and capacitor voltage are improved.

#### 2.3.4. Ripple Capacitor Voltage

From the charge balance equation and further simplification of the above Equation (25) in the steady-state, the capacitor ripple voltage can be calculated as

$$
\Delta v\_{Cchost} = \frac{D\_1 V\_{HV}}{CRF\_S} \tag{29}
$$

From (29), the capacitor value can be sized to minimize the voltage ripple across the capacitor.

**Figure 11.** Step responses of the inductor current, output voltage, load current, and capacitor voltage transfer functions with respect to duty ratio and input voltage of LC-BDC and conventional converter.

#### **3. Experimental Results and Discussion**

The proposed LC impedance bi-directional dc-dc converter has been successfully validated through experiments in both boost and buck modes. The parameters considered for the experimental validations are summarized in Table 1. The experimental setup of the proposed converter is shown in Figure 12. The system performance is evaluated in both steady-state and transient conditions while feeding power to two series-connected 12 V, 50 W lamp load under various test conditions for the 18 V DC to 24 V DC conversion in forwarding boost mode, and 24 V DC to 18 V DC conversion in reverse buck mode. The inductor current, load current, and capacitor voltage waveforms are captured in both boost and buck modes for both conventional and proposed converters. Comparative analysis through experimental results was carried out, as explained below. In the case of the conventional converter, there is a need for two capacitors (*CHV* plays a vital role in boost mode, and *CLV* plays a vital role in buck mode), whereas, in the case of the proposed converter, there is a need for only one capacitor *C*, which can take care of the functionality of the above mentioned two capacitors in the respective modes. It can be observed that two capacitors are used in the realization of the conventional converter, whereas only one capacitor is used for the realization of the proposed LC-BDC, as shown in Table 1.



**Figure 12.** Experimental Setup of Power Converter.

The gate signal of the lower switch, inductor current, load current, and capacitor voltage for four switching cycles are captured and presented in Figures 13–16. From Figure 14, it can be seen that the peak value of the inductor current is 6.17 A in the conventional converter, whereas it is 6.07 A in the proposed converter for the same load current, as shown in Figure 15. From Figure 16, it can be seen that the voltage across the capacitor is 23.20 V in the case of the conventional converter, whereas it is 5.10 V in the case of the proposed converter. Hence, there is 78.02% of capacitor voltage reduction in the proposed converter as compared to the conventional converter for the same input/output voltage conversion.

**Figure 13.** Gate voltage of lower switch (*S1*) in LC-BDC (Blue), and conventional BDC (red).

**Figure 14.** Inductor current in LC-BDC (Blue), and conventional BDC (red).

**Figure 15.** Load current in LC-BDC (Blue), and conventional BDC (red).

**Figure 16.** Capacitor voltages in LC-BDC (Blue), and conventional BDC (Red).

In Figures 17–19, respectively, a zoomed view of respective parameters is presented during both "on" and "off" state. From these figures, peak values during both transient and steady-state can be measured as listed in Table 2.

**Figure 17.** Inductor current of LC-BDC (blue) and conventional BDC (red) converter during (**a**) "on" and (**b**) "off" states.

**Figure 18.** Load current of LC-BDC (blue) and conventional BDC (red) converter during (**a**) "on" and (**b**) "off" states.

**Figure 19.** Capacitor voltage of LC-BDC (blue) and conventional BDC (red) converter during (**a**) "on" and (**b**) "off" states.

**Table 2.** Comparison of the various parameter during both transient and steady state for the proposed and conventional converters.


Form Table 3, it can be observed that the proposed converter not only offer its best performance during steady-state conditions but also the exhibits same best performance during transient conditions in terms of capacitor voltage stresses.

**Table 3.** Comparison of ripple values of capacitor voltage, inductor current and load current for the proposed and conventional converters.


For the ripple content investigation, a zoomed view of the inductor current, load current, and capacitor voltages are presented in Figures 20–23, respectively. The summary of the ripple content for both converter topologies is tabulated in Table 3. From this table, it can be understood that there is a 0.4% reduction of ripple content in capacitor voltages and inductor current for the same content of load current ripples.

**Figure 20.** Zoomed view of inductor current for ripple analysis LC-BDC (blue), and conventional converter (red).

**Figure 21.** Zoomed view of load current for ripple analysis in LC-BDC (blue), and conventional converter (red).

**Figure 22.** Zoomed view of capacitor voltage in the conventional converter for ripple analysis.

**Figure 23.** Zoomed view of capacitor voltage in the proposed converter for ripple analysis.

For the reverse buck mode of operation, the gating signal of the upper switch, inductor current, load current, and capacitor voltage for four switching cycles have been captured, as presented in Figures 24–27. From Figure 25, it can be seen that the peak value of the inductor current is 3.61 A in the case of conventional converter, whereas it is 3.62 A in the case of the proposed converter for the same load current as shown in Figure 26. From Figure 27, it can be seen that the voltage across the capacitor is 16.40 V in the case of conventional converter, whereas it is 7.80 V in the case of the proposed converter. A capacitor voltage reduction of 35.80% can be witnessed in this mode of operation.

**Figure 24.** Gate voltage of upper switch (*S2*) in LC-BDC (Blue), and conventional BDC (red).

**Figure 25.** Inductor current in LC-BDC (Blue), and conventional BDC (red).

**Figure 26.** Load current in LC-BDC (Blue), and conventional BDC (red).

**Figure 27.** Capacitor voltages in LC-BDC (Blue), and conventional BDC (red).

For the critical investigation, results have been captured under various test conditions to assess the proposed converter suitability for various applications like smooth turn-on, faster load turn-off, and converter on- and off-switching with variable duty. During these conditions, captured inductor current, load current, and capacitor voltage are shown in Figures 28–30, respectively. Moreover, in these figures, a zoomed view of respective parameters is presented during both turn-on and turn-off. From these figures, peak values during both transient and steady states can be measured as listed in Table 4. Form the Table 4, and it can be observed that the proposed converter not only offers its best performance during steady-state conditions but also exhibits the same best performance during transient conditions in terms of capacitor voltage stresses.

**Table 4.** Comparison of the various parameter during both transient and steady states for the proposed and conventional converters.


**Figure 28.** Inductor current of LC-BDC (blue) and conventional BDC (red) converter during (**a**) turn on (**b**) turn off.

**Figure 29.** Load current of LC-BDC (blue) and conventional BDC (red) converter during (**a**) turn on and (**b**) turn off.

**Figure 30.** Capacitor voltage of LC-BDC (blue) and conventional BDC (red) converter during (**a**) turn on and (**b**) turn off.

For the ripple content investigation, a zoomed view of the inductor current, load current, and capacitor voltages are presented in Figures 31–34, respectively. The summary of the ripple content in both the converters is tabulated in Table 5. From this table, it can be extracted that there is a 0.3% reduction of ripple content in capacitor voltages ripples for the same load current.

**Table 5.** Comparison of Ripple Values for The Proposed and Conventional Converters in Buck Mode.


**Figure 31.** Zoomed view of inductor current for ripple analysis LC-BDC (blue), and conventional converter (red).

**Figure 32.** Zoomed view of load current for ripple analysis in LC-BDC (blue), and conventional converter (red).

**Figure 33.** Zoomed view of capacitor voltage in the conventional converter for ripple analysis.

**Figure 34.** Zoomed view of capacitor voltage in the proposed converter for ripple analysis.

#### **4. Comparative Analysis**

Another set of simulations is performed to investigate the performance of the above topologies, and simulation parameters are listed in Table 6 as mentioned below. By using these sets of simulations, various performance parameters such as voltage gains, capacitor voltages, and losses were investigated. These simulations are carried out for various output voltages ranging from 36 V to 108 V with the dc input voltage of 24 V, i.e., voltage gain ranging from 1.5 to 4.5. With the help of this data, a comparative analysis is presented in the following subsections.


**Table 6.** Parameters considered for comparative analysis.

#### *4.1. No. of Components*

As mentioned earlier, by changing the impedance network configurations, various topologies are proposed, and hence each topology has a different number of components. The number of components used for different topologies are listed and presented as a bar chart shown in Figure 35. From this chart, it is clear that the proposed converter and conventional BDC require fewer components compared to other topologies.

**Figure 35.** Bar chart of no. of components used in different topology.

#### *4.2. Capacitor Voltage Stress*

Since capacitors used in various topologies are different, total voltage stresses in all capacitors are calculated for comparison purposes. Here in Figure 36, the total capacitor stresses are plotted, while 24 V DC is converted into a range of DC voltage ranging from 36 V to 108 V. From this figure, it is clear that capacitor stresses are low in the case of the proposed converter.

**Figure 36.** Total capacitor stress in different topology.

#### *4.3. E*ffi*ciency Analysis and Loss Comparison*

To perform the converter efficiency analysis, the parasitic resistance of inductors and capacitors, and the diode forward conduction losses are considered in this paper. The parasitic resistance of inductor and capacitor are *rL* and *rC*, respectively, and forward conduction loss of diode due to forward voltage (*VF*) was assumed to be the same in all topologies for comparative analysis. The impact of the parasitic resistances and the forward voltage drop of the main power devices (MOSFETs) are also considered in this manuscript. Equivalent circuits of all the considered buck-boost bi-directional converters with the inclusion of various parasitic components are presented in Figure 37 for the efficiency calculations. Formulas derived for losses and efficiencies are presented in Table 7.

**Figure 37.** Efficiency comparison of various topologies.

**Table7.**Comparisonofvariousparameters(device,inductorandcapacitorRMScurrents,overalllosses,andfficiency)ofexistingandproposedtopologies. **Table7.***Cont.*

 *-*

*" "*

 =

1−2*D*

(1−*D*)

√*D*(1−*D*)

*MVDS* =

⎧

2*DrDs*

(1−2*D*)

⎪⎪⎪⎪⎪⎩

+*FsCoRL* + 1

1−2*D*

*RL* + 4*D*(1−*D*)

(1−2*D*)

2

*RL*

⎪⎪⎪⎪⎪⎭

*rc*

<sup>2</sup> *rL*

⎪⎪⎪⎪⎪⎨

(1−2*D*)

*RL*

(1−2*D*)

*Vo* + (1−*D*)

(1−2*D*)

2

*RL*

⎪⎪⎪⎪⎪⎬

*Po*

2*RF*

⎫

2

+ (1−*D*)

2*Vf*

(1−2*D*) *Io*

*-*

*ICrms* = 2

> *-*

By using the above-derived formulas, the efficiencies and non-ideal voltage conversion ratios of each topology with respect to gain are presented in Figures 37 and 38, respectively. From these results, it can be observed that the efficiency is higher in conventional BDC and LC-BDC compared to other existing topologies. Moreover, it can be seen that the voltage conversion ratio is more linear in the case of conventional BDC, and the proposed converter compared to other existing topologies. In all existing topologies (except conventional BDC), it can be noted that the performance of the converter is becoming poor as the gain is increased further from the designed gain value.

**Figure 38.** Voltage conversion ratio comparison of various topologies.

#### **5. Conclusions**

In this paper, a new LC bi-directional DC–DC converter utilizing small passive components has been proposed and successfully validated. Experimental results proved that there is a reduction of 75% and 35.8% in capacitor voltage for 24 V to 18 V conversion in boost mode, and 18 V to 24 V conversion in buck mode, respectively. Moreover, there is a reduction of one capacitor compared to conventional BDC for the same conversion. In this paper, the proposed converter performance in both transient and steady-state conditions is investigated and presented. This investigation reveals that the proposed converter is able to offer superior performance in both transient and steady-state conditions. Moreover, a comparative analysis of the proposed converter with the conventional BDC, Z-source converter, discontinues current quasi Z-source converter, continues current quasi Z-source converter, improved Z-source converter, switched boost converter, current-fed switched boost converter, and quasi switched boost converter is presented. This comparative analysis proved that the proposed converter offers superior performance compared to existing converters for the same conversion ratio.

**Author Contributions:** Idea, Conceptualization, Formal Analysis, Investigation, Methodology, Software, Validation, and Writing—Original Draft are the main contributions of D.R.; Funding acquisition, Project administration, Resources, Validation, Visualization and Writing—review & editing are the main contributions of R.D.; H.R. and S.M. contributed in terms of Validation and Writing—review & editing. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was supported by the Petrofac Research Chair in Renewable Energy Endowment Fund at the American University of Sharjah.

**Acknowledgments:** The authors are particularly grateful to the Zunik Energies Pvt. Ltd. and authors of the Indian Patent application published on 30 December 2016 (application number: 201641038705) for providing approval to use the patent data for this research work and publications.

**Conflicts of Interest:** The authors declare no conflict of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.

#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

### **Nonlinear Voltage Control for Three-Phase DC-AC Converters in Hybrid Systems: An Application of the PI-PBC Method**

**Federico M. Serra 1,\*, Lucas M. Fernández 1, Oscar D. Montoya 2,3, Walter Gil-González <sup>3</sup> and Jesus C. Hernández <sup>4</sup>**


Received: 17 April 2020; Accepted: 18 May 2020; Published: 20 May 2020

**Abstract:** In this paper, a proportional-integral passivity-based controller (PI-PBC) is proposed to regulate the amplitude and frequency of the three-phase output voltage in a direct-current alternating-current (DC-AC) converter with an LC filter. This converter is used to supply energy to AC loads in hybrid renewable based systems. The proposed strategy uses the well-known proportional-integral (PI) actions and guarantees the stability of the system by means of the Lyapunov theory. The proposed controller continues to maintain the simplicity and robustness of the PI controls using the Hamiltonian representation of the system, thereby ensuring stability and producing improvements in the performance. The performance of the proposed controller was validated based on simulation and experimental results after considering parametric variations and comparing them with classical approaches.

**Keywords:** hybrid system; voltage source converter; passivity-based control; proportional-integral control; voltage regulation

#### **1. Introduction**

#### *1.1. General Context*

Hybrid Renewable Based Systems (HRS) are promising alternatives for electricity supply in remote areas and are also known as stand-alone microgrid systems [1]. The objective of the stand-alone microgrids is to provide energies based on green technologies to people in remote areas, permitting them to augment their productive capabilities and enhance their quality of life [2]. This is possible to implement, thanks to the advances in renewable energy technologies that have allowed the installation of power generations in remote areas, which in turn benefit and cover non-interconnected areas.

Stand-alone microgrid systems can include different types of energy sources (photovoltaic and wind) [3], storage systems (battery banks and supercapacitors) [4], and loads. These elements can be connected through alternating current (AC) or direct current (DC) grids. In Reference [5], a comprehensive review of AC and DC microgrids was presented. DC grids are preferred because they have a higher power density than AC grids; in addition, they do not require synchronization and

incur only minor losses due to the skin effect [6]. Figure 1 shows a representative scheme for an HRS where the elements are connected through a DC grid.

**Figure 1.** Renewable-based hybrid DC system.

Note that the DC network concept comprehends an extensive range of applications, from high-voltage (examples are given in [7,8]) to low-voltage levels (examples are provided in [9,10]).

In addition, DC networks are especially attractive in control applications since the droop controls of reactive power and frequency disappear in these networks, making it easier for power flow control through lines in high-voltage levels or voltage regulation in low-voltage usages [11]. Another important aspect of the DC network is the possibility of providing service to rural or remote areas with renewable source and energy storage devices, as depicted in Figure 1; this helps improve the living conditions in those areas. In Reference [10], a nonlinear controller for a typical configuration of a rural microgrid was presented and in [12] a comparison between DC and AC microgrids implementations for rural social-economic development was performed.

#### *1.2. Motivation*

The interconnection of each part of the system with the DC grid is achieved using power electronics converters, which are responsible for managing the power flow among the sources, storage systems, and loads. In Reference [13,14], examples of the use of the power electronics converters for such interconnections were described. The objective of the power flow control in a hybrid system is to satisfy the energy demand on the loads, maximize the energy extracted from renewable energy sources, and use storage systems efficiently.

The converter is entrusted with controlling the AC voltage applied to the loads, which is usually a DC-AC converter with an LC output filter [15]. This converter can be single-phase or three-phase, depending on the load type. Meanwhile, its control regulates the amplitude and frequency of the output voltage based on a DC voltage applied on the input, which can then be controlled for the remaining HRS [16–18].

The control strategy proposed in this research is motivated by the necessity to have robust and stable control methods for providing sinusoidal voltages in remote areas where conventional power systems are nonexistent [19]. This entails that the opportunity to provide electrical service is by interfacing renewable energy resources (mainly wind turbines and photovoltaic plants) with power electronic converters that can regulate voltage and frequency by tracking sinusoidal references [20]. The approach that uses sinusoidal references is different from a conventional emulation of synchronous generators via virtual inertia control [21] since it is recommended for weak grids with large frequency variations. In these control schemes, active and reactive power measurements are used to define the frequency and voltage references [20]. Nevertheless, the proposed controller in this paper is focusing on supplying electrical service to linear and non-linear loads directly interfaced with VSCs, which implies that the measures of active and reactive power are not efficient in regulating the output voltage. For this reason, our aim is to have a direct voltage control strategy based on trajectory tracking via passivity-based control approach with experimental validations, allowing supporting three-phase balanced voltages in passive and switched loads.

#### *1.3. Brief State-of-the-Art*

In specialized literature, several strategies have been developed for the control of DC-AC converters. Due to their simplicity, the most widely used approaches are based on classic linear controllers, which are proportional-integral (PI) controllers [22]. Even though these strategies are the most used, they cannot guarantee the stability of the system. Additionally, they do not perform well away from the point of operation as in the case of non-linear loads. [16]. Therefore, advanced strategies have been developed to address the poor performance of classic controllers. In Reference [23], a feedback linearization control method was proposed based on a power-balance model between the converter and the load. This method improves the performance of linear and nonlinear loads, but the selection of gains is critical. In Reference [24], a current control algorithm for uninterruptible power supplies based on PID compensator was presented. [25] showed the reduction of voltage distortion caused as a result of slowly varying harmonic currents that use synchronous-frame harmonic regulators.Reference [26] describes an integral resonant controller of the output voltage management arrangement in a three-phase VSI. Reference [27] presented a model predictive control for output voltage regulation of a three-phase inverter with output LC filter feeding linear and nonlinear loads. In addition, authors in [28] use the same control strategy for a single-phase voltage source with linear and nonlinear loads. Despite previous works demonstrating good performance of their objective controls, none of them can guarantee the stability of the system.

On the other hand, the application of passivity-based control (PBC) techniques to power converters has the advantage of providing stable closed-loop controllers with good dynamic behavior. In Reference [16], an interconnection and damping assignment (IDA-PBC) approach was proposed to regulate the output voltage from a DC-AC converter and a comparison was also made with classic controllers. The results of [16] demonstrated the good performance of the proposed controller, even when a nonlinear load was considered. An adaptive robust control method for a DC-AC converter with high dynamic performance under nonlinear and unbalanced loads was also proposed by [29]. In both of these methods, the stability is ensured by the passive properties of the controlled system [30]. The problem with the PBC controllers applied to power converters is that the control laws depend on the system parameters and, so, stable-state errors occur when these parameters vary. The errors caused by variations in the system parameters can be eliminated with different techniques; for e.g., a dynamic extension with an integral action was proposed by [16] but at the expense of increasing the complexity of the system.

PI-PBC controllers have been proposed to combine the advantages (simplicity and robustness) of PI-based designs with the typical stability analysis based on the Lyapunov theorem employed in passive strategies. These controllers have been used in power converters for several applications [31–34].

Authors in [31] have presented the general basis of the PI-PBC theory applied to power electronic converters (switched systems). These authors demonstrate that with PI gains in a Hamiltonian representation of the averaged dynamic of the converter is possible to provide constant direct current–voltage to linear loads. Simulation and experimental results demonstrated that when VSCs are used in conversion mode (sinusoidal input to DC constant output), the PI-PBC method guarantees asymptotic stability if the load is completely linear (i.e., resistive). Observe that the VSC was operated with sinusoidal voltage imposed on the AC side to generate constant DC voltage. This is a different case, compared to the approach presented in this paper as we work with constant DC voltage provided by a combination of batteries and renewables to support three-phase balanced voltage in linear and nonlinear loads, guaranteeing stability conditions in the sense of Lyapunov. In reference [32], a general design using the PI-PBC method was presented for tracking trajectories in power electronic converters (sinusoidal or constant references) if they are bounded and differentiable (i.e., admissible trajectories). The stability in closed-loop is ensured via Barbalat's lemma. The authors of this paper validate their control design in an interleaved boost and the modular multilevel converter, including simulation and experimental validations. Note that the first converter works with AC input to provide a constant DC output, while the second one generates single-phase voltages in linear loads considering a constant DC input. This implies that the application of the developed PI-PBC method is different from our approach since we work with the isolated network applications to generate three-phase voltage signals in linear and nonlinear loads. The authors in [33] presented a methodology based on dynamic power compensation of active and reactive power in transmission systems considering superconducting coils integrated via a cascade connection between DC-DC chopper converter and the VSC. The control for this system is developed with PI-PBC, guaranteeing stability in closed-loop. The main aim of this paper is to compensate subsynchronous oscillations in power systems when faults occur in the power grid. Note that the proposal of these authors works with the VSC connected to the grid by controlling active and reactive power flow; while in our approach, the power grid is non-existent and the objective is to provide voltage service to isolated loads, i.e., we generate the power system node with constant voltage and frequency via PI-PBC design. In [34], standard passivity-based control design for integrating renewable energy resources in power systems was developed. The main idea of [34] is to provide a stable control design via PBC, which is made via energy functions using a Lagrangian formulation. Additionally, it is assumed that the wind generator would be connected to the power grid. This implies that the electrical network supports the voltage on the AC side of the converter. For this reason, the authors of this study focused on active and reactive power control and not on the three-voltage generation for isolated power applications as the case studied in our contribution.

In Reference [35], a general control design of controllers for single-phase network applications was presented via interconnection and damping assignment PBC and PI-PBC approaches. In this work, the authors considered isolated power grids composed of batteries, wind turbines, photovoltaic plants, and energy storage devices composed of superconducting coils and supercapacitors. The main contribution in [35] was to demonstrate stability in single-phase networks under well-defined load conditions. Even if this research uses isolated systems by applying PI-PBC control, it is different from our contribution since, in our work, the grid has a three-phase structure and the loads are strong, nonlinear loads (switched devices), which were not considered in [35]. Note that in [36] the initial design based on PI-PBC and IDA-PBC was complemented with modifications on the controller structure to integrate renewables in single-phase networks. In addition, the difference with our approach is that the authors do not present any experimental test that validates their simulation analysis.

Authors in Reference [37] presented a general stability analysis for single-phase networks feed-through power electronic converters considering constant power load. This analysis was performed assuming a Hamiltonian representation of the system and the perfect operation of the controllers that manage the power flow between the distributed energy resources and the grid. The authors of this work do not mention how this approach is extensible to AC grids with strong nonlinear loads as the case study in our proposal.

Even if controllers based on PI-PBC have been proposed for controlling power, electronic converters in single-phase and three-phase applications. In this paper, we focus on the problem of the voltage generation in three-phase nonlinear loads located in isolated areas by deriving the PI-PBC approach from the classical IDA-PBC method [16], which has not been reported in the scientific literature yet. In addition, our work contains multiple simulation scenarios and some experimental validations that validate the proposed approach, demonstrating its easy implementation in real-life operative cases that combine renewables, batteries, power converters, and nonlinear loads.

It is important to mention that it is necessary to employ optimal tuning of the PI gains so that PI controllers (including classical PI and PI-PBC approaches) perform excellently [38]. Active/passive tuning methods have been reported in the scientific literature. In Reference [39], it was presented an interactive tool for adjusting PI controls in first-order systems from a graphical point of view for first-order systems with time delays, numerical results confirm the efficiency of the tool developed in comparison with other literature reports. In Reference [40], an algorithm for the PID controller based on the gain margin and phase margin concept was presented. However, the controller parameters depend on a single parameter, these parameters are subjected to the desired phase margin, and a minimum required gain margin constraint. The main advantage of these tuning an approach with respect to previous works is that it is easy to implement applicable to any linear as well non-linear model structures. Authors of [41] have presented a simple method to design PI controllers in the frequency domain by proposing an optimization model with constraints. This method uses a single tuning parameter, defined as the quotient between the final crossover frequency and the zero of the controller. This adjusting procedure maximizes the controller gain by considering the equality constraint on the phase margin and an inequality restriction in the gain margin. Numerical results confirm the effectiveness of this proposal in comparison with literature reports. Additional methods for tuning PI controllers have been reported in specialized literature, some of them are particle swarm optimization [42], ant-lion optimizer [43], genetic algorithms [44], and so on. The main feature of these metaheuristic optimization methods is that they work with the minimization of integral indices to find the optimal set of control gains by using sequential programming methods [45].

**Remark 1.** *The selection of control gains is an important task in the design of PI controllers in power converter applications. These methods can be passive or active approaches that work with optimization models or desired performances [46]. Nevertheless, in this research, our focus is on presenting a simple controller based on the properties of the passivity theory combined with classical and well-known PI actions to generate ideal three-phase voltages for non-linear loads in isolated areas. This implies that the focus in the grid performance with load variations and no optimal adjusting of the control gains. In this sense, we employ a basic tuning method based on the root locus design approach [47].*

#### *1.4. Contribution and Scope*

In the present study, a PI-PBC controller is proposed for regulating the amplitude and frequency of the output voltage in a three-phase DC-AC converter with an LC filter, providing a well-defined sinusoidal service to linear and nonlinear loads by transforming the DC signal from the transmission/distribution network to local loads [48].

The main contribution of this research in the literature reports about the control of VSCs for feeding isolated three-phase loads can be summarized as follows:



In addition, the performance of the controller under parametric variation is shown, and a comparison with the classic PI controller demonstrates the superiority of the proposal to mitigate the harmonic content produced by non-linear loads.

Regarding the scope of this research, it is important to mention that in the control design as well as in the simulation, experimental validations will be considered unique voltage source converters that are forced to work as an ideal voltage source to provide sinusoidal voltages to linear and nonlinear loads. For doing so, we consider that the DC side of the converter is fed by a strong DC network (transmission/distribution DC grid) or by a combination of renewable energy resources and batteries [19,28]. In addition, to determine the amount of instantaneous power absorbed by the load (linear and nonlinear), it is considered that there is a current measure at the load side which is important since the amount of current provided by the converter is a linear function of the load consumption. This implies that the existence of this measure is indispensable when no load estimators are implemented, as in the case studied in this research. Note that the implementation of load estimators could be considered for future work since only a few studies have been reported in the scientific literature with experimental validations.

#### *1.5. Organization of the Document*

The remainder of this paper is organized as follows. In Section 2, we describe the configuration of the DC-AC converter and its dynamical modeling using a Hamiltonian representation. In Section 3, we explain the proposed control design based on the PI-PBC approach, which guarantees asymptotic stability according to Lyapunov. In Section 4, we demonstrate the numerical performance of the proposed method based on simulations and experimental validations using a laboratory prototype. Finally, the main conclusions derived based on this study are presented in Section 5.

#### **2. System Configuration and Dynamical Model**

The DC-AC converter comprises of a three-phase voltage source converter with IGBTs (*S*1,... , *S*6) and an LC output filter. Figure 2 shows the structure of the DC-AC converter considered in this study for supplying an arbitrary load (i.e., linear or nonlinear consumption) [49].

**Figure 2.** Three-phase DC-AC converter structure.

The DC-link voltage, *vdc*, is considered to be approximately constant and controlled by other converters involved in the HRS shown in Figure 1, and thus, its dynamics are not considered in our model [16,49].

#### *Dynamical Model*

The DC-AC converter model in *dq* coordinates is as follows [3],

$$\mathrm{Li}\_{\mathrm{d}}^{\ddot{\mathsf{i}}} = -m\_{\mathrm{d}} \upsilon\_{\mathrm{d}\mathfrak{c}} - \mathrm{Ri}\_{\mathrm{d}} - \omega\_{\mathrm{dq}} \mathrm{Li}\_{\mathfrak{q}} - \mathfrak{e}\_{\mathrm{d}\prime} \tag{1}$$

$$\mathrm{Li}\_{q}^{\natural} = -m\_{q}\upsilon\_{\mathrm{dc}} - \mathrm{Ri}\_{q} + \omega\_{\mathrm{dq}}\mathrm{Li}\_{\mathrm{d}} - \mathrm{e}\_{\mathrm{q}}.\tag{2}$$

$$\mathbb{C}\dot{\mathcal{C}}\_d \quad = \ \dot{\mathbf{i}}\_d - \omega\_{d\mathbf{q}}\mathbf{C}\mathbf{e}\_\mathbf{q} - \dot{\mathbf{i}}\_{\mathrm{Ld}\prime} \tag{3}$$

$$\mathbf{C}\dot{\mathbf{e}}\_q = \dot{\mathbf{i}}\_q + \omega\_{dq}\mathbf{C}\mathbf{e}\_d - \dot{\mathbf{i}}\_{Lq} \tag{4}$$

where *ωdq* is the angular frequency of the *dq* reference frame, which is set equal to the desired output-voltage frequency; *id* and *iq* represent the currents in the *dq* frame; *ed* and *eq* are the output voltages; *iLd* and *iLq* correspond to the load currents; and *md* and *mq* are the inverter modulation indexes. All the variables represented in the *dq* reference frame are obtained through Park's transformation from the *abc* variables. The parameters *L*, *C*, and *R* represent the inductance, capacitance of the output filter, and equivalent output resistance, which model the filter inductance losses and converter losses, respectively.

The port-Hamiltonian (pH) model of the system can be written as follows:

$$\dot{\mathbf{x}} = \left[\mathbf{J} - \mathbf{R}\right] \frac{\partial H(\mathbf{x})}{\partial \mathbf{x}} + \mathbf{g}\mathbf{u} + \mathbf{J}\_{\prime} \tag{5}$$

where the state vector is **x**, **J**, and **R** are the interconnection and damping matrices, respectively, *H*(**x**) represents the total energy stored in the system, **g** is the input matrix, **u** is the control input vector, and *ζ* represents the external input.

The pH model of the DC-AC converter is represented by Equation (6).

$$
\begin{bmatrix} L\dot{l}\_d\\ L\dot{l}\_q\\ C\dot{e}\_d\\ C\dot{e}\_q \end{bmatrix} = \begin{bmatrix} -R & -\omega\_{d\eta}L & -1 & 0\\ \omega\_{d\eta}L & -R & 0 & -1\\ 1 & 0 & 0 & -\omega\_{d\eta}C\\ 0 & 1 & \omega\_{d\eta}C & 0 \end{bmatrix} \begin{bmatrix} i\_d\\ i\_q\\ e\_d\\ e\_q \end{bmatrix} + \begin{bmatrix} v\_{d\zeta} & 0\\ 0 & v\_{d\zeta}\\ 0 & 0\\ 0 & 0 \end{bmatrix} \begin{bmatrix} m\_d\\ m\_q \end{bmatrix} + \begin{bmatrix} 0\\ 0\\ -i\_{Ld}\\ -i\_{Lq} \end{bmatrix} . \tag{6}
$$

In this case, the interconnection and damping matrices are defined based on Equation (6) as

$$\mathbf{J} = \begin{bmatrix} 0 & -\omega\_{dq}L & -1 & 0\\ \omega\_{dq}L & 0 & 0 & -1\\ 1 & 0 & 0 & -\omega\_{dq}\mathbb{C}\\ 0 & 1 & \omega\_{dq}\mathbb{C} & 0 \end{bmatrix}^{\prime},\tag{7}$$

$$\mathbf{R} = \begin{bmatrix} R & 0 & 0 & 0\\ 0 & R & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 \end{bmatrix}^{\prime},\tag{8}$$

where **<sup>J</sup>** <sup>=</sup> <sup>−</sup>**J***<sup>T</sup>* is antisymmetric and **<sup>R</sup>** <sup>=</sup> **<sup>R</sup>***<sup>T</sup>* <sup>≥</sup> 0 is symmetric positive semidefinite.

The total energy stored in the system, *H*(**x**), is shown by the sum of the energy stored on the output-filter inductors and capacitors,

$$H(\mathbf{x}) = \frac{1}{2} \left( L\dot{i}\_d^2 + L\dot{i}\_q^2 + \mathbb{C}e\_d^2 + \mathbb{C}e\_q^2 \right). \tag{9}$$

It should be noted that *H*(**x**) is a hyperboloidal function that is convex and positive definite.

#### **3. PI-PBC Approach**

This section presents three main aspects of power electronic converters' control using passivity-based control theory. (1) The design of the proposed PI-PBC approach by transforming the trajectory tracking problem into a regulation problem using the incremental representation. (2) The control objective of the problem, i.e., the definition of the desired sinusoidal trajectory. (3) The stability analysis of the proposed control scheme via Lyapunov's stability theorem for autonomous dynamical systems. In the next subsections, each of these aspects will be discussed.

#### *3.1. Control Design*

The proposed PI-PBC approach depends on the existence of an admissible trajectory, defined as [34],

$$\mathbf{x}^\* = \begin{bmatrix} \ L \dot{x}\_d^\* & L \dot{x}\_q^\* & \mathbb{C}e\_d^\* & \mathbb{C}e\_q^\* \end{bmatrix}^T,\tag{10}$$

such that the dynamical system in Equation (6) is continuous, differentiable, and bounded, which implies that

$$\dot{\mathbf{x}}^{\*} = \left[\mathbf{J} - \mathbf{R}\right] \frac{\partial H(\mathbf{x}^{\*})}{\partial \mathbf{x}^{\*}} + \mathbf{g} \mathbf{u}^{\*} + \mathbf{J}\_{\prime} \tag{11}$$

with some **u**∗ bounded.

If we define **x**˜ = **x** − **x**<sup>∗</sup> and **u**˜ = **u** − **u**∗, the system in Equation (5) can be written as follows:

$$\dot{\mathbf{x}} + \dot{\mathbf{x}}^{\*} = \left[\mathbf{J} - \mathbf{R}\right] \frac{\partial H(\ddot{\mathbf{x}} + \mathbf{x}^{\*})}{\partial(\ddot{\mathbf{x}} + \mathbf{x}^{\*})} + \mathbf{g}(\ddot{\mathbf{u}} + \mathbf{u}^{\*}) + \mathbf{\mathcal{J}}.\tag{12}$$

The energy function of the system given by Equation (9) can be represented as:

$$H(\mathbf{x}) = H(\tilde{\mathbf{x}} + \mathbf{x}^\*) = \frac{1}{2} \mathbf{x}^T \mathbf{P}^{-1} \mathbf{x} = \frac{1}{2} (\tilde{\mathbf{x}} + \mathbf{x}^\*)^T \mathbf{P}^{-1} (\tilde{\mathbf{x}} + \mathbf{x}^\*),\tag{13}$$

and the gradient of this function is,

$$\frac{\partial H(\mathbf{x})}{\partial \mathbf{x}} = \mathbf{P}^{-1} \mathbf{x},\tag{14}$$

then,

$$\begin{split} \frac{\partial H(\ddot{\mathbf{x}} + \mathbf{x}^\*)}{\partial(\ddot{\mathbf{x}} + \mathbf{x}^\*)} &= \mathbf{P}^{-1}(\ddot{\mathbf{x}} + \mathbf{x}^\*) = \mathbf{P}^{-1}\ddot{\mathbf{x}} + \mathbf{P}^{-1}\mathbf{x}^\* \\ &= \frac{\partial H(\ddot{\mathbf{x}})}{\partial \ddot{\mathbf{x}}} + \frac{\partial H(\mathbf{x}^\*)}{\partial \mathbf{x}^\*}. \end{split} \tag{15}$$

Using Equation (15), the system defined by Equation (12) can be written as follows:

$$\dot{\mathbf{x}} + \dot{\mathbf{x}}^\* = \left[\mathbf{J} - \mathbf{R}\right] \frac{\partial H(\ddot{\mathbf{x}})}{\partial \ddot{\mathbf{x}}} + \left[\mathbf{J} - \mathbf{R}\right] \frac{\partial H(\mathbf{x}^\*)}{\partial \mathbf{x}^\*} \quad + \quad \mathbf{g}\ddot{\mathbf{u}} + \mathbf{g}\mathbf{u}^\* + \mathbf{J}\_{\prime} \tag{16}$$

Thus, the following error dynamics are reached.

$$
\dot{\tilde{\mathbf{x}}} = \left[ \mathbf{J} - \mathbf{R} \right] \frac{\partial H(\tilde{\mathbf{x}})}{\partial \tilde{\mathbf{x}}} + \mathbf{g} \tilde{\mathbf{u}}.\tag{17}
$$

In addition, the output of the system can be calculated as follows:

$$\mathbf{y} = \mathbf{g}^T \frac{\partial H(\tilde{\mathbf{x}})}{\partial (\tilde{\mathbf{x}})}.\tag{18}$$

*Electronics* **2020**, *9*, 847

The system in Equation (17) is passive if *<sup>H</sup>*˙ (**x**˜) <sup>≤</sup> **<sup>y</sup>**˜ *<sup>T</sup>***u**˜ [33,34]. Therefore, to obtain a closed-loop dynamic based on Lyapunov theory, the energy function is shown as:

$$H(\bar{\mathbf{x}}) = \frac{1}{2} \bar{\mathbf{x}}^T \mathbf{P}^{-1} \mathbf{x},\tag{19}$$

where *H*(0) = 0 and *H*(**x**˜) > 0, ∀**x**˜ = 0, which satisfy the first and second conditions of the Lyapunov theorem. The time derivative of the proposed energy function is

$$\begin{split} \dot{H}(\ddot{\mathbf{x}}) &= -\ddot{\mathbf{x}}^T \mathbf{P}^{-1} \dot{\mathbf{x}} = -\ddot{\mathbf{x}}^T \mathbf{P}^{-1} \mathbf{R} \mathbf{P}^{-1} \ddot{\mathbf{x}} + \ddot{\mathbf{x}}^T \mathbf{P}^{-1} \mathbf{g} \ddot{\mathbf{u}}, \\ &= -\ddot{\mathbf{x}}^T \mathbf{P}^{-1} \mathbf{R} \mathbf{P}^{-1} \ddot{\mathbf{x}} + \ddot{\mathbf{y}}^T \ddot{\mathbf{u}} \le \ddot{\mathbf{y}}^T \ddot{\mathbf{u}}, \end{split} \tag{20}$$

which proves that the dynamics of the error are also passive.

The proposed controller can be written as follows:

$$
\dot{\mathbf{z}}\_{-} = -\ddot{\mathbf{y}}\_{/} \tag{21}
$$

$$
\ddot{\mathbf{u}}\_r = \begin{array}{c} -K\_P \ddot{\mathbf{y}} + K\_I \mathbf{z}\_r \\ \end{array} \tag{22}
$$

and it has a PI structure, where *KP* and *KI* are the proportional and integral gain matrices, respectively, which are diagonal and positive definite, and ˜**y** is

$$\mathbf{y} = \mathbf{g}^T \frac{\partial H(\tilde{\mathbf{x}})}{\partial \tilde{\mathbf{x}}} = \begin{bmatrix} v\_{dc} \begin{pmatrix} i\_d - i\_d^\* \\ v\_{dc} \end{pmatrix} \\ \upsilon\_{dc} \begin{pmatrix} i\_q - i\_q^\* \\ i\_q - i\_q^\* \end{pmatrix} \end{bmatrix}. \tag{23}$$

Note that **z** represents a set of auxiliary variables that allow the passive output feedback and the inclusion of an integral action to minimize steady-state errors on the control objective. For more details of the PI-PBC approach for non-affine dynamic systems, refer to [32].

Using Equation (11), the references of the *dq* axis currents and the modulation index can be obtained as follows:

$$\dot{\iota}\_d^\* = -\omega\_{dq} \mathbf{C} e\_q^\* + \dot{\iota}\_{\rm{Ld}} \tag{24}$$

$$\mathbf{i}\_q^\* = -\omega\_{d\eta} \mathbf{C} \mathbf{e}\_d^\* + \mathbf{i}\_{\mathbf{L}\eta} \tag{25}$$

$$m\_d^\* \quad = \frac{1}{v\_{dc}} \left( Li\_d^\* + Ri\_d^\* + \omega\_{d\eta} Li\_q^\* + e\_d^\* \right), \tag{26}$$

$$m\_q^\* \quad = \frac{1}{\upsilon\_{dc}} \left( \dot{L}\_q^\* + \dot{R}\_q^\* - \omega\_{d\eta} L \dot{i}\_d^\* + e\_q^\* \right), \tag{27}$$

where we have considered that *e*˙ ∗ *<sup>d</sup>* = *e*˙ ∗ *<sup>q</sup>* = 0 because *e*<sup>∗</sup> *<sup>d</sup>* and *e*<sup>∗</sup> *<sup>q</sup>* are constant values in the Park's reference frame.

Figure 3 shows the proposed control scheme. It can be observed that the modulation indexes are obtained using Equations (22) and (23) with Equation (27), and the references *e*∗ *<sup>d</sup>*, *e*<sup>∗</sup> *<sup>q</sup>* , and *ωdq* are selected by the users. The angle *<sup>θ</sup>dq* used in the Park's transformations is obtained from *<sup>d</sup>ωdq dt* .

**Figure 3.** Block diagram of the proposed controller.

#### *3.2. Control Objective*

The main idea behind implementing VSCs for the integration of non-linear loads in the AC side using the energy provided by the DC network (see Figure 1) is to generate a three-phase sinusoidal signal with constant amplitude and frequency [19]. This implies that the references for the voltage signals in the capacitors in parallel to the load can be defined as follows:

$$\begin{aligned} \begin{aligned} \varepsilon\_a^\*(t) &= \sqrt{2}E\_{rms}\sin(\omega t),\\ \varepsilon\_b^\*(t) &= \sqrt{2}E\_{rms}\sin(\omega t - 2\frac{\pi}{3}),\\ \varepsilon\_c^\*(t) &= \sqrt{2}E\_{rms}\sin(\omega t + 2\frac{\pi}{3}), \end{aligned} \tag{28}$$

where *Erms* is the desired voltage magnitude and *ω* represents the angular frequency, i.e., for this study *ω* = 2*π f* , with *f* = 50 Hz.

From the desired three-phase voltage reference defined in Equation (28), we can observe that the electrical frequency is considered as an input for the Park's transformation (see Figure 3). This implies that if the controller makes its task, the voltage outputs in the load will be sinusoidal waves regardless of the load variations (linear or non-linear behavior) [50].

Note that if we send a constant frequency as input to the Parks' transformation, then the sinusoidal references in Equation (28) become constant values in the *dq* reference frame, i.e.,

$$\begin{aligned} e\_d^\*(t) &= \quad E\_{rms}, \\ e\_q^\*(t) &= \quad e\_0^\*(t) = 0, \end{aligned} \tag{29}$$

which implies that if the controller reaches these values in the *dq* reference frame, then the voltage output will be perfectly sinusoidal in the *abc* reference frame sinusoidal [19].

**Remark 2.** *Note that the voltage and frequency and regulation reported in this paper allows supporting a three-phase sinusoidal signal in load coupling based on the properties of the controller for following constant references in the dq frame [19]. This is different from classical approaches that regulate voltage and frequency by measuring active and reactive power by emulating synchronous machines using the virtual inertia concept [20,21].*

#### *3.3. Stability Analysis*

The stability analysis in modern control applications is an important aspect and should be demonstrated to show that they are suitable for implementation without fails [8,51]; one of the main attractive characteristics of the PBC design is that in a large class of linear and nonlinear system, stability can be guaranteed using Hamiltonian or Lagrangian representations [14,32]. Here, we present basic proof for analyzing the proposed PI-PBC approach, since it was extensively studied in [32,35].

We will enunciate the necessary conditions for guaranteeing stability of a nonlinear autonomous dynamical *x*˙ = *f*(*x*, *u*) system in the sense of Lyapunov around the equilibrium point *x* = *x*- as follows [52]:


Note that to prove these two necessary conditions for the proposed closed-loop dynamical system with state variables *x*˜ and *z*, we employed a quadratic function as recommended in [32] which fulfills the positive definiteness in all the solution space which is zero only in the equilibrium point, as follows:

$$V(\mathbf{\bar{x}}, \mathbf{z}) = H(\mathbf{\bar{x}}) + \frac{1}{2} \mathbf{z}^T K\_l \mathbf{z},\tag{30}$$

where *H*(**x**˜) is the desired Hamiltonian function. Note that the second term is a quadratic expression as function of the integral variables that deal with these new stable variables introduced by the PI-PBC method [33].

Now, if we take the time derivative of the candidate Lyapunov function in Equation (30), then the following result is yielded:

$$
\dot{V}(\ddot{\mathbf{x}}, \mathbf{z}) = \dot{H}(\ddot{\mathbf{x}}) + \mathbf{z}^T K\_I \dot{\mathbf{z}} \tag{31}
$$

where, if it substitutes the expression in Equation (20) by considering the PI-PBC design in Equation (22), then, the following result is reached:

$$\begin{split} \dot{V}(\ddot{\mathbf{x}}, \mathbf{z}) &= \begin{aligned} &-\mathbf{x}^{\mathrm{T}} \mathbf{P}^{-1} \mathbf{R} \mathbf{P}^{-1} \ddot{\mathbf{x}} + \mathbf{x}^{\mathrm{T}} \mathbf{P}^{-1} \mathbf{g} \ddot{\mathbf{u}} + \mathbf{z}^{\mathrm{T}} K\_{I}(-\ddot{\mathbf{y}}) \\ &\leq \ -\ddot{\mathbf{y}}^{\mathrm{T}} K\_{P} \ddot{\mathbf{y}} \leq 0, \end{aligned} \tag{32}$$

which clearly fulfills the second condition of the Lyapunov's theorem [34].

**Remark 3.** *As the derivative of the Lyapunov function with respect to time does not directly contain the state variables of interest, i.e., x*˜ *and z, based on the Lyapunov's stability theorem, we can affirm that the dynamical system* ˙*x*˜ *(see Equation (17)) is stable in a closed-ball that contain the equilibrium point x* = *x*˜*, as demonstrated in [35].*

Regarding the range of application of the aforementioned stability analysis of the real systems and its relation to the parameters of the system, it is important to highlight the following facts:


#### **4. Results**

The proposed PI-PBC controller was validated based on simulations and experimental results. The parameters used in the system in both the simulations and experimental tests are listed in Table 1.


**Table 1.** System parameters.

#### *4.1. Simulation Results*

The simulation results were obtained using a realistic model of the system, including the switching effects, losses, and a detailed transistor model. The simulations were implemented using the SimPowerSystem in MATLAB.

To validate the performance of the proposed controller, a test was performed by changing the load (linear and nonlinear). Figure 4a shows the *dq* axis output voltage and its reference, and Figure 4b shows the output voltage and inductor current for phase *a*. Initially, the system operated with a resistive load of 10 Ω, and the reference of the output voltage was *e*∗ *<sup>d</sup>* = 100 V and *e*<sup>∗</sup> *<sup>q</sup>* = 0 V. When the load was changed from 10 Ω to 5 Ω (at 0.1 s), the *d*-axis voltage exhibited a transient variation; this also occurred for the voltage on the *q*-axis, but within a short time (approximately one cycle of the voltage), the controller was able to regulate the amplitude and frequency of the output voltage in the expected time and with the desired dynamic behavior. At 0.2 s, the nonlinear load was connected to the system. Again, the controller regulated the *dq* axis output voltages in an acceptable time and with suitable behavior. In addition, as shown in Figure 4b, the waveform of the voltage was sinusoidal (free of significant harmonics) even under a nonlinear load.

**Figure 4.** Voltage behavior in the Park's reference frame, and voltage and load current per-phase. (**a**) *edq* profile. (**b**) Voltage and load current in phase *a*.

Figure 5a shows the three-phase output voltage and Figure 5b depicts the currents in the filter inductors. The amplitude of the three-phase output voltage was regulated in advance of the load changes as expected.

**Figure 5.** Dynamical performance of the voltage and currents at the point of load connection. (**a**) Three-phase voltage profile. (**b**) Current profile for linear and nonlinear consumption.

With the objective to show the performance of the proposed controller in the presence of parametric variations, a test with varying inductance and resistance values of the LC filter was performed. Figure 6 shows the *d*-axis output voltage for the same test of Figure 4 when the value of the inductance and resistance of the filter are different and the same are then considered in the controller. The parameter variation consists of a change of 50% in the inductance and resistance of the filter. It can be observed that the *d*-axis voltage exhibited transient differences between the situations with and without parameter variation, while in regimen, the voltage is regulated in the reference value as the integral action of the PI eliminates possible steady-state errors (see [54]).

**Figure 6.** Behavior of *d*-axis voltage for parametric variation.

For comparison purposes with classic approaches (PI controllers), Figure 7 shows the output voltage behavior in *dq*-coordinates for the same test of Figure 4, when a PI controller (in blue) and the proposed PI-PBC controller (in red) are working. It can be observed that the classic PI controller performs similar to PI-PBC when a linear load is considered, but when a non-linear load is connected, the system with the PI controller presents high harmonic content, and therefore the performance is reduced. This test shows the superiority of the proposed strategy to mitigate the harmonics due to non-linear loads.

**Figure 7.** Voltage behavior in the Park's reference frame, comparison between the proposed controller (in red) and the classic PI approach (in blue).

With the aim to demonstrate the performance of the proposed controller under highly demanding conditions, Figure 8 presents a test with a change of the amplitude reference of the output voltage and a change in load when the converter supplies the nonlinear load, as considered in Figure 4. In Figure 8a, the dq axis output voltage and its reference is illustrated and in Figure 8b, the output voltage and inductor current for phase *a* is shown. It can be seen that the proposed PI-PBC controller (in red) regulates the amplitude of the output voltage in the presence of changes of the amplitude reference (0.1 s) and load changes (0.2 s) even when the power of the nonlinear load is increased. Furthermore, in this test, a comparison between the proposed controller (in red) and the classic PI (in yellow) approach was performed. The performance of the classic PI presents high harmonic content as the value of the power of the nonlinear charge increases while the oscillations presented by the PI-PBC are small. It should be noted that the elimination of these oscillations can be accomplished using a multiple reference controller but it will increase the complexity of the control algorithm [24].

**Figure 8.** Test system behavior under very demanding conditions for the proposed controller (in red) and the classic PI approach (in yellow). (**a**) Voltage behavior in the Park's reference frame. (**b**) Phase *a* voltage profile.

Figure 9 depicts the harmonic spectrum of the output voltage using the proposed PI-PBC controller. It can be seen that the amplitude of individual harmonics remains below the required limits by the standards and the total harmonic distortion (THD) is 2.17%. This value is lower than the values established by IEC 62040-3 (less than 8%) [16]. Figure 10 shows the harmonic spectrum of the output voltage for the classic PI controller, the THD is 7.37%, which is very close to the allowed limits. The harmonic spectra were obtained for the system feeding the nonlinear load corresponding to Figure 8 between 0.2 s to 0.3 s.

**Figure 9.** Voltage harmonic spectrums of proportional-integral passivity-based controller (PI-PBC).

**Figure 10.** Voltage harmonic spectrums of Classical PI controller.

Finally, Figure 11 shows a test for an unbalanced load. It can be seen that the three-phase voltage is maintained with constant amplitude and frequency, and without significative harmonics when considering these types of loads.

**Figure 11.** Test system behavior under very demanding conditions for the proposed controller (in red) and the classic PI approach (in yellow). (**a**) Voltage behavior in the Park's reference frame. (**b**) Phase *a* voltage profile.

#### *4.2. Experimental Results*

The experimental results were obtained using a laboratory prototype with the same parameters employed in the simulation tests (see Table 1). The DC-AC converter was constructed using an IGBT module (SEMiX101GD12E4s), and the controller was implemented in a TMS320F28335 floating-point DSP (Texas Instruments).

Figure 12 shows the three-phase voltage generated by the system under a resistive load change (50 Ω to 25 Ω). The controller regulated the amplitude and frequency of the output voltage, and the time response required for the load change was approximately one cycle, thereby agreeing with the previously described simulation results, and this time is acceptable in practical applications as well.

**Figure 12.** Experimental results: three-phase generated voltage for a resistive load change.

The output voltage and inductor current for phase *a* are shown in Figure 13 for the resistive load change and in Figure 14 for a nonlinear load. The results show that dynamic behavior was acceptable in both cases. In addition, when a nonlinear load was considered (Figure 14), the controller generated an output voltage that was free of significant harmonics. The voltage total harmonic distortion is 3.6%. This value is lower than the values established by IEC 62040-3 (less than 8%) [16].

**Figure 13.** Experimental results: output voltage and inductor current for phase *a* with a resistive load change.

**Figure 14.** Experimental results: output voltage and inductor current for phase *a* under a nonlinear load.

#### **5. Conclusions**

In this article, we present a PI-PBC controller to regulate the amplitude and frequency of a three-phase output voltage in a DC-AC converter with an LC filter. Simulations and experimental results show that the proposed controller allowed regulating the amplitude and frequency of the output voltage of the converter under both linear and nonlinear load changes. Additionally, the PI-PBC allowed generating well-defined sinusoidal signals to linear and nonlinear loads. The results of the performance analysis showed that the proposed design combines the simplicity and robustness of PI-based controllers with the stability and performance characteristics of PBC as a powerful tool for the design of power converter controllers used in hybrid power systems.

Derived from this research, some future recommendations have been made as follows: (1) to develop a direct power control approach to operate DC-AC converters in constant power load applications; (2) to combine the advantages of the passivity-based control designs with virtual inertia emulators to develop robust controllers for weak three-phase distribution networks with unbalanced and nonlinear loads; and (3) to employ advanced methodologies for optimal adjustment of PI control gains using active and passive strategies in order to improve the dynamical performance of the proposed PI-PBC method in applications with large variations in the load terminals.

**Author Contributions:** Conceptualization and writing—review and editing, F.M.S., L.M.F., O.D.M., W.G.-G., and J.C.H. All authors have read and agreed to the published version of the manuscript.

**Funding:** This work was partially supported by the Universidad Nacional de San Luis under project PROICO: 142318 "Control de convertidores de potencia para la integración de fuentes de energía renovables y vehículos eléctricos a la red"and by Fondo para la Investigación Científica y Tecnológica (FONCyT) under project PICT-2017-0794 "Cargadores de baterías para vehículos eléctricos: integración con la red y fuentes de energía renovables" and in part by the Universidad Tecnológica de Bolívar under grant CP2019P011 associated with the project: "Operación eficiente de redes eléctricas con alta penetración de recursos energéticos distribuidos considerando variaciones en el recurso energético primario".

**Acknowledgments:** The authors want to express to thanks to Universidad Nacional de San Luis and Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) in Argentina, Universidad Distrital Francisco José de Caldas, Universidad Tecbológica de Bolivar in Colombia and University of Jaén in Spain.

**Conflicts of Interest:** The authors declare no conflicts of interest.

#### **Abbreviations**

The following abbreviations and nomenclature are used in this manuscript:

*Acronyms*



#### **References**


© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

*Article*
