A Mathematical Programming Approach for the Optimal Operation of Storage Systems, Photovoltaic and Wind Power Generation

Abstract

The ever-growing participation of Renewable Energy Sources (RES) in modern distribution networks is replacing an important portion of Conventional Generation (CG), which brings along new challenges in the planning and operation of distribution grids. As RES such as Photovoltaic Energy (PV) and Wind Power Generation (WPG) increase in distribution networks, studies regarding their integration and coordination become more important. In this context, the purpose of this paper is to propose a Multi-period Optimal Power Flow (MOPF) model for the optimal coordination of Battery Energy Storage Systems (BESSs) with PV, WPG, and CG in modern distribution networks. The model formulation was developed in A Modeling Language for Mathematical Programming (AMPL) and solved through the Knitro solver within a time horizon of 24 h. A distinctive feature and one of the main contributions of the proposed approach is the fact that BESSs can provide both active and reactive power. The proposed optimization model reduces power losses and improves voltage profiles. To show the applicability and effectiveness of the proposed model, several tests were carried out on the 33-bus distribution test system and a real distribution system of 141 buses located in the metropolitan area of Caracas. Power loss reductions of up to 58.4% and 77% for the test systems of 33 and 141 buses were obtained, respectively, when BESSs provided both active and reactive power. The results allow us to conclude that the proposed model for optimal coordination of BESSs with RES is suitable for real-life applications, resulting in important reductions of power losses and flattening of voltage profiles.

1. Introduction

1.1. Motivation

In recent years, distribution networks have evolved from passive to active grids through the incorporation of different types of Distributed Generation (DG) along with Demand Response (DR) and BESSs that enable a more active participation of users [1]. Moreover, DGs such as PV and WPG have increased their acceptance around the world due to higher ecological awareness [2]. Nevertheless, the massive penetration of DG leads to profound technical and economic impacts, especially due to their intermittent nature, which poses new challenges to the grid operation [3].

The evolution of the Smart Grid (SG) paradigm has promoted a more complex management of both transmission and distribution systems integrating Distributed Energy Resources (DER) that include not only generation but also storage of energy [4]. In this sense, some studies rely on BESSs to overcome the limitations of DG units [5]. Above all, single PV and WPG are considered non-controllable resources; nonetheless, their coordination with BESS can guarantee a more controllable source in distribution grids [6]. As reported in [7], the combination of PVs and BESSs brings several complexities and extra costs to modern distribution systems. As stated in [8], the correct application of PVs and BESSs can mitigate power losses, increment the system capacity, and improve the voltage profile. There are a bunch of applications for BESSs in transmission and distribution systems. In [9], BESSs are used for arbitrage purposes within an electricity market. In [10], it is described how BESSs can be used for frequency regulation in power grids. In [11], the authors discuss BESSs with spinning reserve for renewable integration. Other applications of BESSs also include peak shaving [12] and DGs sources [13].

According to [14], energy arbitrage considering BESSs consists of monetizing price fluctuations and charging and discharging BESSs when prices are low and high, respectively. In consonance with [15], BESSs represent an interesting alternative to frequency regulation due to their fast response in the energy supply when compared to natural gas plants. In accordance with [16], BESSs can be used as power sources that contribute to efficiency improvement by temporary storage of braking energy in the event of peak shaving. As a general rule, the proper implementation of BESSs constitutes a significant path for improving modern distribution networks [17].

1.2. Literature Review

To spot the relevant literature, a search and later filtering were carried out in Science Direct and IEEE databases using as keywords “storage systems”, “PV and BESS”, “WPG and BESS”, “MOPF” “power losses”, and “BESS in distribution systems”. Only peer-reviewed and high-impact papers were considered in the filtering. Numerous studies have been carried out regarding the implementation of BESSs, PV, WPG, and other sources in distribution networks [18]. Such studies take into account aspects that include optimization [6], planning [19], operation [20,21], and management of electric distribution networks [22]. In [23], the authors propose a methodology to find the optimal number, location, and size of BESSs, with the objective of avoiding over and under-voltages in distribution grids. A heuristic approach is used to place the storage systems, and a MOPF is applied for the sizing problem. Two distribution systems are used: a 34-bus test system and a real Italian low-voltage power grid. The authors of [24] proposed a Grey Wolf Optimization (GWO) algorithm for optimal placement of energy storage in distribution systems with DG. The storage sizing is designed based on load profiles and convenient peak shaving; simulations are carried out using a 34-bus test system, and a comparison is performed with a Genetic Algorithm (GA) approach. In [22], the authors develop a Modified Imperialist Competitive Algorithm (ICA) that coordinates the operation of BESSs, PV, and diesel generators within a 24-h time horizon. The energy management approach proposed by the authors seeks to optimize the charge and discharge of BESSs to minimize operating costs. The authors carry out several tests in the IEEE 33-bus test system considering power flow constraints as well as voltage and current limits. In [25], the authors propose an optimal reactive power control approach that aims to enhance the PV hosting capacity of distribution networks with BESSs. The main objective consists of determining the optimal size, dispatch, and control setting of the Volt/VAr function of smart inverters for PV and BESS units. For this, the optimal location of such devices is also considered. The maximization of the PV hosting capacity is modeled as a mixed-integer nonlinear programming problem also considering the minimization of voltage deviation in the network. A bio-inspired metaheuristic known as Slime Mould Algorithm (SMA) is implemented to solve the optimization problem.

A GA approach is proposed in [20] to optimize BESS capacity and placement in distribution networks. The authors aim at minimizing power losses while taking into account power quality issues. The proposed approach is tested on the IEEE 33-bus test system.

A mathematical approach, guided by two economic metrics, namely the net present value and the internal rate of return, is also developed in [26] for optimal BESS placement. The proposed methodology takes into account the stochastic behavior of PV generation and load. The authors run several tests on the IEEE 33-bus test system showing that the optimal placement of BESS results in voltage profile improvement and peak shaving. Furthermore, they show that the use of one or the other economic metrics leads to different results.

In [27], the authors presented a Mixed Integer Nonlinear Programming (MINLP) approach for integrated planning expecting BESS, DG, and capacitors on medium-size distribution grids. In the proposed model, cost of energy as well as environmental and PV load are considered in a 135-bus test system under different topologies. An exact method, namely Mixed-Integer Second-Order Cone Programming (MI-SOCP) is introduced in [28] to guarantee global optimal solutions in the optimal placement and sizing of DG units in distribution networks. In this case, simulations in a 33-bus and a 69-bus test system validated that exact methods are better than heuristic techniques in terms of quality of solutions and processing time.

In [29], an operation strategy is developed in AC microgrids. The goal consists of minimizing the power variations imposed by diesel generators coordinated with PV generation. A real-life application regarding the optimal operation of BESSs in AC microgrids is presented in [30]. In this case, storage systems are used to minimize the effects of power cuts; therefore, BESSs supply energy to the loads during electricity cuts.

Table 1. Integration of BESS with RES.

Ref.	Resources Considered				Solution	Objective Function
	BESS Power	PV	WPG	CG	Approach
[3]	active/reactive	x	x		ALO	Min power losses
[6]	active	x			HGSO	Min power losses
[20]	active	x			GA	Min power losses
[26]	active	x			MINLP	Min power losses
[33]	active		x		Markov	Min cost energy
[34]	active	x			MOPF	Max net profit
[35]	active	x			SMILP	Min cost operation
[36]	active	x	x		GA	Min cost operation
[37]	active	x			MOPF	Min battery degradation
[38]	active	x			Fuzzy	Min fluctuation of PV
[39]	active		x		HOA	Min power losses
[40]	active	x	x		MOPF	Min cost operation
[41]	active/reactive		x		OPF	Min cost energy
[42]	active	x	x		PSO	Min cost energy
[43]	active	x			OPF	Min cost energy
[44]	active/reactive			x	MOPF	Min cost generation
[45]	active	x			MOPF	Min power losses
[46]	active	x			JA	Min cost energy
Authors	active/reactive	x	x	x	MOPF	Min power losses

presents a summary of several research works dealing with the optimal operation of storage systems and several types of DG in distribution networks. In general, every approach presents advantages and disadvantages. For example, metaheuristics may be trapped in locally optimal solutions, especially in large-scale applications [31]. These techniques are easy to implement and may work with discrete and continuous variables; however, they require the fine-tuning of different parameters. Moreover, due to the possibility of finding sub-optimal solutions, it is recommended to run the algorithm several times to extract good quality solutions [32].

According to [47], the use of an Optimal Power Flow (OPF) approach is effective in electrical grids since it facilitates the control of variables and adds operational constraints that allow for representation of a more realistic operation. Nevertheless, as reported by [48], a conventional single-period OPF approach is unsuited for today’s requirements, principally by virtue of the advent of BESSs that impose inter-temporal constraints; therefore, a MOPF is a more suitable option. In this line, the authors in [44] proposed a MOPF considering the operation limits of BESS in its formulation and providing a more realistic modelling of modern distribution networks. Furthermore, the massive penetration of alternative sources as well as BESS, added with typical network constraints, require powerful new analysis tools that consider the dynamic nature of the system [45].

Although a lot of studies have been carried out to explore the optimal operation of storage systems in combination with RES, the research on this issue is far from being exhaustive, since only a few studies aim at minimizing total active power losses within a 24-h horizon. Therefore, a knowledge gap is identified: as evidenced in Table 1, there are no studies that combine PV, WPG, and CG with BESSs providing active and reactive power. Thus, the following research question is proposed: “Is it possible to optimize the operation of BESS providing both active and reactive power with renewable and non-renewable generation within a MOPF framework?”

1.3. Article Contributions and Organization

Taking into account the knowledge gap identified in the literature review, this paper proposes a MOPF approach for the optimal operation of BESS in distribution systems integrated with PV, CG, and WPG. The objective function consists of minimizing power losses while improving the voltage profile within a 24-h time horizon; furthermore, reactive power compensation is provided by BESS. The problem formulation was developed in AMPL [49] and solved through the Knitro solver [50]. To sum up, the main features and contributions of this paper are provided below.

• Apart from conventional generation, the proposed approach also considers the interaction of PV and WPG through their typical production curves.
• BESS is modelled to provide both active and reactive power depending on grid requirements.
• The objective function is formulated to reduce active power losses in a 24-h time horizon considering the constraints of the systems as well as the charge/discharge capacities of BESSs.
• The proposed MOPF was developed in commercially available software (AMPL) and can be applied in real-size systems. Because it is a deterministic methodology with great solution capacity, it is suitable for analysis of real networks.

The organization of this paper is as follows. Section 2 provides a brief introduction to storage systems along with PV and WPG. Section 3 describes the mathematical model formulation based on MINLP. Section 4 describes the methodology implemented to solve the proposed model. Tests and results are presented and discussed in Section 5. Finally, the concluding statements are given in Section 6.

2. Battery Energy Storage Systems, Photovoltaic and Wind Power Generation

As mentioned previously, storage devices have become a well-known topic in distribution grids. They can help reduce greenhouse emissions and alleviate congestion in peak hours [4]. According to [51], diverse types of BESSs can be found in the industry, each with their own established purposes highlighting the lithium-ion (Li-ion) [52] and vanadium redox batteries.

Usually, the configuration of the BESSs is used to provide active power in the electric system [7]; however, they can also be used to deliver reactive power [53]. In [54], a real application is presented in a microgrid to verify the performance of BESSs providing active and reactive power. The results indicate a better performance when both operation modes are implemented. The study conducted by [55] considers smart inverter functions and takes into account the real operation of an electrical system, determining reactive power injection or absorption from the grid. In this sense, this strategy provided reduced losses, alleviated the power curtailment, and improved home profits. In [56], the smart inverter controller is recommended to deal with short-term transients. Most BESSs are adjusted to operate based on the voltage/Var function [25], which means that they provide a certain amount of reactive power depending on voltage limits. In these cases, the smart inverter does not inject any reactive power if the voltage limits stay between 0.95 to 1.05 p.u. If the voltage is below 0.95 p.u., the inverter operates in capacitive mode; otherwise, when the voltage is above 1.05 p.u., it operates in inductive mode. In this work, the smart inverter is adjusted to control the active and reactive power dispatch, considering its limitations, as illustrated in Figure 1. When the inverter operates with a Power Factor (PF) other than one, a part of the active power is cut to supply reactive power. The control is carried out aiming to optimize the operation of the grid.

In [25], the BESS is applied to reduce total active power losses for the best operation of the electrical system. When BESS is charging, the inverter works with a PF equal to 1. When BESS supplies energy to the grid, the power factor can vary within specified limits, allowing both active and reactive power to be injected to the grid.

Photovoltaic energy plays an important role in climate change combat. It can be defined as a distributed energy source that can be used for self-consumption in distribution systems [57] or in large-scale power plants. Besides, PV can be used to improve grid security, reduce energy losses, and favor the local economy. According to [58], PV generation is the most used RES, since solar radiation around the world is abundant. Equally important has been the constant reduction of costs of photovoltaic modules and other required components as pointed out in [59]. Normally, the schematic configuration of PV systems is connected on grid [60] or off grid [61]. The expansion of PV in distribution grids brings operational challenges for management due to uncertain generation, requiring coordination with other devices such as BESSs [5]. Usually, the output power generation of PV is related to solar irradiance [6] and temperature. In this paper, a curve of irradiance of a typical sunny day in Brazil is considered, and temperature variance is not integrated in the model.

WPG is also viewed as friendly to the environment and as a resource that can be used for reducing gas emissions [62]. Similarly, it can be installed in electrical grids as in remote places. In [63], a sensitivity analysis is carried out to evaluate the power system behavior when WPG is inserted into the power grid. Several tests in distribution systems of 34, 70, and 126 buses showed that power losses are reduced and voltage profiles are improved when wind turbines are inserted in the studied systems.

The energy produced by WPG depends on the speed of the wind. Furthermore, random and rapid changes in this resource may cause several technical issues regarding protection, power quality, and generation dispatch control. In [64], the authors classify and analyze several intermittent smoothing approaches used to minimize power fluctuations caused by wind energy. It was found that BESSs are the most effective smoothing approach.

As indicated by [65], WPG is a promising energy source due to its low environmental impact and the reduction of installation costs in the last decade. WPG is considered a variable power source by virtue of wind variation. In accordance with [66], these variations can produce short-term fluctuation, as well as voltage and frequency variations. In this context, from the operation point of view, strategies are needed to support manageability facing the intermittency inherent in WPG; in this sense, storage systems represent an applicable option [67]. CG can be considered resources such as diesel generation, coal, hydro power plants, and gas turbine generators [68]. This type of generation does not depend on the variability of wind or irradiation and provides firm energy to the system.

3. Problem Formulation

The proposed approach for optimal coordination of BESSs with RES and CG uses the forecast of load and generation for the next day as input data. If there is a divergence in forecasts, the quality of the result may be compromised. These uncertainties can be treated by probabilistic methods, which is out of the scope of this research.

The proposed MOPF is a non-linear, non-convex, large-scale problem involving discrete and integer variables. These characteristics make it a challenging problem to solve by conventional iterative methods [69]. Equation (1) represents the objective function that corresponds to the minimization of energy losses in a 24-h period denoted as T

$$minf\left(x\right)=\sum _{t=1}^T\sum _{i=1}^{N_L}{R}_i{I}_i^2$$

(1)

where:

$f\left(x\right)$—Total active power losses (W);

$N_L$—Number of lines of the system;

$R_i$—Branch resistance i;

$I_i$—Electric current circulating in branch i;

T—Period composed of hours t.

The objective function given by Equation (1) is subject to the following constraints.

3.1. Load Flow Constraints

Equations (2) and (3) represent the balance of active and reactive power. Note that generation sources are composed of CG, PV, and WPG. Except for the substation, the generators supply only active power; nonetheless, BESSs can supply both active and reactive power to the grid.

$$P_k^t+{P_{CG}}_k^t+{P_{PV}}_k^t+{P_{WPG}}_k^t+{P_B}_k^t-{P_L}_k^t=V_k^t\sum _{m\in {\mathsf{\Omega }}_{\kappa }}{V}_m^t(G_{km}cos{\theta }_{km}^t+B_{km}sin{\theta }_{km}^t)$$

(2)

$$Q_k^t+{Q_B}_k^t-{Q_L}_k^t=V_k^t\sum _{m\in {\mathsf{\Omega }}_{\kappa }}{V}_m^t(G_{km}sin{\theta }_{km}^t-B_{km}cos{\theta }_{km}^t)$$

(3)

where:

$P_k^t$—Active power generated by substation k in time t;

$Q_k^t$ — Reactive power generated by substation k in time t;

${P_{CG}}_k^t$—Active power generated by CG at bus k in time t;

${P_{PV}}_k^t$—Active power generated by PV generators at bus k in time t;

${P_{WPG}}_k^t$—Active power generated by WPG at bus k in time t;

${P_B}_k^t$—Active power supplied or absorbed by BESS at bus k in time t;

${Q_B}_k^t$—Reactive power supplied or absorbed by BESS at bus k in time t;

$P_{Lk}^t$—Active power load connected to the bus k in time t;

$Q_k^t$—Reactive power generated by conventional generators at bus k in time t;

$Q_{Lk}^t$—Reactive power load connected to the bus k in time t;

$G_{km}$—Real part of the admittance matrix for the branch $k-m$;

$B_{km}$—Imaginary part of the admittance matrix for the branch $k-m$;

$V_k^t$—Voltage on the bus k at time t;

$V_m^t$—Voltage on the bus m at time t;

${\mathsf{\Omega }}_{\kappa }$—Set of all buses connected to the bus k.

3.2. Charge and Discharge Operation of BESS

The BESS model describes the charging and discharging of the battery. In this case, a linear model is used, where the charging and discharging efficiency values are considered as presented in [20], which is adjusted through the parameters $n_c$ and $n_d$, respectively. The BESS model can be expressed by Equation (4)

$${B_S}_k^{t+1}=\left\{\begin{array}{c}{B_S}_k^t+n_c{P_B}_k^t\mathrm{\Delta }t\forall {P_B}_k^t\ge 0\hfill \\ {B_S}_k^t-\frac{1}{nd}{P_B}_k^t\mathrm{\Delta }t\forall {P_B}_k^t<0\hfill \end{array}\right.$$

(4)

where:

${B_S}_k^t$—Energy stored in the BESS at bus k in time t;

$n_c$; $nd$—The charging and discharging efficiencies, respectively.

3.3. Charge and Discharge Capacities of BESS

The capacity of BESSs to supply or absorb energy is given by Equation (5), which is related to the technology of the battery.

$${P_B}_k^{min}\le {P_B}_k\le {P_B}_k^{max}$$

(5)

where:

${P_B}_k^{min}$—Minimum value of active power injected into BESS at bus k;

${P_B}_k^{max}$—Maximum value of active power injected into BESS at bus k.

3.4. Storage BESS System

The BESS system has a storage capacity that must be considered throughout the operation period and is given by Equation (6).

$${B_S}_k^{min}\le {B_S}_k\le {B_S}_k^{max}$$

(6)

where:

${B_S}_k^{min}$—Minimum BESS storage capacity at bus k;

${B_S}_k^{max}$—Maximum BESS storage capacity at bus k;

3.5. Voltage Constraints

Equation (7) represents the minimum and maximum limits imposed on the voltage profile.

$$V_k^{min}\le V_k^t\le V_k^{max}$$

(7)

where:

$V_k^{min}$—Minimum voltage limit at bus k;

$V_k^{max}$—Maximum voltage limit at bus k;

3.6. PV Generation Limits

Equation (8) represents the minimum and maximum limits imposed by PV generation.

$${P_{PV}}_k^{min}\le {P_{PV}}_k^t\le {P_{PV}}_k^{max}$$

(8)

${P_{PV}}_k^{min}$—Minimum value of active power injected by PV at bus k;

${P_{PV}}_k^{max}$—Maximum value of active power injected by PV at bus k;

3.7. WPG Generation Limits

Equation (9) represents the minimum and maximum limits imposed by WPG generation.

$${P_{WPG}}_k^{min}\le {P_{WPG}}_k^t\le {P_{WPG}}_k^{max}$$

(9)

${P_{WPG}}_k^{min}$—Minimum value of active power injected by CG at bus k;

${P_{WPG}}_k^{max}$—Maximum value of active power injected by CG at bus k;

3.8. CG Limits

Equation (10) represents the minimum and maximum limits imposed by CG generation.

$${P_{CG}}_k^{min}\le {P_{CG}}_{k}t\le {P_{CG}}_k^{max}$$

(10)

${P_{CG}}_k^{min}$—Minimum value of active power injected by WPG at bus k;

${P_{CG}}_k^{max}$—Maximum value of active power injected by WPG at bus k;

In this work, the generations CG, PV, and WPG followed the curves presented in the respective case studies.

4. Solution Approach

An important element of renewable sources and the proliferation of devices such as BESSs bring several operational challenges to modern distribution grids. In most studies, BESSs are predominantly used for ancillary services [53] or integrated with DG [6,20]. Only a few studies recognize the minimization of power losses by means of integrating BESSs with other generation resources [70].

Hence, this paper investigates the operation of CG, PVs, WPG, and BESSs within a 24-h period, specifically when the BESSs are capable of providing both active and reactive power in presence of renewable sources. The proposed model was developed in AMPL. This software is a modelling language for mathematical programming and was designed by [49].

AMPL is an effective and robust tool for solving large-scale mathematical programming problems that works with continuous and discrete variables [71]. The model was formulated considering continuous values of the variables, which means that they can take any value within the ranges defined by their limits. The solver used was Knitro. This solver features three algorithms: interior point algorithm with conjugate gradient, direct interior point algorithm, and active set method, offering different programming strategies options and also allowing the possibility of interaction of these methods during the solution (crossover) [50].

Several studies use AMPL to solve optimization problems considering storage systems. In [72], the authors presented a new method set up on cost–benefit for optimally sizing BESS in microgrids in the presence of PV and WPG. The formulation of the problem was modeled in AMPL and solved by CPLEX solver. In [73], the authors proposed a Mixed Integer Linear Programming (MILP) approach considering RES, BESS, and Electric Vehicle (EV) charging stations using AMPL and solved by CPLEX. Tests were carried out on a 24-bus test system. The paper proposed by [74] shows the coordinated management of RES, BESS, and EVs modeled using AMPL and solved with MINOS solver.

It was found that the application of commercial solvers, such as Knitro, to approach the OPF problem is still little explored. The great advantage lies in being a reliable tool with several solution strategies and the ability to overcome numerical difficulties during the convergence process.

5. Tests and Results

This section presents the results of the proposed approach considering two test systems of 33 and 141 buses. All simulations were performed on an Intel^® Core™ i7 CPU @ 1.8 GHz computer with 4 GB of RAM and Windows 10 Home 64 bits.

5.1. Results with the 33-Bus Test System

The 33-bus test system used in this paper is depicted in Figure 2. The operating voltage of the system is 12.66 kV. Voltage limits are considered from 0.90 to 1.05 p.u. in all buses. The network was modified with the addition of three BESSs and seven PVs. Aiming to evaluate the adjustment of the PF of the inverters, the reactive loads were limited to 60%. In accordance with information provided in [75], the BESSs are considered to have a loading and unloading capacity of 1 MWh per hour and storage of 5 MW. The loading and unloading efficiency of BESS is supposed to be 100%.

The optimal operation of BESS is considered within a 24-h time horizon at intervals of 1 h. A typical photovoltaic generation curve $\left(P_{PV}\right)$ and a typical residential load curve $\left(P_{load}\right)$ were adopted in the analyses. Figure 3 shows the total $P_{PV}$ generation and active load in the system as well as the net active power expressed as $\left(P_{load}\right)-\left(P_{PV}\right)$. Note that the injection of active power by the PV generation is greater than the load at noon, resulting in a negative net power from 11:00 to 13:00, which may cause reverse power flows.

Seven test cases were carried out considering the operational modes of BESS (providing only active power or active and reactive power combined). In the base case, the grid has no distributed energy resources. In the other cases, in addition to the PV generation, the insertion of 1, 2, and 3 BESSs was considered. When BESS provides only active power, it is identified as $BES{S}_P$, and if it provides both active and reactive power, it is identified as $BES{S}_{PQ}$. The 7 cases are summarized in

Table 2. Cases analyzed in the 33-bus test system.

Cases	$\mathit{PV}$	$\mathit{BESS}{1}_{\mathit{P}}$	$\mathit{BESS}{1}_{\mathit{PQ}}$	$\mathit{BESS}{2}_{\mathit{P}}$	$\mathit{BESS}{2}_{\mathit{PQ}}$	$\mathit{BESS}{3}_{\mathit{P}}$	$\mathit{BESS}{3}_{\mathit{PQ}}$
Base case
Case a	X
Case b	X	X
Case c	X		X
Case d	X	X		X
Case e	X		X		X
Case f	X	X		X		X
Case g	X		X		X		X

.

5.1.1. Analysis of BESS Operation in the 33-Bus Test System

Figure 4 depicts the total active power losses for the 7 cases under study. Note that power losses for the Base Case (without DG and BESSs) are 4.1139 MWh. Nonetheless, Case a, which only considers PV, has 3.0282 MWh of power losses. This represents a reduction of 26.3% compared to the Base Case, which shows that PV alone is able to provide important savings on power losses.

Case b, which considers a single BESS providing active power (plus PV generation) presents power losses of 2.3466 MWh, which represents a further reduction of 21.7% with respect to Case a, and a total power loss reduction of 42.9% with respect to the Base Case. In Case c, when active and reactive power are provided by a single BESS unit, power losses are further reduced by 26.0% with respect to Case a and 46.0% with respect to the Base Case. This shows that greater power loss reductions are achieved when BESSs provide both active and reactive power. This is observed as well for the rest of the cases; every time the BESS units were allowed to provide active and reactive power, the impact on power loss reduction was greater. Note that the smallest amount of power loss is achieved in Case g when three BESSs provide active and reactive power. In this case, total power losses are 1.7083 MWh, representing a reduction of 58.4% with respect to the Base Case and 43.5% with respect to Case a when only PV generation is considered. The results obtained in this system show that, although PV generation allows for reducing power losses, its combination and proper coordination with BESSs results in greater power losses along with other positive impacts that are presented and discussed in the next subsections.

5.1.2. Analysis of Voltage Profile

Figure 5 shows the voltage profile for the seven cases under study. For the sake of simplicity, the displayed value is the average of the voltage magnitudes at all buses. In the Base Case, the voltage profile is barely over the minimum allowed, especially in peak hours; nonetheless, when PV generation is introduced (Case a), the voltage profile significantly improves. This happens basically from 8:00 to 16:00, which is when PV generation is providing energy to the network; nonetheless, in peak hours, there is still a significant drop in the voltage profile since there is no coincidence with PV generation. In this sense, it can be seen that when the participation of BESS increases, the voltage profile flattens; that is to say, voltage magnitudes tend to reduce around noon and increase their magnitude at night, thus avoiding voltage peaks. This is due to the fact that BESSs charge when PV presents its generation peak (which surpasses the total demand) and discharges when most needed (at load pick hours when there is no PV generation).

5.1.3. Analysis of the Slack Bus Generation

Figure 6 shows the slack bus generation (the substation bus) for the time horizon under study. Note that in Case a there is a negative net power injection from 11:00 to 13:00. This is due to the fact that total PV generation is greater than the system demand at these hours, which leads to the presence of reverse power flows in the distribution system. The main issue of concern lies in the fact that reverse power flows may require a protection coordination re-adjustment as pointed out in [76,77]. This undesired situation is avoided when BESSs are properly integrated and coordinated with PV generation through the proposed MOPF approach. Also note that in Case f and Case g, when a maximum of three BESSs are integrated in the distribution system, the power supplied by the substation diminishes its variability over time.

5.1.4. Energy Management Analysis of BESSs

Figure 7 depicts the energy stored by BESS 1 in Case c for active power management as well as active and reactive power management. As can be seen in Figure 7a, from 7:00 to 17:00, the energy stored (labelled as ESB1p) increases progressively from 0 to 5 MW every hour. Afterwards, the energy stored steadily decreases to 0 MW in hour 23:00. Note that the BESS unit begins to supply active power (labelled as PB1) in hour 8:00 until hour 15:00; then it takes power from the grid from hour 18:00 (peak hour) to hour 23:00. In Figure 7b, it can be seen that the BESS supplies both active and reactive power in different periods of the day. From hour 0:00 to 8:00, the BESS unit barely provides any power and stores a small amount of energy; then from hour 8:00 to 18:00, the energy stored (labelled as ESB1pq) increases steadily to then decrease and reach 0 MW at hour 23:00. Note also that the BESS takes both active and reactive power from the network from hour 18:00 to hour 23:00.

Figure 8 illustrates the energy management when two BESS units are considered in the system and provide either active or both active and reactive power. When only active power is considered as shown in Figure 8a,c, BESS 1 stores energy to full capacity while BESS 2 does not; furthermore; the active power injection follows the same pattern for both BESS units form hour 8:00 to hour 23:00. When both BESS units manage active and reactive power, their patterns are slightly different, with BESS 2 being the unit with lower charging and discharging.

Figure 9 depicts the energy management when three BESS units are considered. It can be seen that BESS 1 and BESS 3 store more energy than BESS 2 and follow a quite similar pattern. The difference in energy management of these units with respect to BESS 2 can be attributed to their location in the network; it is also worth mentioning that introducing three BESS units resulted in the highest reduction of power losses as already indicated.

Table 3. Adjustment of power factor according to number of BESSs.

Hours	(1) ${\mathit{BESS}}_{\mathit{PQ}}$	(2) ${\mathit{BESS}}_{\mathit{PQ}}$	(3) ${\mathit{BESS}}_{\mathit{PQ}}$
0:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
1:00	1.0	1.0; 1.0	1.0; 1.0; 0.95
2:00	0.93	0.93; 0.95	0.94; 0.95; 1.0
3:00	1.0	1.0; 1.0	1.0; 0.95; 1.0
4:00	1.0	1.0; 1.0	0.96; 1.0; 1.0
5:00	1.0	1.0; 1.0	1.0; 1.0; 0.92
6:00	0.90	0.90; 0.93	0.90; 0.93; 0.90
7:00	0.90	0.90; 1.0	0.90; 1.0; 1.0
8:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
9:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
10:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
11:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
12:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
13:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
14:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
15:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
16:00	1.0	1.0; 1.0	1.0; 1.0; 1.0
17:00	1.0	1.0; 1.0	0.90; 0.93; 0.90
18:00	0.93	0.93; 0.93	0.93; 0.95; 0.90
19:00	0.93	0.93; 0.95	0.94; 0.95; 0.90
20:00	0.93	0.93; 0.95	0.94; 0.95; 0.90
21:00	0.93	0.93; 0.95	0.94; 0.95; 0.90
22:00	0.93	0.93; 0.95	0.94; 0.95; 0.90
23:00	0.93	0.93; 0.95	0.93; 0.95; 0.90

shows the adjustment of PF at each hour of the day, taking into account the number of BESS units considered in the system. Note that at peak demand, the power factor is different from the unity, which means that the BESS manages reactive power. When the BESSs are charging, the power factor is 1.0.

5.2. Results with the 141-Bus Test System

To show the applicability of the proposed approach, several tests were also carried out with a real test system of 141 buses. This network, depicted in Figure 10, is located in the metropolitan region of Caracas in Venezuela [78]. It presents a nominal voltage of 12.47 kV and a load of 11.9029 MW and 7.375 MVAr. On average, the loads present a lagging power factor of 0.85. In this case PV, WPG, and CG were integrated in the network to supply active power to the load. For the sake of simplicity, it is assumed that all generators have a capacity of 1 MW and are located as indicated below:

• 5 PV units at buses 50, 60, 70, 80, and 90;
• 2 WPG units at buses 106 and 118;
• 1 CG unit at bus 136.

Furthermore, 5 BESS units are considered at buses 10, 20, 45, 55, and 65. The storage capacity of each BESS unit is assumed to be 5 MW with a maximum charge/discharge power of 1 MWh. The sizing and location of these units, as with the previous test system, were chosen empirically. For the 141-bus test system, 4 cases were analysed:

• Base case without DGs (all power is supplied through the substation);
• Considering all types of DGs: PV, WPG, and CG;
• Considering all DGs and BESS units only supplying active power;
• Considering all DGs and BESS units supplying active and reactive power.

Figure 11 shows the load curve of the system (labelled as $P_{Load}$). Note that the maximum load is presented at 21:00. The generation profiles of PV, WPG, CG and all units combined are also depicted.

5.2.1. Analysis of the BESS Operation

Figure 12 illustrates the total energy losses for the cases under study. Note that the Base Case presents 16.5347 MWh of power losses. However, when DGs are included in the system (the combination of PV, WPG, and CG) in the locations and quantities previously described, the losses drop to 6.1720 MWh, which represents a reduction of 62% with respect to the Base Case. Initially, the DGs in the system are not coordinated with BESS; nonetheless the amount of losses is further reduced another 12% when BESSs are integrated into the systems providing only active power. Finally, when BESSs provide both active and reactive power, the power losses are 3.7928 MWh, which represents a reduction of 77% with respect to the base case. This shows the positive impact of the optimal coordination of resources carried out by the proposed MOPF.

5.2.2. Voltage Profile Analysis

Figure 13 illustrates the voltage profile of the 141-bus test system for all the cases under study. The average voltage magnitudes of all buses are indicated for each hour. Note that in the Base Case, several buses present voltages below 0.95 p.u., especially from hour 18:00 to 23:00, which indicates the peak load of the grid. When DGs are introduced into the system, the voltage profile presents a significant improvement, while keeping essentially the same pattern. Nonetheless, when BESSs are also integrated and properly coordinated with the DG units, the voltage profile improves and gets flatter. It was also observed that the best voltage profile was achieved when BESS units provide both active and reactive power.

5.2.3. Analysis of Slack Bus Generation

Figure 14 illustrates the power provided by the substation at every hour of the time horizon. Note that when only DGs are introduced, there are reverse power flows in hours 4:00 and 5:00. Nonetheless, this is corrected as BESS units are introduced into the system. Also note that the energy provided through the substation remains almost constant when BESSs are coordinated with the DGs.

5.2.4. Energy Management of BESSs

Figure 15 illustrates the energy management of the 5 BESS units allocated in the system when they provide both only active power as well as active and reactive power. Note that BESS 1 (Figure 15a) stores energy from 1:00 to 8:00, reaching its full capacity to later provide energy until hour 14:00, when it begins to charge again until hour 18:00; from this hour on (peak load), BESS 1 provides active power to the network until hour 23:00. The other BESS units present a similar pattern to match the intermittency characteristics of PV and WPG (see left column of Figure 15). When BESS units are capable of handling both active and reactive power, the charging and discharging pattern is more dynamic, as can be seen in the right column of Figure 15. In this case, the greatest power loss reduction and flattest voltage profile are obtained; furthermore, there are no reverse power flows. Figure 16 illustrates the sum of all power delivered to the network by the 5 BESS units in both operational modes. Note that the pattern of each mode remains similar to those of the individual cases, presenting a more dynamic operation when both active and reactive power are delivered to the network. In Figure 16a, it can be seen that the BESS units store energy from around 1:00 to 8:00; then they provide some energy to the network from 9:00 to 14:00 to later recharge from 15:00 to 17:00 and finally deliver all the energy stored during the peak hours. A similar pattern is seen in Figure 16b, with a more intense discharge from hours 9:00 to 14:00.

5.2.5. Methodology Analysis

The tests carried out showed that the methodology can be used for the management of BESSs in networks with conventional and intermittent DGs, such as solar and wind power. The MOPF, in all tests performed, responded satisfactorily, even with several BESSs and DGs. It was evident that, when adding one more BESS to the network, the operation of the grid is changed. The possibility of the BESSs supplying reactive power has added another possibility of control to the network, helping in the reduction of power losses and control of voltage levels.

The solver Knitro uses a precision of 1.0 × 10${}^{-8}$ in the iterative convergence process. Unlike metaheuristic algorithms, the proposed approach is deterministic, always guaranteeing the same solution to the problem. The convergence process can be changed if some internal parameter of Knitro is modified. However, seeking to generalize the application, we leave all the parameter adjustments automatically, performed by the solver, which makes the adjustments consider the type of problem, number of variables, and constraints.

Figure 17 presents an analysis of the convergence process in p.u. (base of 10 MVA) for two cases: (a) with DGs and BESSs providing active power; and (b) with DGs and BESSs providing both active and reactive power. For case “a”, the number of iterations was 866 and the algorithm spent 51.18 s. For case “b”, the algorithm converged in 1038 iterations in 67.35 s. In both cases, the number of variables and constraints exceeded 7000. Due to the deterministic nature of the model, even with many variables and constraints, the results obtained can be considered of good quality, since the MOPF guarantees the enforcement of all network constraints and the accuracy of results.

6. Conclusions

This paper presented a model and implementation of a MOPF for the optimal management of BESSs in distribution networks that also integrate CG, PV, and WPG. The proposed model was solved in AMPL through the commercial solver Knitro. Several tests were carried out in a distribution network of 33 buses and a real system comprising 141 buses. The benefits of integrating BESSs were analysed, considering two operational modes: when managing only active power and when managing active and reactive power.

The first analyses were carried out considering 7 PV generations plus up to 3 BESS units within a 24-h horizon. When only PV is considered in the 33-bus test system, a power loss reduction of 26.3% is achieved with respect to the base case. When a single BESS provides active power, such losses further decrease by 21.7%. When 3 BESSs are considered in the system providing active and reactive power, power losses reduce to 58.4%. As regards voltage magnitudes, these are improved due to the presence of BESSs. Furthermore, the voltage profile is flattened, avoiding voltage peaks.

In the second test system, the optimal management of BESSs was explored considering five BESSs units in the presence of CG, PV, and WPG generation. This resulted in a reduction of energy losses of up to 77% (when all generation units are used and BESSs deliver both active and reactive power) followed by an important flattening and improvement of the voltage profile. It was also shown in both test systems that the proper management of BESSs prevents reverse power flows in distribution networks that otherwise appear at noon due to the presence of PV generation. Furthermore, the solution of the MOPF problem demanded low computation time (in the range of a few seconds).

The main limitation of the proposed model lies in the dependence of a proper prediction of generation through PV and WPG. For further increasing the integration of RES with BESS in modern power grids, it is recommended to adopt new strategies to properly size and locate renewable resources, paying special attention to the emergence of reverse power flows that may cause technical problems. This can be carried out through proper RES–BESS coordination strategies. It is also important to choose the correct placement of storage systems, which can be considered in future studies. Finally, other objectives and constraints related to the use of BESSs in distribution networks such as arbitration and frequency control can be modeled.