Optimal Operation of Isolated Microgrids Considering Frequency Constraints

Abstract

Isolated microgrids must be able to perform autonomous operation without external grid support. This leads to a challenge when non-dispatchable generators are installed because power imbalances can produce frequency excursions compromising the system operation. This paper addresses the optimal operation of PV–battery–diesel-based microgrids taking into account the frequency constraints. Particularly, a new stochastic optimization method to maximize the PV generation while ensuring the grid frequency limits is proposed. The optimization problem was formulated including a minimum frequency constraint, which was obtained from a dynamic study considering maximum load and photovoltaic power variations. Once the optimization problem was formulated, three complete days were simulated to verify the proper behavior. Finally, the system was validated in a laboratory-scaled microgrid.

1. Introduction

The integration of distributed generation requires the development of new concepts for active grid operation, where microgrids are the most promising one [1]. Microgrids can operate in grid connected as well as isolated modes [2,3]. In isolated mode, the active power balance to maintain the grid frequency has become one of the main challenges. The integration of large amount of photovoltaic (PV) generation can further stress the power balance due to the lack of inertia and the fast power variations of the resource. One possible solution to avoid frequency deviations produced by PV power generation is its curtailment [4]. Frequency deviations can also be limited by increasing the grid inertia, which can be achieved by connecting rotating machines [5]. The main drawback is that these solutions have and adverse effect on the operation cost.

To solve the power balance problems while minimizing the operation cost, a hierarchical control architecture is commonly used [6,7,8,9]. The primary control layer stabilizes the voltage and frequency deviations due to power imbalances by adjusting the active and reactive power references in a time frame of milliseconds. Then, the secondary control is responsible for recovering the voltage and frequency to their reference values. Commonly, it is done by using PI based closed loop controllers in a slower time scale than the primary control response time. Finally, The tertiary control determines the power references to perform the optimal operation of the microgrid.

1.1. Literature Review

Different methods have been considered for designing energy management systems (EMS), i.e., the tertiary control layer, for microgrids. These methods mainly consist of: (i) formulating an objective function; (ii) defining a set of constraints to ensure the proper system behavior; and (iii) applying an algorithm to find the optimal solution.

In [10], a mixed integer linear program (MILP) is formulated to minimize the microgrid operation cost. The microgrid includes critical and controllable loads, energy storage, controllable generation and renewable generation. Because the system under study is connected to the utility grid, any power imbalance is considered to be compensated by the external network producing a very small frequency deviation. Accordingly, power reserves are not considered. Even though the problem formulation does not consider forecast errors, its periodical execution, similar to the rolling horizon process, permits redefining periodically the operation plan compensating unpredicted deviations.

The works presented in [11,12,13] prove the real implementation of different EMSs for the minimum price or minimum cost of the isolated microgrid operation. These papers solve the optimization problems using MILP, multi-layer ant colony optimization and multi-period gravitational search algorithms, respectively. These studies consider perfect forecast. Thus, the hierarchical control structure is not implemented and power reserves are not considered. Consequently, power imbalances and frequency deviations are not studied. Thus, the grid stability cannot be ensured.

The study performed in [14] proposes a heuristic method, based on genetic algorithms, for solving the cost minimization problem for the microgrid operation. It first develops a forecasting method and then formulates the problem and the generic algorithm. The problem formulation differs depending on whether the microgrid operates connected or disconected form the main grid, considering load and generation forecast for the power balance equations. As power reserves are considered, the power imbalances due to forecast errors may be compensated, but this may lead to a suboptimal operation point of the microgrid. In addition, the transient response when imbalances due to forecast errors occur is not analyzed.

To avoid operating in a suboptimal operation point in microgrids due to forecast errors, different studies propose the formulation of stochastic optimization problems [15,16,17]. In this method, a set of forecasted scenarios is generated. Then, the decision variables are optimized for all scenarios, where the objective function is the sum of the objective function of each scenario.

In [18], an EMS for minimizing the use of diesel generation in a PV–wind–diesel–battery-based isolated microgrid is developed. The optimization problem is formulated as a MILP and executed using the rolling horizon technique to reduce the effects of the uncertainties of forecasted variables. In addition, the primary control layer (particularly the droop curves) vary depending if diesel generation is turned on or off. This fact can affect the transient performance, but a transient study is not performed.

The authors of [11,12,13] did not consider forecast errors. This issue is solved in [14,18] by considering power reserves. To improve the average optimal operation point against the uncertainty, the authors of [15,16,17] proposed a stochastic optimization method. These previous studies do not analyze dynamic and transient behavior. This gap is treated in [19]. This study develops a multi agent EMS for an isolated microgrid. One of the particularities and not studied in the previous cited papers, is that the transient response considering the primary and secondary control layers is analyzed. The tertiary control layer (EMS), which is the objective of the study, determines not only the scheduled setpoints but also the required reserves to compensate photovoltaic and load forecasting errors, avoiding frequency deviations. These frequency deviations are analyzed later in a real time dynamic simulator platform.

The local controls of generation units will react to frequency deviations to achieve a power balance and to maintain the grid frequency. In [20], the system frequency is introduced into the optimization problem. Particularly, the frequency-power (f-P) droop control is considered and a the maximum frequency deviation is constrained. These constraints apply for the steady state, but they do not consider the transient behavior. An optimal power flow (OPF) problem that includes the frequency transient behavior is presented in [21], explaining the need to limit its deviations. However, the main assumption is that the frequency decreases linearly during the first few seconds until reaching the steady state. The typical frequency transient behavior usually presents an overshoot, as shown in [22]. Hence, the maximum frequency deviation during the transient may be greater than the deviation in the steady state. This effect is not considered in [21].

1.2. Required Improvements in the EMS Development for Isolated Microgrids

As shown above, EMSs for isolated microgrids are commonly designed without analyzing their dynamic behavior. The primary and secondary control layers are responsible for stabilizing the microgrid after disturbance, but the EMS must consider their necessities to perform the operation properly. This issue has been previously solved by incorporating power reserves constraints in the optimization problems of the EMSs [14,19]. Nevertheless, few little dynamic considerations have been performed when designing EMSs. In addition to the power up/down regulation capacity, there are dynamic aspects that should be considered by the EMSs, which are not studied yet.

Utility grids are usually characterized by incorporating many rotating machines and, consequently, have large inertia. During power imbalances, and until the primary and secondary controls react, the required energy is obtained from the rotating machines leading to frequency variations. Due to the big inertia, these frequency variations are usually small. Accordingly, in grid connected microgrids, it can be assumed these deviations are not relevant [10]. In contrast, grid isolated microgrids present low inertia, and even lower when large amount of photovoltaic power is installed. Accordingly, these assumptions can no longer be accepted. Power reserves will determine whether the inner control loops can compensate the microgrid imbalances. However, due to the low inertia, the transient frequency deviations can reach unacceptable levels collapsing the system. Even though the study performed in [19] considers the up/down regulation and analyzes the dynamic response, a required inertia to ensure the frequency does not exceed the acceptable limit is not studied. Hence, in this case, the EMS developed in [19] disconnects to match the rotating machines, and the system stability could be compromised. Similarly, if frequency transients present overshoots, the stability of the system is not ensured by the proposed methods in [20,21].

According to the above issues, for designing a reliable EMS, it is still necessary to incorporate dynamic constraints into the problem formulation. Particularly, in addition to the power reserves, the minimum grid inertia to ensure an stable operation should be considered on the tertiary control layer of isolated microgrids.

1.3. Paper Contributions

This paper focuses on the above-mentioned issue. In particular, an EMS for ensuring that transient frequency deviations do not exceed a defined limit is developed. Accordingly, the main contribution of this paper are:

• The parameters that, being available by the EMS, may influence the frequency deviations are analyzed.
• The the maximum frequency deviation in front of the maximum power imbalance is formulated. This formulation uses the above-mentioned parameters.
• An EMS including a frequency constraint is formulated.
• The validation of the proposed EMS using dynamic simulation and laboratory platform is presented.

Particularly, this paper proposes a power dispatch optimization algorithm for PV–battery–diesel-based microgrids including demand and PV forecasting. To deal with uncertainty, the problem is based on stochastic optimization and computed on-line, in a similar way to the rolling horizon technique. The algorithm, which maximizes the PV generation, considers a frequency variation constraint obtained by analyzing multiple off-line dynamic simulations and performing a statistical study. The result shows that the minimum system frequency depends on the number of connected diesel generators, the battery power generation/consumption and the PV power generation. The algorithm was tested using simulation software (MATLAB-SIMULINK for simulation; and GAMS for solving the MILP optimization problem, using the SCIP solver) and validated in a laboratory platform. Particularly, three different days (based on real second-by-second data) were simulated. Then, one of the simulated days was tested in a laboratory-scaled microgrid platform.

2. System Description

The system under study is depicted in Figure 1. The microgrid consists of several diesel generators ($N_d$), where each unit i has a rated power $P_{di}$; a PV power plant, where the rated power is P_pv-nom; and a battery with rated power and capacity P_bat-nom and $C_{bat}$, respectively. Finally, all these generation and storage units feed the total power demanded by the loads ($P_c$). The layout is based on a real stand alone system. It has the particularity that all generation and storage units (controllable units) are connected to the same bus. Thus, the load side can be treated as a single aggregated load. Each controllable unit has its local controller (LC) which is in charge of managing each resource separately:

• LC for diesel generation power plant: The local controller is in charge of controlling the frequency of the grid. A proportional–integral (PI) controller, where the input is the frequency error (filtered by a low pass filter), computes the mechanical torque setpoint of each diesel generator. This local controller also receives the required number of connected diesel generators and accordingly sends orders of connection/disconnection to each diesel unit. Each diesel generator has its internal controller in charge of reaching the torque setpoint and to perform its connection and disconnection according to the LC requirements. A similar control architecture is found in [23]. The main difference is that in the present paper the PI is a central controller that coordinates all the diesel units, while in [23] a single unit is considered.
• LC for the PV power plant: This LC implements a power–frequency droop curve to provide support to the grid. Reducing the active power will always be possible, but increasing it (under frequency events) will depend on the available active power. The controller can perform power curtailments. A maximum PV power setpoint is received externally and a PI controller computes the active power setpoint of each PV inverter. This controller is defined in [24], but the ramp rate limitation is not taken into account.
• LC for the battery: This controller receives externally an active power sepoint and applies a power–frequency droop curve to provide grid support. The output is the droop modified setpoint. The inner control loops will be in charge of reaching this value of active power. The dynamic model is simplified as in [25], but the local frequency droop has been included.

3. Methodology

3.1. EMS Design Requirements

The purpose of this section is to describe the steps followed for designing the EMS. The process is depicted in Figure 2. It shows that the EMS requirements are mainly determined by the characteristics of the system it will operate (system definition), the usage of the forecasting information (system data processing) and the identified operational requirements (system operation requirements).

First, the system characteristics are gathered—mainly the electrical characteristics and the available forecasting data—assuming grid isolated operation. Then, a statistical analysis of the forecasting for PV generation and demand is performed to identify the probability distribution of their errors. This allows generating random forecast scenarios (as detailed in Section 3.5). Next, the operation for the storage system is defined considering long-term variability of PV generation and demand. The minimum number of diesel generating units needed to face the largest demand change expected in the system is also determined. Finally, the EMS is designed, with two main purposes. On the one hand, the optimization problem is formulated based on the steady state equations determining the power balances in the system and limiting system variables. On the other hand, a frequency constraint, which is included in the optimization problem, is formulated (based on dynamic simulation results) relating the PV power generated, the battery power and the number of connected diesels with the minimum allowed frequency after a maximum power imbalance in the system.

The EMS performance is described in Section 3.2. The execution cycle of the EMS is detailed in Section 3.3. The procedure to determine the frequency constraint is explained in Section 3.4. For the stochastic optimization problem, it is required to generate a number of random scenarios, which is explained in Section 3.5. Finally, the whole optimization problem formulation is addressed in Section 3.7.

3.2. EMS Performance

The objective is to achieve the optimal utilization of the PV energy while achieving a generation–demand balance maintaining the grid frequency. In addition, it ensures that the minimum frequency $\left(f^{mn}\right)$ reached after a severe generation–load imbalance is within the limits (see Section 3.4 and the frequency constraint explained later for more detail).

The output variables (the setpoints to the generation and storage units) of the EMS are: (i) number of diesel generators to be connected ($D_{con}^{\ast }$); (ii) the setpoint to the battery ($P_{bat}^{\ast }$); and (iii) the maximum PV power setpoint ($P_{P{V}_{max}}^{\ast }$). They are calculated for the remaining of the day at each optimization execution period. On the other hand, the inputs are: (i) the load forecast ($L^c$); (ii) the available PV power forecast ($L^{PV}$); and (iii) the initial state of charge (SOC). Forecasts include the mean and standard deviation.

Figure 3 shows the time periods used. $\left(T_{for}\right)$ represents the time periods when forecasts are updated. $\left(T_{EMS}\right)$ is the period between EMS executions. Finally, $T_{intra}$ is the optimization problem time resolution. When the EMS is executed, the output variables (decision variables) are calculated for the rest of the day. While $P_{bat}^{\ast }$ and $P_{\mathit{\text{PV-max}}}^{\ast }$ are calculated with a time resolution of $T_{intra}$, the resolution of $D_{con}^{\ast }$ is $T_{EMS}$.

3.3. Execution Cycle

The optimization algorithm and its execution considers the daily sun period. Thus, the horizon of each execution is end of the day. This can be observed in Figure 3, where the execution cycle during the day d is depicted.

EMS period T execution: At period $T\in \{1,\dots ,n{T}_{EMS}\}$, the $P_{T,p}^{ba{t}^{\ast }}$ and $P_{T,p}^{P{V}_{max}^{\ast }}\forall p\in \{1,\dots ,n{T}_{intra}\}$ are sent to their respective converters. These values are calculated in previous EMS executions (see Figure 3). Then, the SOC at the beginning of the EMS period $T+1$ is estimated using the current SOC and the battery setpoints for the current execution period.

Using the estimated SOC at the EMS period $T+1$ and the forecast for the rest of the day d, the optimization problem is solved, and $P_{t,p}^{ba{t}^{\ast }}$ and $P_{t,p}^{P{V}_{max}^{\ast }}\forall p\in \{1,\dots ,n{T}_{intra}\},\forall t\in \{T+1,\dots ,n{T}_{EMS}\}$, and $D_t^{co{n}^{\ast }}\forall t\in \{T+1,\dots ,n{T}_{EMS}\}$ are calculated.

The solution must be reached before the beginning of the EMS period $T+1$. Otherwise, the setpoints calculated for the EMS period $T+1$ by the EMS execution at the period $T-1$ are sent to the respective converters.

3.4. Modeling Frequency Deviations

As explained above, one of the requirements of the isolated microgrid is the need to maintain the frequency in the required range. The frequency deviations depend on the grid inertia (i.e., the number of connected rotating machines) among other factors. One possible solution to ensure the frequency requirements is to connect the maximum number of rotating machines (diesel generators) providing large amount of inertia. However, these machines usually have a minimum active power generation. (The industry has reported that during low load condition diesel engines suffer from the “slobbering” effect. This effect is related to the low heat in the cylinder, allowing unburned fuel and oil to leak through the slip joints. This eventually leads to power losses, accelerated ageing and high maintenance costs). Thus, this strategy leads to a costly (fuel cost) and pollutant (CO₂ emissions) solution. Accordingly, the optimal solution is to connect the minimum number of rotating machines that ensures that, after a maximum power imbalance, the grid frequency will be kept in the required range.

Thus, the approach of this paper is to obtain an empirical linear equation determining the minimum frequency reached after a maximum power imbalance. This expression is then used in the optimization algorithm.

To obtain this expression, the worst case was first defined. The load and PV production of a real microgrid were monitored with 1 s resolution during six days and with 30 s resolution during one year. Using load data, a maximum load variation of 1.5 MW in 1 s was identified. This severe variation could have been produced due to the disconnection of a big load. For the case of PV data, it registered a maximum power variation of 1 MW in 1 s. According to the available recorded data, these changes will not occur simultaneously. Thus, the worst case considered is that the maximum power imbalance will occur after a sudden load variation of 1.5 MW, representing the situation when the maximum frequency deviation will occur.

Then, a simulation model of the microgrid was created. The model of the diesel generators are described in [23] while simplified PV and battery models are described in [25].

Using the simulation model, a bundle of scenarios varying $D_{con}$ from $N_d$ to $N_{dmin}$ (being $N_{dmin}$ the minimum number of diesel units connected to supply the maximum power imbalance), varying the $P^{pv}$ from the rated PV power to 0 and varying the $P^{bat}$ from $P^{mxB}$ (maximum battery power) to $P^{mnB}$ (minimum battery power) were simulated. In these simulations, the worst case (maximum load variation) was tested and the frequency response was analyzed, storing the minimum frequency reached for each simulation. From the analysis, a relation between the EMS output variables and the minimum frequency was performed (this analysis is explained below). To keep the optimization problem solvable using mixed integer linear programming (MILP), a linear regression is proposed for that purpose as Equation (1):

$$f^{mn}={\theta }_{ind}+{\theta }_d·D_{con}+{\theta }_{pv}·P_{PV}+{\theta }_{bat}·P_{bat}$$

(1)

where ${\theta }_x$ are the coefficients of linear regression.

The minimum frequency reached after the maximum power variation is represented in Figure 4 as a box plot against the $O{N}^{dies}$, $P^{bat}$ and $P^{P{V}_{max}^{\ast }}$. For each of the decision variables, it is possible to observe the tendency of the minimum frequency reached. The lower is the $P^{bat}$ and $P^{P{V}_{max}^{\ast }}$, the higher (in absolute values) is the maximum frequency deviation reached. On the other hand, the lower is the $O{N}^{dies}$, the lower is maximum frequency deviation reached. Figure 5 shows the summary of performing a linear regression; it can be observed that the coefficients for $P^{P{V}_{max}^{\ast }}$ and $P^{bat}$ are negative and the coefficient for $O{N}^{dies}$ is positive. The p-values for all the coefficients are lower than ${10}^{-8}$ and hence the obtained coefficients can be taken as significant.

3.5. Scenarios Generation

The forecasting system updates the forecasts for the rest of the day with a period $T_{for}$. The forecasts are based on a mean value and an error following a normal distribution with mean value $({\mu }_{err}=0)$ and a standard deviation $\left({\sigma }_{err}\right)$. Using these values, the EMS generates a number of random scenarios $N_s$ defined by the the pair $L_{t,p,s}^c$ and $L_{t,p,s}^{PV},\forall t\in \{To,\dots ,n_{TEMS}\},\forall p\in \{1,\dots ,n_{Tintra}\};\forall s\in \{1,\dots ,N_s\}$ being $To$ the actual $T^{EMS}$ period.

3.6. Stochastic Formulation Approach

The forecast errors are considered by using stochastic formulation. Particularly, $N_s$ scenarios are generated (more details are given in Section 3.5). Then, the decision variables are constant for all scenarios, i.e., devices receive the same setpoints in all scenarios. In contrast, the rest of the variables are computed depending on each scenario. This way, the optimization problem ensures finding decision variables that fulfill the problem constraints for all scenarios generated. Then, the global objective function is the sum of the objective functions of all scenarios. Note that more probable scenario are generated more times and thus are counted more times in the global objective function, i.e., the most probable scenarios have higher weights in the objective function.

3.7. Formulation of the Optimization Algorithm

The optimization problem is stochastic. It means that, from the forecast (mean and deviation values), different scenarios are generated. The solution (the battery setpoints, the maximum PV power setpoints and the number of connected diesel generator setpoints) is unique independently of the scenario, but the constraints must be accomplished for all scenarios. The objective function is the sum of the objective functions of all scenarios. This way, we obtain an optimal solution considering forecast errors. In this section, the different optimization sets, decision variables and restrictions required to define the optimization problem are detailed.

3.7.1. Sets

The sets defining the EMS executions and the time resolution are shown in Equations (2) and (3), respectively.

$$T^{EMS}=\{1,\dots ,n_{TEMS}\}$$

(2)

$$T^{intra}=\{1,\dots ,n_{Tintra}\}$$

(3)

where $n_{TEMS}$ is the number of the remaining executions of the optimization algorithm until the end of the day and $n_{Tintra}$ is the number of periods of $T_{intra}$ s between two executions of the optimization algorithm.

The index of the diesel generators are defined by the set in Equation (4), where $N_d$ is the total number of diesel generators.

$$N^{diesel}=\{1,\dots ,N_d\}$$

(4)

Stochastic optimization is used to account for the forecast errors. Hence, each optimization execution considers $N_s$ scenarios which are generated from the forecast inputs (mean and deviation). The set of the different scenarios is defined in Equation (5).

$$S=\{1,\dots ,N_s\}$$

(5)

3.7.2. Decision Variables

The decision variables are those that the optimization algorithm finds to optimize the objective function.

The battery power setpoint is defined as Equation (6), where positive values of power mean that the battery is discharging. It is also distinguished if the battery is charging or discharging. The battery charging and discharging powers are defined as Equations (7) and (8), respectively. To prevent obtaining a solution where the battery could simultaneously charge and discharge, a binary variable is defined in Equation (9). The SOC is shown in Equation (10). In the EMS algorithm, it is assumed that the battery setpoint is the same as the real battery power generation/consumption.

$$P_{t,p}^{ba{t}^{\ast }},\forall t\in T^{EMS},\forall p\in T^{intra}$$

(6)

$$P_{t,p}^{ba{t}_{char}},\forall t\in T^{EMS},\forall p\in T^{intra}$$

(7)

$$P_{t,p}^{ba{t}_{disch}},\forall t\in T^{EMS},\forall p\in T^{intra}$$

(8)

$$X_{t,p}^{char},\forall t\in T^{EMS},\forall p\in T^{intra};X_{t,p}^{char}\in \{0,1\}$$

(9)

$$SO{C}_{t,p}^{bat},\forall t\in T^{EMS},\forall p\in T^{intra}$$

(10)

The diesel connection/disconnection setpoint and the power generation of each diesel generator are denoted as Equations (11) and (12), respectively. $O{N}_{t,p}^{dies}$ is 1 if the diesel generator d at the EMS period t and the intra period d is connected (and 0 otherwise).

$$O{N}_{t,p}^{dies},\forall t\in T^{EMS},\forall d\in N^{Diesel},O{N}_{t,p}^{dies}\in \{0,1\}$$

(11)

$$P_{t,p,s,d}^{dies},\forall d\in N^{Diesel},\forall s\in S,\forall t\in T^{EMS},\forall p\in T^{intra}$$

(12)

The PV power generation of each scenario is written as Equation (13), while the maximum PV power setpoint is expressed as Equation (14).

$$P_{t,p,s}^{pv},\forall s\in S,\forall t\in T^{EMS},\forall p\in T^{intra}$$

(13)

$$P_{t,p}^{P{V}_{max}^{\ast }},\forall t\in T^{EMS},\forall p\in T^{intra}$$

(14)

3.7.3. Parameters

The load and PV scenarios are generated according to the forecast mean values and deviations. These scenarios are expressed as Equations (15) and (16), respectively. They represent the active power of load and the available PV power.

$$L_{t,p,s}^c,\forall s\in S,\forall t\in T^{EMS},\forall p\in T^{intra}$$

(15)

$$L_{t,p,s}^{PV},\forall s\in S,\forall t\in T^{EMS},\forall p\in T^{intra}$$

(16)

The battery capacity, the initial SOC, the battery efficiency and the maximum and minimum battery active power are written as $Ca{p}^{bat}$, $SO{C}^i$, ${\eta }^{bat}$, $P^{mxB}$ and $P^{mnB}$, respectively. The maximum and minimum active power of each diesel unit are expressed as $P^{mxD}$ and $P^{mxD}$, respectively. The minimum frequency is expressed as $f^{mn}$. The diesel generators performs a frequency control through a PI controller. To provide a power reserve for frequency regulation, a power margin of diesel generators is reserved. This power margin is denoted as $marg{e}_{dies}$.

3.7.4. Objective Function

The objective function it to maximize the PV power generation. To do so, the battery is charged or discharged according to the forecast and the problem requirements. During the charging and discharging process, there are some power losses. Thus, the real useful PV power must take into account them. Accordingly, the objective function is written as Equation (17).

$$\left[MAX\right]Z=\sum _{t,p,s}{P}_{t,p,s}^{PV}-n_S\left(1-{\eta }_{bat}\right)abs\left(P_{t,p}^{ba{t}^{\ast }}\right)$$

(17)

To linearize this function, it can be re-written as Equation (18).

$$\left[MAX\right]Z=\sum _{t,p,s}{P}_{t,p,s}^{PV}-n_S\left(1-{\eta }_{bat}\right)(P_{t,p}^{ba{t}_{char}}+P_{t,p}^{ba{t}_{disch}})$$

(18)

3.7.5. Constraints

The objective function has been linearized but. to prevent obtaining simultaneous charge and discharge of the battery, the following constrains are included (Equations (19)–(23)):

$$P_{t,p}^{ba{t}^{\ast }}=P_{t,p}^{ba{t}_{char}}-P_{t,p}^{ba{t}_{disch}}\forall t\in T^{EMS},\forall p\in T^{intra}$$

(19)

$$P_{t,p}^{ba{t}_{char}}\le P^{mxB}{X}_{t,p}^{char}\forall t\in T^{EMS},\forall p\in T^{intra}$$

(20)

$$P_{t,p}^{ba{t}_{disch}}\le P^{mxB}(X_{t,p}^{char}-1)\forall t\in T^{EMS},\forall p\in T^{intra}$$

(21)

$$P_{t,p}^{ba{t}_{char}}\ge 0\forall t\in T^{EMS},\forall p\in T^{intra}$$

(22)

$$P_{t,p}^{ba{t}_{disch}}\ge 0\forall t\in T^{EMS},\forall p\in T^{intra}$$

(23)

Then, the power balance at each period must be accomplished. This is forced by the restriction in Equation (24).

$$P_{t,p,s}^{pv}+\sum _{d\in N^{Diesel}}{P}_{t,p,s,d}^{dies}+P_{t,p}^{ba{t}^{\ast }}-L_{t,p,s}^c=0\forall t\in T^{EMS},\forall p\in T^{intra},\forall s\in S$$

(24)

Then, as commented above, a margin of diesel generation is reserved for frequency regulation. Thus, the maximum diesel generation is limited (Equation (25)):

$$\sum _{d\in N^{Diesel}}{P}_{t,p,s,d}^{dies}\le \sum _{d\in N^{Diesel}}O{N}_{t,d}^{dies}{P}^{mxD}-marg{e}_{dies}\forall t\in T^{EMS},\forall p\in T^{intra},\forall s\in S$$

(25)

The relationship between the SOC at instant t and the SOC at instant $t-1$ is shown in Equation (26). The SOC is between 0 and 1 p.u. This constraint is formulated as Equation (27). On the other hand, the battery power limits constraint is Equation (28).

$$\begin{array}{c}\text{-If}{T}^{EMS}=1\mathrm{and}{T}^{intra}=1\hfill \\ SO{C}_{t,p}^{bat}=SO{C}^{initial}-P_{t,p}^{bat}\frac{\Delta t}{Ca{p}^{bat}}\forall t\in T^{EMS},\forall p\in T^{intra}\hfill \\ \text{-If}{T}^{EMS}\ge 1\mathrm{and}{T}^{intra}=1\hfill \\ SO{C}_{t,p}^{bat}=SO{C}_{t-1,\left|p\right|}^{bat}-P_{t,p}^{bat}\frac{\Delta t}{Ca{p}^{bat}}\forall t\in T^{EMS},\forall p\in T^{intra}\hfill \\ \text{-If}{T}^{intra}\ne 1\hfill \\ SO{C}_{t,p}^{bat}=SO{C}_{t,p-1}^{bat}-P_{t,p}^{bat}\frac{\Delta t}{Ca{p}^{bat}}\forall t\in T^{EMS},\forall p\in T^{intra}\hfill \end{array}$$

(26)

$$0\le SO{C}_{t,p}^{bat}\le 1\forall t\in T^{EMS},\forall p\in T^{intra}$$

(27)

$$P^{mnB}\le P_{t,p}^{bat}\le P^{mxB}\forall t\in T^{EMS},\forall p\in T^{intra}$$

(28)

Then, the PV power cannot be greater than the available PV power of the corresponding scenario. Thus, Equation (29) must be included into the optimization algorithm. The PV power must also be lower than the maximum PV power setpoint of Equation (30).

$$P_{t,p,s}^{PV}\le L_{t,p,s}^{PV}\forall t\in T^{EMS},\forall p\in T^{intra},\forall s\in S$$

(29)

$$P_{t,p,s}^{PV}\le P_{t,p}^{P{V}_{max}^{\ast }}\forall t\in T^{EMS},\forall p\in T^{intra},\forall s\in S$$

(30)

Each diesel unit has a maximum and a minimum power at each scenario, which is formulated as Equation (31).

$$O{N}_{d,t,s}^{dies}{P}_d^{mnD}\le P_{t,p,s}^{dies}\le P^{mxD}O{N}_{d,t,s}^{dies}\forall t\in T^{EMS},\forall p\in T^{intra},\forall s\in S$$

(31)

Finally, the minimum frequency constraint is included in the optimization model. In the previous section, it has been shown how to express the minimum frequency reached in the microgrid after a maximum power imbalance. This constraint is written as Equation (32).

$$f^{mn}\le {\theta }_{ind}+{\theta }_d\sum _{d}O{N}_{t,d}^{dies}+{\theta }_{bat}{P}_{t,p}^{ba{t}^{\ast }}+{\theta }_{pv}{P}_{t,p}^{P{V}_{max}^{\ast }}\forall t\in T^{EMS},\forall p\in T^{intra}$$

(32)

4. Case Study

Based on a real case, the microgrid included: 9 × 1.2 MVA diesel units and 2 × 560 kWh batteries, which are interconnected through 4 × 550 kVA inverters (2 inverters per battery). The total battery power was then 2.2 MVA. The rated power of the PV plant was 10 MW, similar to the one presented in [24]. The minimum accepted frequency was $f^{mn}=49.0$ Hz.

Table 1. Parameters for the EMS optimization problem.

Parameter	Value	Parameter	Value	Parameter	Value
$n_{TEMS}$	288	$Ca{p}^{bat}$	1120 kWh	$P^{mnB}$	−2200 kW
$n_{Tintra}$	10	$SO{C}^i$	0.9	$P^{mnD}$	0.3·1100 kW
$N_d$	9	${\eta }^{bat}$	0.9	$P^{mxD}$	1100 kW
$N_s$	5	$P^{mxB}$	2200 kW	$marg{e}_{dies}$	2000 kW

shows the problem parameters.

Three scenarios were simulated. The load consumption was the same for all scenarios and shown in the result plots. The difference between the three scenarios was reflected in the available PV power profile. In the first case, after 12:30, the available PV profile presented large variations. The second scenario had lower PV variability, but it was not a fully sunny day. Finally, the last case consisted of a sunny day with not appreciably fast PV power variations. The simulation results are shown in Figure 6 for the first case, in Figure 7 for the second case and in Figure 8 for the last case. Note that the simulation considered the execution cycle explained in Section 3.3 and the EMS outputs were introduced to the dynamic model.

For each scenario, the top plot depicts the active power of microgrid’s devices as well as the power demand and the available PV power. In the middle plot, the SOC and the connections of diesel units can be observed. Then, the bottom plot shows the frequency response of the microgrid in blue, being the green lines the frequency droop dead-band (out of this range, the PV plant and the batteries provide frequency support). It can be observed that, for the three scenarios, the battery was discharged at the beginning of the day to be able to charge during the hours of high PV power. In addition, as could be expected, the active power of diesel generators and the connected units followed a trend complementary to the PV power generation. Thus, during the peak PV production hours, the amount of connected diesel generators was lower, as well as their production. It is also shown that the frequency deviations were kept inside the acceptable range. Comparing the total PV energy generated to the available PV energy for the three scenarios, the relative amount of used PV energy was 94.57%, 84.46% and 94.98%, respectively. The second scenario had the lowest PV profitability, but note that, in this case, the maximum available PV power was higher than the load in some periods.

During the times 13:00–15:00, the frequency exceeded the droop dead-band several times. Thus, the PV and battery provided frequency support. This happens because, during this period, the number of connected diesel generators was small (low inertia). Hence, either the large PV fluctuations or the connection of new generators injecting active power produced a frequency transient. While the frequency may have exceeded the frequency droop dead-band (green lines), it did not exceed the minimum value of 49 Hz.

5. Experimental Validation

5.1. Platform Description

An emulated microgrid was used for the experiments. As described in [26], an emulator consists of a platform able to convert software processed variables into real magnitudes. Accordingly, real equipment can be tested by its interconnection to the emulator platform. Hence, the system presented above can be tested properly through the emulation concept.

The layout of the laboratory microgrid (emulated microgrid) and its physical devices are depicted in Figure 9a,b, respectively. The emulated devices (diesel units, PV generators, storage, and loads) mimicked the behavior of the real device they are representing and form the emulated subsystem of the experimental setup. They were configured using a dedicated PC and a communication network. On the other hand, the real devices of the experimental setup were the PV and battery inverters, the power transformers, the EMS (which was implemented on a dedicated PC) as well as the communication network and the SCADA system. Because it is desired to emulate the isolated operation, the switch interconnecting the real system with the external grid was opened.

5.2. Emulation Results

The simulated results were validated using the first test case (the one presenting the highest PV power variability) and the emulation platform under a real time emulation test. The input data were scaled-down according the emulator’s power ratings. The outputs of the EMS were sent, periodically ($T_{EMS}$ = 5 min), to the devices (emulated). In Figure 10, the experimental results can be observed, showing how the response was very similar to the simulation results. In particular, the same tendency in the diesel units connections and disconnections as well as in the battery utilization can be observed. An important observation is that, generally, the generation was greater than the load because the emulated inverters had power losses.

6. Conclusions

A new methodology for the optimal operation of isolated microgrids has been proposed. This methodology is based on stochastic optimization to consider the forecast errors. In addition, a minimum frequency constraint has been formulated and included the optimization algorithm to ensure the secure operation of the microgrid. To maintain the optimization problem as a mixed integer linear problem, this constraint has been defined using a linear regression.

Three different scenarios, based on real data, were tested using a dynamic model of the microgrid. The results show a good behavior with a stable grid frequency and high rate of PV energy used.

After proving the proper response of the EMS using a simulation model, it was implemented to manage a laboratory scale microgrid, where real time limitations, communication delays and measurement errors occur. It has been shown that the system can also operate properly with real platforms having similar behavior to the simulated system.