Goal-Programming-Based Multi-Objective Optimization in Off-Grid Microgrids

Abstract

Renewable-based off-grid microgrids are considered as a potential solution for providing electricity to rural and remote communities in an environment-friendly manner. In such systems, energy storage is commonly utilized to cope with the intermittent nature of renewable energy sources. However, frequent usage may result in the fast degradation of energy storage elements. Therefore, a goal-programming-based multi-objective optimization problem has been developed in this study, which considers both the energy storage system (battery and electric vehicle) degradation and the curtailment of loads and renewables. Initially, goals are set for each of the parameters and the objective of the developed model is to minimize the deviations from those set goals. Degradation of battery and electric vehicles is quantified using deep discharging, overcharging, and cycling frequency during the operation horizon. The developed model is solved using two of the well-known approaches used for solving multi-optimization problems, the weighted-sum approach and the priority approach. Five cases are simulated for each of the methods by varying weight/priority of different objectives. Besides this, the impact of weight and priority values selected by policymakers is also analyzed. Simulation results have shown the superiority of the weighted-sum method over the priority method in solving the formulated problem.

1. Introduction

The energy sector contributes a major portion of global greenhouse gas emissions and a 46% increase is predicted in the next four decades. Meanwhile, the energy demand is also increasing and it is expected to increase by 56% by 2040 as compared to the demand in 2010 [1]. Similarly, a large portion of the global population is still deprived of basic energy needs [2]. Renewable energy is considered as a potential solution to deal with the increasing energy demand in an environment-friendly manner [2]. Especially, wind and solar energy are considered as promising sources due to their extensive availability and lower operating costs [2,3]. Wind energy is more abundant in coastal areas while solar energy is considered as a major driving factor for rural electrification. The integration of locally available renewable energy to fulfill local energy demand is considered as a viable option instead of extending the national grid to remote/rural areas [4]. These small-scale power systems are generally known as off-grid microgrids, due to the absence of connection with the central utility grid. Renewable-based off-grid microgrids are beneficial for fulfilling the energy needs of remote communities considering the availability of local resources [5]. However, planning and management of off-grid microgrids are challenging due to the dependence on intermittent renewable sources and the lack of connection with the utility grid.

The integration of battery energy storage system (BESS) is considered a potential solution to cover the uncertainties associated with renewable energy sources. BESS can enhance system flexibility while mitigating power variations and enhancing power quality [6]. Similar to BESS, the integration of electric vehicles (EVs) is also considered for reducing carbon emissions while absorbing renewable variations in remote areas [7] and high renewable penetration areas [8]. Frequent utilization is required to achieve the abovementioned benefits from storage technologies (BESS and EVs). However, the frequent utilization results in fast degradation of BESS [9] and EVs [10]. On the other hand, load shedding and renewable curtailment increase with reduced utilization of storage technologies. Therefore, a trade-off needs to be determined to minimize the load shedding and renewable curtailment while enhancing the lifetime of the energy storage elements.

The existing literature on the energy management of energy storage integrated off-grid/islanded microgrids can be broadly grouped into two categories. In the first category, the reduction of load shedding and/or renewable curtailment is considered without considering the degradation of energy storage elements [11,12,13,14]. Both optimal sizing and control of isolated microgrids have been considered in [11] for minimizing load shedding and renewable curtailment. A control mechanism for reducing the risk of load shedding and renewable curtailment over a finite horizon has been proposed in [12] for isolated microgrids. Load shedding has been integrated into the sizing phase of BESS in [13] to reduce the operation cost of the microgrid, where swarm optimization has been utilized for optimal sizing of BESS. An effort-based power-sharing mechanism among networked microgrids having BESS has been proposed in [14] for enhancing fairness during power-sharing among community microgrids.

In the second category, the degradation of BESS is considered while ignoring load shedding and/or renewable curtailment [15,16,17,18]. Lifetime optimization of off-grid renewable-based energy systems has been proposed in [15] considering the aging model of Li-ion batteries. In [16], various dynamic factors including battery degradation have been considered for life-cycle analysis of BESS in off-grid microgrids. Battery capacity loss and lifetime targets have been considered in [17] for the sizing of multi-energy microgrids using a mixed-integer linear programming model. A practical degradation model for Li-ion batteries has been developed in [18] and it has been utilized to optimize the battery scheduling to achieve realistic operation cost.

Few studies have considered both the degradation of energy storage elements and curtailment of loads and/or renewables [19,20,21]. In [19], the battery degradation model has been developed and it has been integrated into the optimization model of off-grid energy systems. This study has considered a single objective to minimize the operation cost of the microgrid and battery degradation cost. BESS degradation and load shedding have been considered in [20] for community microgrids, where the problem has been modeled as a single objective problem. In addition, the curtailment of renewables has not been considered in this study. In [21], BESS degradation has been considered along with the curtailment of loads and renewables. However, EVs have not been considered in this study and a linear-programming-based model is utilized.

It can be observed from the literature survey that most of the existing studies either focus on the degradation of batteries only or curtailment of loads and renewables only. Few studies have been reported that have considered both of the aspects simultaneously. However, these studies have modeled the problem as a single optimization problem [19,20], which is not suitable due to contradicting objective functions. Similarly, multi-objective models have also been developed using linear or mixed-integer linear programming models [21]. These models result in infeasible solutions if even a single constraint is not met. In the case of off-grid renewable microgrids, it might not be possible to meet all the constraints during the entire operation horizon. Therefore, we argue that goal-programming is more suitable for multi-objective optimization of renewable-based off-grid microgrids as compared to the linear or mixed-integer linear programming models. Finally, these studies have only considered static BESS while the penetration of EVs is increasing, and these can be utilized instead of BESS in many cases. Therefore, a comprehensive study is required that considers the curtailment of renewables and loads together with the degradation of static BESS and EVs using a suitable multi-objective optimization technique for conflicting objectives.

In order to address the abovementioned limitations, a multi-objective optimization model is developed in this study for renewable-based off-grid microgrids using goal-programming. The objectives considered in this study are the degradation of energy storage elements (BESS and EVs) and curtailment of load and renewables. Deep discharging, overcharging, and cycling frequency are considered as indicators for energy storage system degradation. Goals are set for each of the parameters and the objective of the developed optimization model is to minimize the deviation from those set goals. In order to assign different precedences to different goals, based on their relative importance, two solution methods are analyzed in this study, which are the weighted-sum method and the priority-based method [22]. In the former, higher weights are assigned to more significant parameters and vice versa, while in case of the latter, higher priorities are assigned to more significant parameters and vice versa. Five cases are simulated for both methods to analyze the sensitivity of each method and compare operation results under different weight/priority combinations. Similarly, the impact of policymaker decisions, in terms of weights and priorities, on the overall operation is also analyzed.

2. System Configuration and Multi-Objective Optimization

Off-grid microgrids are considered as a potential solution to deal with energy issues in remote and rural areas, where extension of the central grid is not economical. Especially, renewable-based off-grid microgrids have gained attention due to their environmental benefits and abundant availability of energy resources (wind, solar, etc.). Therefore, in this study, a renewable-based off-grid microgrid is considered, where both BESS and EVs are used for storing energy.

2.1. System Configuration

The configuration of the renewable-based off-grid microgrid system considered in this study is shown in Figure 1. It can be observed that the energy source of the microgrid is only renewables, which can be utilized to fulfill the energy demand, load. In order to address the uncertainties in renewables, an energy storage system is required. Therefore, BESS and EVs are considered as energy storage elements in this study, as shown in Figure 1. In contrast to stationary storage elements (BESS), EVs can only be used during their parking period. Typical parking periods (EV useable period) of EVs in offices or residential buildings are shown in Figure 2. The arrival time (t_a) and departure time (t_d) of EVs in offices and residential areas are different. The arrival time in offices is early morning and departure is around evening. In the case of residential apartments, the arrival time will be evening, and departure time will be early morning on the following day. The time between t_a and t_d is named as EV useable period in this study and this information needs to be provided to the optimization algorithm. Both BESS and EVs can be used to absorb the excess of renewables after feeding loads. Similarly, during peak load or low renewable intervals, they can be discharged to suffice the energy demand of the microgrid. EVs generally need to be charged to a certain level, known as target level, before their departure time to make them able to travel to their destinations [23]. The target energy is generally set by the EV users considering their travel patterns and target destinations. An energy management system (EMS) is utilized to collect the information of microgrid components and determine optimal set-points for each component, considering an operation objective. EMS is also responsible for informing the optimal set-points to the microgrid components after a defined time interval.

2.2. Multi-Objective Optimization and Goal Programming

In multi-objective optimization, even for a simple problem, it is unlikely to find a single solution that will optimize all the objective functions simultaneously, i.e., there is no global optimal solution [24]. Contrarily, there exist several equally significant solutions, which are commonly known as Pareto fronts, and are a trade-off between the objectives. Therefore, the objective of multi-objective optimization is to find the trade-off between these contradictory objectives, and several methods are available in the literature to achieve this goal [22]. The solution methods used for solving multi-objective problems are broadly categorized into two categories, i.e., methods with the posterior articulation of preferences and methods with a priori articulation of preferences. The advantage of the former method is in the utilization of the totality of the Pareto front but the solution times are often prohibitive, due to a large number of sub-problems [22]. The latter considers the preferences of the decision-makers before the optimization and any solution found is within the Pareto front. Due to a reduction in solution time, the latter is generally preferred over the former [22,25].

Another class of algorithms used for solving multi-objective optimization problems is searching algorithms, where different heuristic or meta-heuristic optimization methods are employed to find the optimal solution [26,27,28]. However, the major drawback of these searching algorithms is the convergence time, which increases exponentially with an increase in the problem size. Besides this, these algorithms cannot guarantee a Pareto-optimal solution, due to the possibility of trapping in local optima. In order to address the issues of searching algorithms, goal-programming has been used in this study to solve the proposed multi-objective optimization problem. The merits/advantages of goal-programming are as follows.

Goal-programming is a multi-objective optimization approach that falls under the methods with a priori articulation of preferences, and thus inherits its merits. Goal-programming is more beneficial for such problems that have conflicting objective functions [25]. All the parameters of objective functions are assigned a goal or a target value to be accomplished. The objective of goal-programming is to minimize the deviations from the set targets. This is opposite to the linear and mixed-integer linear programming models, where the objective is to minimize or maximize the parameter itself. Goal programming is especially more beneficial in cases where all the constraints are difficult to meet. Goal programming problems can be solved by either assigning weights to individual deviation parameters of the objective function or by defining priorities for each deviation. In contrast to the searching algorithms, it can guarantee a Pareto-optimal solution. Details about these solution methods are presented in the next section.

3. Problem Formulation

The overview of the problem formulation and solution technique developed for solving the multi-objective optimization problem of off-grid microgrids is shown in Figure 3. After obtaining the input values, the worst-case values of load and renewables are computed using Equations (1) and (2). It can be observed from Equation (1) that the worst-case for the load ($L_t^{}$) occurs when the load is more than the forecasted value ($L_t^f$). Similarly, Equation (2) implies that the worst case for renewables ($R_t^{}$) is when renewable power is lower than its forecasted value ($R_t^f$). In these equations, $\Delta$ represents the uncertainty factor and both load and renewables in theses equations are in kWh. The next step is to formulate the linear-programming-based optimization model, which is discussed in the following sub-section.

$$\begin{array}{cc}{L}_t^{}=L_t^f+L_t^f·{\Delta }_t^l& \forall t\in T\end{array}$$

(1)

$$\begin{array}{cc}{R}_t^{}=R_t^f-R_t^f·{\Delta }_t^r& \forall t\in T\end{array}$$

(2)

3.1. Linear-Programming-Based Model

The objective of the linear programming model is to minimize the load shedding of load, curtailment of renewables, and minimize the degradation of energy storage systems. Load shedding and curtailment of renewables can be directly computed but different factors need to be considered to estimate the degradation of the energy storage systems. The three major factors responsible for the degradation of storage elements (BESS and EVs) are deep discharging, over-charging, and frequency of charging/discharging (cycling frequency) [29,30]. In this study also, these three factors are considered to model the degradation of energy storage elements in the off-grid microgrid. Detailed mathematical modeling is carried out in the following sections.

3.1.1. Objective Function

The objective of the formulated problem is to minimize the degradation of the BESS and EVs while minimizing the curtailment of loads and renewables, as given by Equation (3). In this equation, $SO{C}_t$ and $SO{C}_{t,v}$ respectively, represents the state-of-charge (SOC) of BESS and vth EV at time t. Similarly, $R_t^{cur}$ and $L_t^{cur}$ respectively represent the curtailment amount of renewables and loads. The first term of the objective function in line 1 represents the deep discharging of BESS, the second term represents the overcharging of BESS, and the third term represents the cycling frequency of the BESS. The fourth and fifth terms in line 1 represent the curtailment amount of loads and renewables, respectively. Similarly, the first, second, and third terms in the second line of the objective function represent deep discharging, overcharging, and cycling frequency of EVs, respectively. All the quantities in the objective function are taken in kWh for the sake of uniformity in units.

$$\begin{array}{l}Min{\displaystyle \sum _{t=1}^T((SO{C}_t-SO{C}_{\min })+(SO{C}_t-SO{C}_{\max })}+(SO{C}_{t-1}-SO{C}_t)+R_t^{cur}+L_t^{cur})\\ +{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{v=1}^V((SO{C}_{t,v}-SO{C}_{v,\min })+(SO{C}_{t,v}-SO{C}_{v,\max })+}(SO{C}_{v,t-1}-SO{C}_{v,t}))}\end{array}$$

(3)

3.1.2. Battery Energy Storage System (BESS) Constraints

Equations (4)–(7) represent the constraints of BESS in the off-grid microgrid. In these equations, $P_t^{bc}$ and $P_t^{bd}$ represent the charging and discharging amount while ${\eta }^{bc}$ and ${\eta }^{bd}$ represent the charging and discharging efficiency of BESS. Similarly, P^cap represents the capacity of the BESS in kWh. Equation (4) represents the charging constraint while Equation (5) represents the discharging constraint of BESS. These equations imply that charging and discharging amount depends on the state-of-charge (SOC) at the previous interval, the capacity of BESS, and charging/discharging efficiencies. The SOC of BESS at any interval t can be computed using Equation (6). Finally, Equation (7) represents the operation bounds of SOC in BESS.

$$\begin{array}{cc}0\le P_t^{bc}\le P_{}^{cap}\cdot \left(1-SO{C}_{t-1}^{}\right)\cdot \frac{1}{{\eta }^{bc}}& \forall t\in T\end{array}$$

(4)

$$\begin{array}{cc}0\le P_t^{bd}\le P_{}^{cap}\cdot SO{C}_{t-1}^{}\cdot {\eta }^{bd}& \forall t\in T\end{array}$$

(5)

$$\begin{array}{cc}SO{C}_t^{}=SO{C}_{t-1}^{}-\frac{1}{P_{}^{cap}}\cdot \left(\frac{1}{{\eta }^{bd}}\cdot P_t^{bd}-P_t^{bc}\cdot {\eta }^{bc}\right)& \forall t\in T\end{array}$$

(6)

$$\begin{array}{cc}SO{C}_{\min }^{}\le SO{C}_t^{}\le SO{C}_{\max }^{}& \forall t\in T\end{array}$$

(7)

3.1.3. Electric Vehicles (EV) Constraints

Similar to BESS, Equations (8) and (9) represent the charging and discharging constraints of EVs. In these equations, $P_{t,v}^{ec}$ and $P_{t,v}^{ed}$ represent the charging and discharging amount while ${\eta }^{ed}$ and ${\eta }^{ed}$ represent the charging and discharging efficiencies of vth EV, respectively. Similarly, $P_v^{cap}$ and $SO{C}_{t,v}^{tar}$ represent the capacity and target SOC (before departure) of vth EV, respectively. Equation (10) can be used to compute the SOC at time t and Equation (11) represents the constraints of SOC. In these equations, $\tau$ represents the useable period of EVs, i.e., intervals during which EVs are parked. Equation (12) implies that EVs cannot be used outside of the useable period. Finally, Equation (13) represents EVs need to be charged to the required level (target SOC) before their departure time, which is set by the EV owner.

$$\begin{array}{cc}0\le P_{t,v}^{ec}\le P_v^{cap}\cdot \left(1-SO{C}_{t-1,v}^{}\right)\cdot \frac{1}{{\eta }^{ec}}& \forall t\in \tau ,v\in V\end{array}$$

(8)

$$\begin{array}{cc}0\le P_{t,v}^{ed}\le P_v^{cap}\cdot SO{C}_{t-1,v}^{}\cdot {\eta }^{ed}& \forall t\in \tau ,v\in V\end{array}$$

(9)

$$\begin{array}{cc}SO{C}_{t,v}^{}=SO{C}_{t-1,v}^{}-\frac{1}{P_v^{cap}}\cdot \left(\frac{1}{{\eta }^{ed}}\cdot P_{t,v}^{ed}-P_{t,v}^{ec}\cdot {\eta }^{ec}\right)& \forall t\in \tau ,v\in V\end{array}$$

(10)

$$\begin{array}{cc}SO{C}_{v,\min }^{}\le SO{C}_{t,v}^{}\le SO{C}_{v,\max }^{}& \forall t\in \tau ,v\in V\end{array}$$

(11)

$$\begin{array}{cc}SO{C}_{t,v}^{}=P_{t,v}^{ed}=P_{t,v}^{ec}=0& \forall t\notin \tau ,v\in V\end{array}$$

(12)

$$\begin{array}{cc}SO{C}_{t,v}^{}\ge SO{C}_{t,v}^{tar}& \forall t=t_{t,v}^d,v\in V\end{array}$$

(13)

3.1.4. Power Balancing Constraint

The power balancing equation implies that at any interval t, the power generation and consumption should be balanced. Equation (14) shows that the off-grid microgrid load ($L_t^{}$) can be fulfilled using BESS charging ($P_t^{bc}$) and discharging ($P_t^{bd}$), EVs charging ($P_{t,v}^{ec}$) and discharging ($P_{t,v}^{ed}$), or renewable power ($R_t^{}$). Similarly, renewable power can be utilized to charge the BESS and EVs in addition to fulfilling the local load demand. Finally, renewables can be curtailed ($R_t^{cur}$) during excess intervals and load can be curtailed ($L_t^{cur}$) during shortage intervals.

$$\begin{array}{cc}{P}_t^{bd}-P_t^{bc}+{\displaystyle \sum _{v=1}^V(P_{t,v}^{ed}-P_{t,v}^{ec})}+L_t^{cur}+R_t^{}=L_t^{}+R_t^{cur}& \forall t\in T\end{array}$$

(14)

3.2. Goal-Programming-Based Model

The second step is to transform the developed linear model into a goal programming model, as depicted in Figure 3. In goal-programming, goals/targets are set for each of the objectives, and upward deviations are represented with plus (+) superscripts while negative (-) superscripts are used to represent downward deviations.

3.2.1. BESS Degradation Goals

The goals for all the objective functions related to the BESS degradation are set to zero, i.e., deep discharging (DD), overcharging (OC), and frequent cycling frequency (CF) is undesirable. Equations (15)–(17) represent the goals for BESS degradation factors (deep discharging, overcharging, and cycling frequency, respectively). In these equations, (+) superscript and negative (−) superscript represent the upward and downward deviations, respectively. Upward and downward deviations may occur simultaneously during simulations and produce unwanted results. Therefore, binary variable ($\lambda$) is defined, and Equations (18) and (19) are modeled to avoid the simultaneous occurrence of upward and downward deviations of respective parameters.

$$\begin{array}{cc}SO{C}_t-SO{C}_{\min }=D{D}_{{}^t}^{b+}·{\lambda }_{t,11}^b-D{D}_{{}^t}^{b-}·{\lambda }_{t,12}^b& \forall t\in T\end{array}$$

(15)

$$\begin{array}{cc}SO{C}_t-SO{C}_{\max }=O{C}_{{}^t}^{b+}·{\lambda }_{t,21}^b-O{C}_{{}^t}^{b-}·{\lambda }_{t,22}^b& \forall t\in T\end{array}$$

(16)

$$\begin{array}{cc}SO{C}_{t-1}-SO{C}_t=C{F}_t^{b+}·{\lambda }_{t,31}^b-C{F}_t^{b-}·{\lambda }_{t,32}^b& \forall t\in T\end{array}$$

(17)

$$\begin{array}{cc}{\lambda }_{t,11}^b+{\lambda }_{t,12}^b=1,{\lambda }_{t,21}^b+{\lambda }_{t,22}^b=1,{\lambda }_{t,31}^b+{\lambda }_{t,32}^b=1& \forall t\in T\end{array}$$

(18)

$$\begin{array}{cc}{\lambda }_{t,11}^b+{\lambda }_{t,12}^b+{\lambda }_{t,21}^b+{\lambda }_{t,22}^b,{\lambda }_{t,31}^b+{\lambda }_{t,32}^b\in \left\{0,1\right\}& \forall t\in T\end{array}$$

(19)

3.2.2. Load and Renewable Curtailment Goals

The goals of load and renewable curtailment are also set to zero, as given by Equations (20) and (21), respectively. Since any positive deviation in load curtailment ($L{C}_t^{+}$) will result in the discomfort of the consumers and positive deviation in renewable curtailment ($R{C}_t^{+}$) will result in the loss of resources. The same notation of signs is used to represent the upward and downward deviations in the goals of load and renewable curtailment, i.e., + and - superscripts for upward and downward deviations, respectively. Negative deviations of load ($L{C}_t^{-}$) and renewable ($R{C}_t^{-}$) curtailment are not possible physically; therefore, binary variables are not defined and negative deviations will be ignored in the objective function.

$$\begin{array}{cc}{R}_t^{cur}=R{C}_t^{+}-R{C}_t^{-}& \forall t\in T\end{array}$$

(20)

$$\begin{array}{cc}{L}_t^{cur}=L{C}_t^{+}-L{C}_t^{-}& \forall t\in T\end{array}$$

(21)

3.2.3. EV Degradation Goals

Similar to BESS, the EV degradation goals are also set to zero. Equations (22)–(24) respectively represent the deep discharging (DD), overcharging (OC), and cycling frequency (CF) goals of EVs. In these equations, (+) superscript and negative (−) superscript represent the upward and downward deviations, respectively. Equations (25) and (26) constrain the simultaneous occurrence of upward and downward deviations for EVs.

$$\begin{array}{cc}SO{C}_{t,v}-SO{C}_{v,\min }=D{D}_{{}^{v,t}}^{e+}·{\lambda }_{t,v,11}^e-D{D}_{{}^{v,t}}^{e-}·{\lambda }_{t,v,12}^e& \forall t\in \tau ,v\in V\end{array}$$

(22)

$$\begin{array}{cc}SO{C}_{t,v}-SO{C}_{v,\max }=O{C}_{t,v}^{e+}·{\lambda }_{t,v,21}^e-O{C}_{t,v}^{e-}·{\lambda }_{t,v,22}^e& \forall t\in \tau ,v\in V\end{array}$$

(23)

$$\begin{array}{cc}SO{C}_{t-1,v}-SO{C}_{t,v}=C{F}_{t,v}^{e+}·{\lambda }_{t,v,31}^e-C{F}_{v,t}^{e-}·{\lambda }_{t,v,32}^e& \forall t\in \tau ,v\in V\end{array}$$

(24)

$$\begin{array}{cc}{\lambda }_{t,v,11}^e+{\lambda }_{t,v,12}^e=1,{\lambda }_{t,v,21}^e+{\lambda }_{t,v,22}^e=1,{\lambda }_{t,v,31}^e+{\lambda }_{t,v,32}^e=1& \forall t\in \tau ,v\in V\end{array}$$

(25)

$$\begin{array}{cc}{\lambda }_{t,v,11}^e+{\lambda }_{t,v,12}^e+{\lambda }_{t,v,21}^e+{\lambda }_{t,v,22}^e,{\lambda }_{t,v,31}^e+{\lambda }_{t,v,32}^e\in \left\{0,1\right\}& \forall t\in \tau ,v\in V\end{array}$$

(26)

3.2.4. Goal-Programming-Based Objective Function Formulation

The objective of goal programming is to minimize the deviations from the set goals for all the objectives. The goals for load and renewable curtailment are set to zero. However, positive deviations are possible to determine any feasible solution. Similarly, the goal for all the factors of BESS and EV degradation is also set to zero with possible positive and negative deviations.

The goal-programming objective functions are formulated by considering the severity of deviating from the set goals. In the case of energy storage elements, negative deviation of deep discharging and positive deviation of overcharging result in fast degradation. Therefore, the two terms for BESS and EV are grouped as the first objective function, as given by Equation (27). Negative deviations for load and renewables are not possible physically. Therefore, only positive deviations of load and renewables are formulated as the second objective function, as given by Equation (28). Cycling frequency in both positive and negative direction are grouped in the third objective function, as given by Equation (29). Finally, the positive deviation of deep discharging and negative deviation of overcharging have the lowest impact on energy storage degradation. Therefore, they are grouped in the fourth objective function, as given by Equation (30). The alpha ($\alpha$) parameters multiplied with each of the terms are scalers and are used to define the precedence of individual parameters.

$$Ob{j}_1={\displaystyle \sum _{t=1}^T({\alpha }_{11}^b\cdot D{D}_t^{b-}+{\alpha }_{12}^b\cdot O{C}_t^{b+})+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{v=1}^V({\alpha }_{v,11}^e\cdot D{D}_{t,v}^{e-}+{\alpha }_{v,12}^e\cdot O{C}_{t,v}^{e+})}}}$$

(27)

$$Ob{j}_2={\displaystyle \sum _{t=1}^T({\alpha }_{21}^b\cdot L{C}_t^{+}+{\alpha }_{22}^b\cdot R{C}_t^{+})}$$

(28)

$$Ob{j}_3={\displaystyle \sum _{t=1}^T({\alpha }_{31}^b\cdot C{F}_t^{b+}+{\alpha }_{32}^b\cdot C{F}_t^{b-})}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{v=1}^V({\alpha }_{v,31}^e\cdot C{F}_{t,v}^{e+}+{\alpha }_{v,32}^e\cdot C{F}_{t,v}^{e-})}}$$

(29)

$$Ob{j}_4={\displaystyle \sum _{t=1}^T({\alpha }_{41}^b\cdot D{D}_t^{b+}+{\alpha }_{42}^b\cdot O{C}_t^{b-})}+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{v=1}^V({\alpha }_{v,41}^e\cdot D{D}_{t,v}^{e+}+{\alpha }_{v,42}^e\cdot O{C}_{t,v}^{e-})}}$$

(30)

3.2.5. Weighted-Sum-Based Modeling Approach

After problem formulation, the goal-programming problems can be solved using two methods, which are priority-based and weighted-sum methods. In the weighted-sum method, weights are defined for each of the objectives, as depicted in Figure 4. In this method, all the objectives are summed to form a single objective function, as given by Equation (31). The objective function is subjected to all the constraints, i.e., Equations (4)–(26). The weight parameters play a major role in deciding the preference of different objective functions.

$$Min\left({\beta }_1·Ob{j}_1+{\beta }_2·Ob{j}_2+{\beta }_3·Ob{j}_3+{\beta }_4·Ob{j}_4\right)$$

(31)

Subject to: Equations (4)–(26)

3.2.6. Priority-Based Modeling Approach

In the priority-based solution method, priorities are defined for each of the objectives, where most significant objectives are set with the highest priorities. In this method, the objective functions are solved one by one starting with the highest priority objective function. It can be observed from Figure 4 that initially, objective 1 is solved with standard constraints as given by (32). In the second step, objective two is solved and in addition to standard constraints (33), the first objective function is set as additional constraints as given by (34). This process is repeated until all the objective functions are solved or a unique optimal solution is found.

$$\begin{array}{ccc}MinOb{j}_1\hfill & & \\ & & \mathrm{Subject}\mathrm{to}:\mathrm{Equations}\left(4\right)–\left(26\right)\end{array}$$

(32)

$$\begin{array}{ccc}MinOb{j}_2\hfill & & \\ & & \mathrm{Subject}\mathrm{to}:\mathrm{Equations}\left(4\right)–\left(26\right)\end{array}$$

(33)

$${\displaystyle \sum _{t=1}^T({\alpha }_{11}^b\cdot D{D}_t^{b-}+{\alpha }_{12}^b\cdot O{C}_t^{b+})+{\displaystyle \sum _{t=1}^T{\displaystyle \sum _{v=1}^V({\alpha }_{v,11}^e\cdot D{D}_{t,v}^{e-}+{\alpha }_{v,12}^e\cdot O{C}_{t,v}^{e+})}}}\le Ob{j}_1^{*}$$

(34)

4. Numerical Simulations

In order to evaluate the performance of the proposed method for solving multi-objective optimization in off-grid microgrids, a test system, the same as in Figure 1, is considered in this study. Both weighted-sum and priority-based solution methods have been utilized to solve the target problem. In this section, different possible combinations of weight and priority factors are analyzed. However, in the next section, the impact of relative weight and priorities of different objectives, considering choices of policymakers, are discussed. The performance of both methods is analyzed in the following sections.

4.1. Input Data

The hourly forecasted values of load and renewable power are taken as inputs for this study. The worst-case scenario values of load and renewables are shown in Figure 5 and the forecasted values were taken from [31]. One BESS and two EVs are considered in this study as energy storage elements and the parameters related to BESS and Evs are shown in

Table 1. Parameters of energy storage elements.

Storage Element	Capacity (kWh)	Departure Time (h)	Arrival Time (h)	Efficiency (%)	SOC (%)
					Initial	Min.	Max.	Target
BESS	750	-	-	95	20	20	80	-
EV1	70	18	9	95	20	20	80	50
EV2	60	20	8	93	20	20	80	50

. The data of EVs considered this study are taken from [32] and Hyundai Kona and Kia e-Niro are selected, which have the useable battery range in 60-70 kWh. Both these cars use Li-ion batteries, which have efficiencies in the range of 90% to 95% [33]. Therefore the size of the battery is chosen as 60 and 70kWh along with efficiencies of 93% and 95%. Similarly, BESS is also assumed to be a Li-ion battery and its size is the same as in [27] (due to usage of the same input data) and efficiency is taken as 95% [33]. The developed model has been solved in Java NetBeans [34] with the integration of CPLEX 12.7 [35]. The performance of the proposed algorithm has been tested for a 24-hour scheduling horizon with a resolution of 1 hour.

4.2. Weighted-Sum Approach Results

In order to analyze the performance of the developed algorithm, five cases are analyzed in the weighted-sum approach. The values of weights assigned to each objective function in different cases are tabulated in

Table 2. Weight and priorities defined for different simulation cases.

Cases	Weights				Priorities
	Obj 1	Obj 2	Obj 3	Obj 4	Obj 1	Obj 2	Obj 3	Obj 4
Case 1	1	1	1	1	1	1	1	1
Case 2	4	1	3	2	4	1	3	2
Case 3	3	4	2	1	3	4	2	1
Case 4	1	4	1	1	4	3	2	1
Case 5	1	20	1	1	4	3	1	1

. In Table 2, higher weight values imply higher precedence of the objective function and vice versa. Case 1 is considered as the base case, where the weights for all the objective functions are set to 1, i.e., all objectives are equally important.

It can be observed in Figure 6 and Figure 7a that, in Case 2, renewable curtailment has increased during several intervals as compared to other cases due to the lowest weight for load and renewable curtailment. The SOC of BESS remains at 20% and BESS is never used across the day (Figure 7b). Similarly, EVs have just been charged to the target SOC level at the beginning interval and were not used, as shown in Figure 8a,b.

In Case 3, the highest weight is assigned to load and renewable curtailment and is followed by deep discharging/overcharging. Therefore, load shedding and renewable curtailment are reduced as compared to the previous case as shown in Figure 6 and Figure 7. BESS has crossed the over-charging limit once during interval 13, as shown in Figure 7b. However, EVs have operated within the bounds for this case, as shown in Figure 8.

In Case 4, weights for energy storage elements are set to 1 while keeping the weight for load and renewable curtailment as 4. Due to this difference, load shedding and renewable curtailment have been further reduced as shown in

Table 3. Summary of weighted-sum method results.

Cases	Load Curtailment		Renewable Curtailment
	Amount (kWh)	Increase/Decrease (%)	Amount (kWh)	Increase/Decrease (%)
Case 1	310	0.00%	448.78	0.00%
Case 2	536	72.90%	1173.47	161.46%
Case 3	73.94	−76.15%	382.25	−14.82%
Case 4	14.66	−95.27%	288.88	−35.63%
Case 5	0	−100%	157.27	−64.96%

. However, BESS crossed the upper SOC bound during two intervals (12 and 13) while EV1 and EV2 also crossed the upper SOC bounds once each (interval 9), as shown in Figure 7 and Figure 8.

In Case 5, the difference between energy storage degradation and load/renewable curtailment is further increased to analyze the impact on BESS/EV SOC profiles. Due to this significant difference, load shedding is reduced to zero and renewable curtailment is also reduced to the lowest level. However, the cycling frequency of BESS and EVs has significantly increased. Similarly, energy storage elements are charged to their fullest to absorb renewables and reduce load shedding.

It can be observed from the results in Table 3 that proper setting of weight values is necessary to achieve a trade-off among the contradicting objectives. Case 2 and Case 5 are not desirable since these cases focus more on one aspect while ignoring others. Case 1 and Case 3 could be considered as potential solutions where Case 3 is more focused on reducing load shedding and renewable curtailment with slight violations in energy storage elements.

4.3. Priority Approach Results

Similar to the previous weighted-sum approach, five cases are considered for the priority approach in this section. The priorities assigned to each objective function in different cases are tabulated in Table 2. The numeric values represent the priority of the objective functions, i.e., 1 represents the least significant function and 4 represents the most significant function. Case 1 is considered as the base case, and results of other cases are compared with the base case, where the priorities of all the objective functions are set to 1, i.e., all objectives are equally important.

In Case 2, load and renewable curtailments are set with the lowest priority while energy storage elements are set with higher priorities. Load shedding and renewable curtailment have increased in this case while BESS is not utilized, as shown in Figure 9 and Figure 10. Both EVs have been charged to meet the target SOC (50%) only and are not utilized for absorbing renewables or shedding loads, as shown in Figure 11. The results of this case are similar to the corresponding case of the weighted-sum method.

In Case 3, the priority of load and renewables is set higher than the degradation of energy storage elements. In this case, load shedding is reduced to zero and the curtailment of renewables is also minimized, as shown in

Table 4. Summary of priority method results.

Cases	Load Curtailment		Renewable Curtailment
	Amount (kWh)	Increase/Decrease (%)	Amount (kWh)	Increase/Decrease (%)
Case 1	310	0.00%	448.78	0.00%
Case 2	536	72.90%	1173.36	161.46%
Case 3	0	−100.00%	124.36	−72.29%
Case 4	96.31	−68.93%	325.72	−27.42%
Case 5	96.31	−68.93%	325.72	−27.42%

. However, BESS and EVs are frequently charged/discharged to absorb renewables and reduce load shedding. Besides this, energy storage elements violated their upper SOC bounds during several intervals, as shown in Figure 10b and Figure 11. The results of this case are different from the corresponding case of the weighted-sum approach. It shows the sensitivity of the priority method to the assigned priorities.

In Case 4, the over-charging and deep-discharging violation is given the highest priority and is followed by load and renewable curtailment priority. SOC of energy storage elements is not violated across the day, as shown in Figure 10b and Figure 11. However, load shedding and renewable curtailment have been significantly reduced in comparison with the Case 1, as shown in Table 4.

In Case 5, the priority of cycling frequency is further reduced while keeping the priorities of other elements the same as the previous case. Due to the lower priority of cycling frequency, energy storage elements are charged/discharged more frequently. However, SOC bounds are not violated across the day. Interestingly, the load shedding amount and renewable curtailment amount is the same as the previous case, as shown in Table 4. This is due to the significance of the highest priority objective in the priority-based approach.

It can be observed from the results in Table 4 that the priority-based solution method is more sensitive to the highest priority objective function. The significance of lower priority functions is lower, and thus the impact of lower priority functions on the overall result of the optimization problem is not significant. The decision-makers need to be more conscious about selecting the priorities, especially selecting the highest priority objective function.

5. Discussion and Analysis

The result of the proposed goal-programming-based multi-objective optimization technique for off-grid microgrids is subjected to the selection of weight factors and priorities of individual objective functions. Generally, the values of weights and priorities are determined by the policymakers by keeping in mind their specific targets. For example, policymakers who are more concerned about the life of energy storage elements will choose a higher weight factor or higher priority for avoiding the degradation of BESS and EVs. Similarly, those who are more concerned about the shedding of loads and renewables will set higher priorities and weight values to mitigate load and renewable curtailment. Policymakers need to make choices among different objectives and these choices are generally relative.

Therefore, in this section, the relative impact of different weight and priority values on the final goal of each objective is analyzed. Degradation of BESS and EVs is considered as one objective and curtailment of load and renewables is considered as another objective. A range of scenarios are simulated in each case to analyze the impact of different possible choices of policymakers. An appropriate range of weight and priority values are suggested for different policymakers, considering their different ranges of objectives/goals.

5.1. Impact of Weight Factor

In this section, the weight factor for energy storage elements is taken as α and the weight factor for power curtailment (load and renewables) is taken as β. Policymakers need to choose between relative values of α and β. A higher value of α implies higher importance to battery and EVs and a higher value of β implies higher importance of load/renewable curtailment. Simulations are conducted for different values of the weight ratio β/α. A weight ratio value less than one implies that the weight of renewable/load curtailment is higher than the degradation of energy storage elements and vice versa. The weight ratio of value 1 implies that both the parameters are given equal importance.

It can be observed from Figure 12a that when the weight ratio decreases from 1, the load and renewable curtailment amount is also reduced. However, the violation in BESS and EVs also increases with the decrease in weight ratio (Figure 12b). This is due to higher importance of load/renewable curtailment with lower weight ratio values, i.e., the lower the ratio, the higher the importance of the load and renewable curtailment. When the weight ratio increases from 1, the violations have reduced to zero and load/renewable curtailment remains the same from weight ratio factor 2 onwards. This is due to the inability of the BESS/EVs to absorb more renewables or reduce load shedding without violating the safety bounds of deep discharging and overcharging.

It can be concluded that a policymaker who is more concerned about the energy storage lifetime will choose a higher weight ratio, i.e., greater than or equal to 2, while a policymaker who is more concerned about load/renewable curtailment will choose a lower weight ratio, i.e., less than or equal to 0.2. However, a tradeoff between these objectives can be achieved by choosing the weight ratio in the range of 0.5 and 1. This range can provide lower load shedding/renewable curtailment while maximizing the lifetime of BESS and EVs by reducing violations.

5.2. Impact of Priority

In this section, the impact of different possible priorities by policymakers is analyzed. The priority of energy storage elements is represented as ρ₁ and the priority of load/renewable curtailment is represented as ρ₂. Similar to the previous section, different possible choices of priority ratio (ρ₂ /ρ₁) are analyzed. A higher than 1 priority ratio implies that the priority of energy storage degradation is higher than load and renewable curtailment, and vice versa. A priority ratio of 1 implies that both have equal importance. For the sake of comparison, the same range of priority factors is considered with the range of weight factor in the previous section.

It can be observed from Figure 13a that when the priority ratio is less than 1, load curtailment is reduced to zero and renewable curtailment is also reduced. However, severe violations in BESS and EVs can be observed from Figure 13b. Contrarily, when the priority ratio is greater than 1, load and renewable curtailments have increased while reducing the violations in BESS and EVs, as evident from Figure 13. It is worth noticing that in the case of the priority method, the magnitude of priority is meaningless, only the relative magnitude of priority is considered and objective with higher priority is satisfied first. Therefore, the results for the priority ratio in the range of 0.05 to 0.5 are the same and the results of the priority ratio in the range of 1 to 20 are the same.

It can be concluded that policymakers who are more concerned about the energy storage lifetime will choose a priority ratio greater than 1 while policymakers who are more concerned about load/renewable curtailment will choose a priority ratio less than 1. In contrast to the weighted method, the tradeoff between these objectives cannot be achieved in the case of the priority method, i.e., policymakers need to choose one of the objectives.

5.3. Comparative Analysis

After analyzing different weight ratios and priority ratio factors that could be selected by different policymakers, it can be concluded that the priority method is more suitable for policymakers who are more interested in one of the goals. Since, in case of priority method, policymakers do not need to worry about the choice of priority value, i.e., any value greater than the priority of other objectives will give the same results. Contrarily, the weighted-sum method is more suitable for policymakers who are more interested in finding a trade-off among different objectives. However, the policymakers need to find an appropriate value of weight ratios (relative weights), which was in the range of 0.5 to 1 in this case.

6. Conclusions

A goal-programming-based multi-objective optimization model has been developed in this study considering the degradation of energy storage elements and curtailment of loads and renewables. The developed model has been solved using two well-known approaches, i.e., weighted-sum approach and priority-based approach. Simulation results have shown that the priority-based solution method is more sensitive to the priorities assigned to the objective functions. Especially, the selection of the highest priority function significantly changes the overall optimization results, while lower priority objectives have a minute impact on the overall results. Therefore, the selection of the highest priority could be a challenging task. The weighted-sum approach appears to be promising in solving this particular problem due to its lower sensitivity to the weights assigned to different objective functions. By selecting preferred weights for different objective functions, based on the preferences of decision-makers, desired results can be obtained.