Stochastic Economic–Resilience Management of Combined Cooling, Heat, and Power-Based Microgrids in a Multi-Objective Approach

Abstract

The primary goal of a microgrid (MG) operator is to provide electricity to consumers while minimizing costs. For this aim, the operator must engage in the cost-effective management of its resource outputs, which can encompass electrical, thermal, or combined cooling, heat and power (CCHP) systems. Conversely, there has been a growing emphasis on enhancing the resilience of MGs in response to low-probability high-impact (LPHI) incidents in recent years. Therefore, MG-associated energy management strategies have to factor in resilience considerations. While resilience improvement activities increase the operational cost, they lead to a reduction in lost load, and subsequently, a decrease in the MG outage costs, making these activities economically viable. This paper focuses on MGs’ energy management with the primary goals of enhancing resilience, minimizing operational costs, and mitigating active power losses as well as environmental pollution. To attain this goal, various means like renewable resources (specifically photovoltaic (PV) and wind turbine (WT) systems), CCHP, and energy storage devices are integrated. Additionally, for reaching the solution, a genetic algorithm (GA) is implemented. MG operation considers the resilience concept, and according to the obtained results, it is observed that the cost of operation and environmental pollution, respectively, experience an increase about 6.31% and 2.8%. However, due to the reduction in outage costs by an average of 13.91% and power losses by 0.5%, the overall cost is diminished about 5.93%. This cost reduction is achieved through increased CCHP generation and a decreased outage duration during emergencies.

1. Introduction

Renewable energy resources (RESs) have been widely utilized, and there have been international agreements aimed at expanding their integration into power systems in recent decades. The purpose of these agreements is to increase the share of RESs in the world [1,2,3,4,5]. Despite the numerous benefits of RESs in generating clean energy, their inherent uncertainties lead to the development of new technologies and energy management methods. To address these uncertainties, storage devices, demand response programs (DRPs), and reserve resources are employed in various studies. Additionally, the concept of MGs has emerged as a means to manage these uncertainties [6,7,8,9,10,11,12,13]. Among other advantages of MG development, it includes the enhancement of control and stability in distribution systems [14,15,16,17,18,19,20,21]. MGs consist of controllable loads, distributed generation (DG) resources, storages, and RESs, which manage energy and engage in power trading with the main grid to reduce the total cost of operation [22,23,24,25,26,27,28]. A comprehensive overview of the latest research is provided in [29] for MGs’ energy management. In this reference, the management of MGs’ power is classified into two approaches. The first approach focuses on minimizing MG operation costs by reducing generation expenses and purchasing cost-effectively from the retail market [30,31,32,33,34,35,36,37,38,39,40,41]. The second approach involves the integrated operation of MGs, referred to as the networked operation [42,43,44]. In [45], power losses and loading capacity are improved through the energy and structural management of MGs. Another study aims to optimize MGs to reduce losses, operational costs, and environmental pollution [46]. Reference [46] also addresses static voltage stability and considers uncertainties associated with load and RES generation. Optimal power flow is implemented in [47,48] to minimize the MG operational costs with Monte Carlo simulation (MCS), accounting for wind and load uncertainties. In [49], MGs’ power management is studied considering PV units to mitigate operational costs per year, power losses, and environmental impact. The uncertainties related to PV and WT generation, as well as loads, are addressed in this paper. In [50], MGs’ power management is conducted on a day-ahead basis, incorporating responsive loads for social welfare maximization. The issue of energy management in DC MGs with discrete-time analysis is explored in [51]. The power management in AC-DC hybrid MGs is also investigated in [52], where RESs and battery energy storage systems (BESSs) are employed. An online bi-level system is used in [53] for MG power management. Multi-energy MGs are studied in [54], in which energy management is addressed using the game-theoretic model. In [55], the reliability and resilience costs of MGs are minimized through energy management and grid reconfiguration. These options are also integrated in [56] to diminish the losses and voltage imbalances and enhance the voltage profile. The voltage stability in MGs is enhanced in [57] by utilizing these two tools. In [58], the energy management of MGs is achieved through incentive-based DRPs and grid reconfiguration. The optimal expansion of capacitors, along with energy management, is discussed in [59]. In [60], the optimum location and capacity of capacitors and DGs, as well as the optimal grid reconfiguration, is determined using the fuzzy method. A Pareto front-based GA is employed in [61] for sizing RESs and capacitors to reduce the operational, planning, and environmental costs while enhancing static voltage stability. Several papers delve into enhancing the resilience of MGs. A multi-objective method is presented in [62] for improving economic resilience in MG planning, employing a bi-level stochastic programming approach with MILP to enhance resiliency. In [63], the reliability and resilience indices are improved in MGs using DGs and storage systems from the MG owner’s perspective. In this research, loads are considered during the occurrence of LPHI events. A statistical approach is introduced in [64] to model the MG resilience and utilizes an optimization algorithm to control the MG during islanding events to enhance MG resilience under the islanded mode. Finally, a three-stage model is proposed in [65] to determine the optimum location and capacity of mobile storage for improving the resilience indices. The results demonstrate that resilience is improved through the use of mobile storage. In [66], a novel deep learning strategy is proposed to control an integrated wind–photovoltaic power system for improving generation efficiency and damping oscillations. The results indicated that a wind turbine can respond to sudden changes to maximize total power generation. References [67,68] analyze the energy management strategy of MGs from the perspective of economy and sustainable development, respectively. The role of MGs in energy transition is also studied, and the challenges and opportunities of MG energy management are put forward [69]. An MG energy management method is proposed in [70] based on multi-objective optimization, which can effectively improve the operational efficiency of MGs. A unit commitment optimization method is proposed in [71] for integrating solar photovoltaic plants in MGs. In [72], an economic dispatch is addressed for MGs with a mathematical programming approach. The results show that by including storage and DRPs, it is possible to save between 18% and 75% of the costs. Another energy management study on MGs was carried out by considering DRPs and uncertainties in [73].

Studies on the energy management of MGs can be classified into four categories [74,75,76,77,78,79,80,81,82,83]. These categories encompass various aspects, including types of problem data, both stochastic and deterministic; problem objectives covering both technical and economic goals; the employed devices, such as renewable and non-renewable units, storage, DRPs, and capacitors; and the methods used for solving the problems like metaheuristic and non-metaheuristic algorithms. Also, there are several methods that can be utilized in future studies [84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,103,104,105,106,107]. Despite the numerous papers published in the field of the energy management of MGs, there are still research gaps in this field. The main research gap in the field of energy management in CCHP-based MGs is the problem of the resource planning and operation of this type of MG, taking into account the issue of resilience, which is addressed in this paper, and its modeling is developed. In this paper, the first innovative contribution is the modeling of energy management for MGs with hybrid energy systems and capacitive banks to enhance resilience. The second innovation in this paper is the modeling of the problem from an economic perspective. The objective functions of resilience, losses, and environmental impact are economically modeled by considering the respective costs. As a third innovation, the solution of the MG energy management problem is considered using the Pareto front approach. The contributions are portrayed in the summary below:

• Modeling power management in hybrid MGs with capacitive banks to improve their resiliency;
• Modeling the MG energy management problem from an economic perspective, considering the respective costs;
• Modeling the problem by implementing a Pareto front-based multi-objective optimization strategy.

The hybrid system considered in this paper includes CCHP, WT, and PV units, along with thermal and battery energy storage systems (TESSs and BESSs), as well as an auxiliary boiler (AB). The discussed problem is a multi-objective optimization problem, managing the energy within MGs, which involves determining the optimal allocation of renewable and non-renewable resources, as well as managing the charge and discharge cycles of BESSs and TESSs and the AB’s generation at each hour. The objectives include reducing the operational costs, environmental pollution, power losses, and MG resilience costs. A model of the problem is presented in the second section in full detail. The third section explains the method employed to solve the problem. The case studies and analysis of the energy management results are provided in the fourth section. Finally, the fifth section gives the conclusions.

2. Problem Modeling

The hybrid system includes PV and WT units, CCHP, AB, BESS, and TESS. This system is depicted in Figure 1.

2.1. Objective Functions

The objectives include the minimization of resilience cost, operational cost, environmental pollution, and power losses within the MG. Each of these objectives are modeled economically, taking into account the associated costs as outlined below.

Improving reliability in power grids means reducing outages against the occurrence of incidents with a high probability of occurrence but with a low impact intensity, such as a short circuit in the network. This term does not meet the needs of the network in order to reduce outages against the occurrence of events with a low probability of occurrence but with a high impact intensity, such as earthquakes, floods, and storms. Therefore, with the aim of reducing outages during natural disasters, the term resilience was defined. Therefore, a reduction in outages against the occurrence of natural disasters and, consequently, a reduction in network costs considering the possibility of natural disasters in the network are studied under the title of resilience. Therefore, the cost of resilience is the same as the cost of outages during natural disasters. In other words, in this article, in order to improve the resilience of the network, energy management in microgrids is carried out in such a way that the costs of network outages are minimized when natural disasters occur in the network. In this article, we have four objective functions, which include the network cost function in normal conditions, the network cost function in the event of natural disasters, power losses, and the environmental objective function. These four objective functions are finally converted into one economic objective function by considering the weight coefficients. Therefore, the issue of security in energy supply is also discussed in this article in an economic way.

2.1.1. Resilience Enhancement

The resilience component of the objective function is modeled in (1) and (2), wherein it is modeled as the summation of energy purchase costs for supplying MG loads during emergency conditions, along with the outage costs. The value of $w_t$ in (1) is 1 if an emergency occurs; otherwise, it will be zero. $C_{Res}\left(t,s\right)$ in (2) represents the resilience costs at hour t and scenario s. The terms 1 and 2 in (2) signify the costs associated with WT and PV generations, respectively; term 3 is the cost of CCHP generation, the term 4 shows the cost of the capacitive bank operation, terms 5 and 6 are the BESS and TESS operation costs, respectively, and term 7 represents the outage cost during emergencies.

$$f_1\left(t,s\right)=w_t{C}_{Res}\left(t,s\right), \forall t\in T, \forall s\in S$$

(1)

$$\begin{array}{l}{C}_{Res}\left(t,s\right)={\displaystyle \sum _{i=1}^{N_{WT}}}{C}_{WT,c}\left(i,t,s\right)+{\displaystyle \sum _{i=1}^{N_{PV}}}{C}_{PV,c}\left(i,t,s\right)+{\displaystyle \sum _{i=1}^{N_{CCHP}}}{C}_{CCHP,c}\left(i,t,s\right)+{\displaystyle \sum _{l=1}^{N_{CB}}}{C}_{CB,c}\left(l,t,s\right)+{\displaystyle \sum _{i=1}^{N_{TESS}}}{C}_{TESS,c}\left(i,t,s\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{N_{BESS}}}{C}_{BESS,c}\left(i,t,s\right)+P_{bo}\left(t,s\right)\times {\rho }_{bo}, \forall t\in T, \forall s\in S\end{array}$$

(2)

2.1.2. MG Operational Cost Minimization

The MG operational costs include the energy cost traded with the upstream grid, the operational costs of BESSs and TESSs, the generation costs of RESs and CCHP, and the capacitors’ reactive power costs under normal operating conditions. The MG operation cost is formulated in (3).

$$\begin{array}{l}{f}_2\left(t,s\right)=C_{Grid}\left(t,s\right)+{\displaystyle \sum _{i=1}^{N_{WT}}}{C}_{WT}\left(i,t,s\right)+{\displaystyle \sum _{i=1}^{N_{PV}}}{C}_{PV}\left(i,t,s\right)+{\displaystyle \sum _{i=1}^{N_{CCHP}}}{C}_{CCHP}\left(i,t,s\right)+{\displaystyle \sum _{l=1}^{N_{CB}}}{C}_{CB}\left(l,t,s\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{N_{TESS}}}{C}_{TESS}\left(i,t,s\right)+{\displaystyle \sum _{i=1}^{N_{BESS}}}{C}_{BESS}\left(i,t,s\right), \forall t\in T, \forall s\in S\end{array}$$

(3)

The energy costs traded with the main grid can be modeled as (4). Additionally, the operation costs of WT, PV, and CCHP units can be shown as (5)–(7), respectively. Also, (8) and (9) describe the coefficients introduced in (5)–(7). The capacitor reactive power costs and its coefficients are presented in (10)–(12), respectively. The costs of BESSs and TESSs are then provided in (13) and (14), respectively. The respective coefficients are given in (15)–(18) [108].

$$C_{Grid}\left(t,s\right)=P_{Grid}\left(t,s\right)\times {\rho }_{Grid}\left(t\right), \forall t\in T, \forall s\in S$$

(4)

$$C_{WT}\left(i,t,s\right)=a_{WT}\left(i\right)+b_{WT}\left(i\right)\times P_{WT}\left(i,t,s\right), \forall t\in T, \forall s\in S, \forall i\in N_{WT}$$

(5)

$$C_{PV}\left(i,t,s\right)=a_{PV}\left(i\right)+b_{PV}\left(i\right)\times P_{PV}\left(i,t,s\right), \forall t\in T, \forall s\in S, \forall i\in N_{PV}$$

(6)

$$C_{CCHP}\left(i,t,s\right)=a_{CCHP}\left(i\right)+b_{CCHP}\left(i\right)\times \left(\frac{P_{CCHP}\left(i,t,s\right)}{{\eta }_{ele}}+\frac{H_{ab}\left(i,t,s\right)}{{\eta }_{the}}\right), \forall t\in T, \forall s\in S, \forall i\in N_{CCHP}$$

(7)

$$a_{CCHP}\left(i\right)=a_{WT}\left(i\right)=a_{PV}\left(i\right)=\frac{{Cost}_{Capital}^{CCHP}\times P_{Capacity}^{CCHP}\times Gr}{T_{Life}\times 365\times 24\times {CF}_{CCHP}\left(i\right)}, \forall i\in N_{CCHP}$$

(8)

$$\begin{array}{l}{b}_{CCHP}\left(i\right)={Cost}_{CCHP}^{O\&M}+{Cost}_{CCHP}^{Fuel}={Cost}_{CCHP}^{O\&M}+{\beta }_{gas}\times {\rho }_{gas}, \forall i\in N_{CCHP}\\ b_{WT}\left(i\right)={Cost}_{WT}^{O\&M}, \forall i\in N_{WT}\\ b_{PV}\left(i\right)={Cost}_{PV}^{O\&M}, \forall i\in N_{PV}\end{array}$$

(9)

$$C_{CB}\left(l,t,s\right)=a_{CB}\left(l\right)+b_{CB}\left(l\right)\times Q_{CB}\left(l,t,s\right), \forall t\in T, \forall s\in S, \forall l\in N_{CB}$$

(10)

$$a_{CB}\left(l\right)=\frac{{Cost}_{Capital}^{CB}\times P_{Capacity}^{CB}\times Gr}{T_{Life}\times 365\times 24\times {CF}_{CB}\left(l\right)}, \forall l\in N_{CB}$$

(11)

$$b_{CB}\left(l\right)={Cost}_{CB}^{O\&M}, \forall l\in N_{CB}$$

(12)

$$C_{BESS}\left(i,t,s\right)=a_{BESS}\left(i\right)+b_{BESS}\left(i\right)\times \left|P_{BESS}\left(i,t,s\right)\right|+{\rho }_{grid}\left(t\right)\times P_{BESS}\left(i,t,s\right), \forall t\in T, \forall s\in S, \forall i\in N_{BESS}$$

(13)

$$C_{TESS}\left(i,t,s\right)=a_{TESS}\left(i\right)+b_{TESS}\left(i\right)\times \left|H_{TESS}\left(i,t,s\right)\right|+{\rho }_{gas}\left(t\right)\times H_{TESS}\left(i,t,s\right), \forall t\in T, \forall s\in S, \forall i\in N_{TESS}$$

(14)

$$a_{BESS}\left(i\right)=\frac{{Cost}_{Capital}^{BESS}\times P_{Capacity}^{BESS}\times Gr}{T_{Life}\times 365\times 24\times {CF}_{BESS}\left(i\right)}, \forall i\in N_{BESS}$$

(15)

$$a_{TESS}\left(i\right)=\frac{{Cost}_{Capital}^{TESS}\times P_{Capacity}^{TESS}\times Gr}{T_{Life}\times 365\times 24\times {CF}_{TESS}\left(l\right)}, \forall i\in N_{TESS}$$

(16)

$$b_{BESS}\left(i\right)={Cost}_{BESS}^{O\&M}, \forall i\in N_{BESS}$$

(17)

$$b_{TESS}\left(i\right)={Cost}_{TESS}^{O\&M}, \forall i\in N_{TESS}$$

(18)

2.1.3. Minimization of Environmental Pollution

The total environmental pollution produced by the upstream grid and the CCHP can be formulated as (19) [61].

$$f_3\left(t,s\right)=P_{Grid}\left(t,s\right)\times {ER}_{Grid}+\left[\sum _{i=1}^{N_{CCHP}}\left(\frac{P_{CCHP}\left(i,t,s\right)}{{\eta }_{ele}}\right)+\frac{H_{ab}\left(i,t,s\right)}{{\eta }_{the}}\right]\times {CF}_{CCHP}\left(i\right)\times ER\left(i\right), \forall t\in T, \forall s\in S$$

(19)

2.1.4. Power Loss Minimization

The loss of power can be written as (20) [108].

$$f_4\left(t,s\right)=P_{loss}=\sum _{k=1}^{N_{BR}}R\left(k\right)\times {\left|I(k,t,s)\right|}^2, \forall t\in T, \forall s\in S$$

(20)

2.2. Final Objective Function

The objective function can be developed based on the summation of the four aforementioned objectives, which ultimately evaluate the objective functions. The final version can be obtained through the weighted summation of the four objectives (21). Since the nature of the first and second objectives is economic, they are included in (21) without coefficients.

$$f_T=\sum _{t=1}^{N_t}\sum _{s=1}^{N_s}{f}_1\left(t,s\right)+f_2\left(t,s\right)+{\rho }_3\times f_3\left(t,s\right)+{\rho }_4\times f_4\left(t,s\right)$$

(21)

2.3. Constraints

The constraints of the problem, including those associated with the grid and resources’ generated energy, are presented as follows.

2.3.1. Power Balance at Each Bus

The energy generated at each MG bus should equal the sum of the consumption and the energy transferred from that bus to other buses. These constraints are presented for active and reactive energy in (22) and (23), respectively [108].

$$\begin{array}{c}PG\left(z,t,s\right)-PD\left(z,t,s\right)=\sum _{r=1}^{N_{Bus}}\left|V\left(z,t,s\right)\right|\times \left|V\left(r,t,s\right)\right|\times \left|Y\left(z,j\right)\right|\times \cos \left(\delta \left(z,t,s\right)-\delta \left(r,t,s\right)-\phi \left(z,r,t,s\right)\right),\\ \forall t\in T, \forall s\in S, \forall z\in N_{Bus} for z\ne r\end{array}$$

(22)

$$\begin{array}{c}QG\left(z,t,s\right)-QD\left(z,t,s\right)=\sum _{r=1}^{N_{Bus}}\left|V\left(z,t,s\right)\right|\times \left|V\left(r,t,s\right)\right|\times \left|Y\left(z,j\right)\right|\times \sin \left(\delta \left(z,t,s\right)-\delta \left(r,t,s\right)-\phi \left(z,r,t,s\right)\right),\\ \forall t\in T, \forall s\in S, \forall z\in N_{Bus} for z\ne r\end{array}$$

(23)

2.3.2. Line Current Limit

The following constraint illustrates the current passing through the lines, which remains below the upper bound as defined in (24) [108].

$$\left|I(k,t,s)\right|\le I_k^{max}, \forall t\in T, \forall s\in S,\forall k\in N_{BR}$$

(24)

2.3.3. Bus Voltage Limit

The magnitude of the buses’ voltage should also fall in the permissible range, based on (25) [108].

$$V_{min}\le \left|V(z,t,s)\right|\le V_{max}, \forall t\in T, \forall s\in S,\forall z\in N_{Bus}$$

(25)

2.3.4. CCHP Generation Limit

The CCHP generation should fall within the allowable range, as expressed in (26) and (27), respectively [108].

$$P_{CCHP}^{min}\left(i\right)\le P_{CCHP}(i,t,s)\le P_{CCHP}^{max}\left(i\right), \forall t\in T, \forall s\in S,\forall i\in N_{CCHP}$$

(26)

$$H_{CCHP}^{min}\left(i\right)\le H_{CCHP}(i,t,s)\le H_{CCHP}^{max}\left(i\right), \forall t\in T, \forall s\in S,\forall i\in N_{CCHP}$$

(27)

2.3.5. AB Operation Limit

The heat power generated by the AB must fall within the permissible range based on (28) [108].

$$H_{AB}^{min}\left(i\right)\le H_{AB}(i,t,s)\le H_{AB}^{max}\left(i\right), \forall t\in T, \forall s\in S,\forall i\in N_{AB}$$

(28)

2.3.6. Reactive Power Limit of Capacitor

The limit of reactive power generation for a capacitive bank is also presented in (29) [108].

$$Q_{CB}^{min}\left(l\right)\le Q_{CB}(l,t,s)\le Q_{CB}^{max}\left(l\right), \forall t\in T, \forall s\in S,\forall l\in N_{CB}$$

(29)

2.3.7. BESS

The constraints related to BESSs are given in (30) to (34) [108].

$$0\le P_{BESS}^{ch}(i,t,s)\le P_{BESS}^{ch-max}\left(i\right)\times U_{BESS}^{ch}(i,t,s), \forall t\in T, \forall s\in S,\forall i\in N_{BESS}$$

(30)

$$0\le P_{BESS}^{dis}(i,t,s)\le P_{BESS}^{dis-max}\left(i\right)\times U_{BESS}^{dis}(i,t,s), \forall t\in T, \forall s\in S,\forall i\in N_{BESS}$$

(31)

$$U_{BESS}^{dis}\left(i,t,s\right)+U_{BESS}^{ch}(i,t,s)\le 1, \forall t\in T, \forall s\in S,\forall i\in N_{BESS}$$

(32)

$$E_{BESS}\left(i,t,s\right)=E_{BESS}\left(i,t-1,s\right)-P_{BESS}^{dis}\left(i,t,s\right)\times {\eta }_{BESS}^{dis}+\left(\frac{P_{BESS}^{ch}\left(i,t,s\right)}{{\eta }_{BESS}^{ch}}\right), \forall t\in T, t>1, \forall s\in S,\forall i\in N_{BESS}$$

(33)

$$E_{BESS}^{min}\left(i\right)\le E_{BESS}(i,t,s)\le E_{BESS}^{max}\left(i\right), \forall t\in T, \forall s\in S,\forall i\in N_{BESS}$$

(34)

2.3.8. TESS

The constraints associated with TESSs are presented in (35) to (39) [108].

$$0\le H_{TESS}^{ch}(i,t,s)\le H_{TESS}^{ch-max}\left(i\right)\times U_{TESS}^{ch}(i,t,s), \forall t\in T, \forall s\in S,\forall i\in N_{TESS}$$

(35)

$$0\le H_{TESS}^{dis}(i,t,s)\le H_{TESS}^{dis-max}\left(i\right)\times U_{TESS}^{dis}(i,t,s), \forall t\in T, \forall s\in S,\forall i\in N_{TESS}$$

(36)

$$U_{TESS}^{dis}\left(i,t,s\right)+U_{TESS}^{ch}(i,t,s)\le 1, \forall t\in T, \forall s\in S,\forall i\in N_{TESS}$$

(37)

$$E_{TESS}\left(i,t,s\right)=E_{TESS}\left(i,t-1,s\right)-H_{TESS}^{dis}\left(i,t,s\right)\times {\eta }_{TESS}^{dis}+\left(\frac{H_{TESS}^{ch}\left(i,t,s\right)}{{\eta }_{TESS}^{ch}}\right), \forall t\in T, t>1, \forall s\in S,\forall i\in N_{TESS}$$

(38)

$$E_{TESS}^{min}\left(i\right)\le E_{TESS}(i,t,s)\le E_{TESS}^{max}\left(i\right), \forall t\in T, \forall s\in S,\forall i\in N_{TESS}$$

(39)

2.3.9. Cooling and Heating Loads

Heat and cooling loads are supplied by CCHP, the AB, and the TESS. The constraints for heating and cooling loads are defined in (40) and (41) [108].

$$H_{CCHP}\left(i,t,s\right)=\frac{P_{CCHP}\left(i,t,s\right)}{{\eta }_{EH}}, \forall t\in T, \forall s\in S,\forall i\in N_{CCHP}$$

(40)

$$H_{CCHP}\left(i,t,s\right)+H_{ab}\left(i,t,s\right)+H_{TESS}^{dis}\left(i,t,s\right)-H_{TESS}^{ch}\left(i,t,s\right)=D^{heating}\left(t,s\right)+D^{cooling}\left(t,s\right), \forall t\in T, \forall s\in S,\forall i\in N_{CCHP}$$

(41)

2.3.10. Sensitive Loads

During emergencies, sensitive loads of MGs, like those in hospitals and security systems, must be supplied. The corresponding constraint is outlined in (42) [108].

$$\begin{array}{l}{\displaystyle \sum _{i=1}^{N_{WT}}}{C}_{WT,c}\left(i,t,s\right)+{\displaystyle \sum _{i=1}^{N_{PV}}}{C}_{PV,c}\left(i,t,s\right)+{\displaystyle \sum _{i=1}^{N_{CCHP}}}{C}_{CCHP,c}\left(i,t,s\right)+{\displaystyle \sum _{l=1}^{N_{CB}}}{C}_{CB,c}\left(l,t,s\right)+{\displaystyle \sum _{i=1}^{N_{TESS}}}{C}_{TESS,c}\left(i,t,s\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^{N_{BESS}}}{C}_{BESS,c}\left(i,t,s\right)\ge \gamma \times PD\left(z,t,s\right), \forall t, \forall s, \forall z\end{array}$$

(42)

2.4. Modeling of Wind and Solar Systems’ Generation

The powers produced by WT and PV systems are calculated in (43) and (44), respectively [108].

$$P_w\left(v\right)=\left\{\begin{array}{c}\begin{array}{c}\begin{array}{cc}0,& 0\le v\le v_{ct}\end{array}\\ \begin{array}{cc}{P}_{rated}\times \frac{\left(v^2-v_{ct}^2\right)}{\left(v_r^2-v_{ct}^2\right)},& v_{ct}\le v\le v_r\end{array}\end{array}\\ \begin{array}{cc}{P}_{rated},& v_r\le v\le v_{co}\end{array}\\ \begin{array}{cc}0,& v_{co}\le v\end{array}\end{array}\right.$$

(43)

$$P_{PV}\left(si\right)={\eta }^{PV}\times S^{PV}\times si, \forall t\in T, \forall s\in S,\forall i\in N_{CCHP}$$

(44)

After fully developing the formulations for the modeling problem, the solving strategy for the problem is described in the following section.

3. Problem-Solving Method

The developed model is the power management model for CCHP-based MGs, focusing on economic and resilience improvement. The studied MG has a hybrid energy system including PV and WT units, a BESS, an AB, CCHP, and a TESS. The model presented here is approached from the perspective of the microgrid owner and aims to minimize resilience costs, operational costs, environmental pollution, and power losses.

3.1. Scenario Generation

A scenario generation approach is applied here for modelling the uncertainty. The first variable for which the uncertainty is modeled using this method is the generation of the WT unit, which can be considered using the Weibull distribution function. The other two variables, the generation of the PV unit and the MG’s load, are modeled using the normal distribution function.

3.2. Problem-Solving Flowchart

The GA is employed to tackle the optimization problem in this study. To this end, the objective function expressed as (21) is minimized when the constraints expressed in (22)–(42) are considered. The corresponding flowchart is depicted in Figure 2. The problem-solving process is comprehensively outlined below:

Step 1: Input grid data, generation, demand, prices, and GA initial data.

Step 2: Produce the initial population. In this step, members of the initial population are randomly generated, as specified in step 1.

Step 3: Consider variables’ uncertainty based on the scenarios, run power flow calculations, and compute objectives for each scenario. In this step, all possible scenarios are examined by considering the uncertainties.

Step 4: Calculate the mean value of the objective for each member. As mentioned, several scenarios are considered for each population member. Therefore, all the objectives of the considered scenarios must be averaged to compute the objective of each member. Naturally, by averaging the objective functions of each member for all scenarios, the objective function of that member is improved on average, while many of the considered scenarios may not occur in the desired time. This issue causes the average value of the expected objective function to not occur. It depends on how the risk is managed. In the averaging approach, the minimum and maximum value of the objective function is ignored, while the minimum and maximum values are very important because these minimum and maximum values may ultimately determine the costs of the problem. In more recent studies, the conditional value at risk (CVaR) index is used to manage risk and consider uncertainties, which has a better capability than other indices. In this article, for the sake of simplification, the averaging approach is used.

Step 5: Prioritize the members based on their values. Then, select 50% of the members as parents of the next generation,

Step 6: Apply GA operators to parents and generate offsprings. Since half of the members of the original GA population are removed due to fitness selection, offspring must be generated to maintain the population size. This process of offspring generation is realized using crossover and mutation operators.

Step 7: Repeat steps three to seven until the stopping criterion is reached.

Step 8: Select the best objective function as the optimum solution.

In the subsequent sections of the paper, a case study is considered. In this section, the data of the problem, as well as simulation results, are delineated. Eventually, an analysis of the simulation results is presented.

4. Numerical Study

As mentioned, the understudied MG consists of a hybrid energy system including PV and WT units, a BESS, an AB, CCHP, and a TESS. The developed model minimizes operational cost, resilience cost, environmental pollution, and power losses from the MG owner’s viewpoint. The data for the problem are given in the next section.

4.1. Problem Data

A scenario generation method is employed for modeling the uncertainty variables. The first uncertainty variable that is modeled using this method is the generation of the WT unit, which is characterized using the Weibull distribution function. The other two variables are the generation of the solar system along with the MG’s load; both are considered based on the normal distribution function. The last variable is the occurrence of an emergency condition, which has two occurrence and non-occurrence modes. The considered MG is depicted in Figure 3. The MG’s rated voltage equals 12.66 kV, and its real and reactive demands are 3715 kW + j 2300 kVAr. Three capacitors are placed in buses 7, 13, and 29, and the hybrid energy systems are connected to buses 4 and 14. The peak hour load quantities are given in Figure 4 and the daily load percentages compared to the peak load are presented in Figure 5. It should be noted that the electrical load uncertainty in the grid is considered using the normal distribution function. Figure 6 and Figure 7 display the quantities of the heating and cooling loads connected to buses 4 and 14, as well as the electricity price, respectively. The coefficients considered for power loss and environmental pollution in the main objective function are assumed to be 1.5 USD/kW and 10 USD/kg, respectively. Additionally, the probability of an emergency occurrence is 1%, and sensitive loads constitute 30% of the total. The outage cost is also considered as 30 USD/kWh. Further details and data are presented in

Table 1. Specifications of hybrid energy system’s equipment [108].

CCHP		AB		WT
Size	Parameter	Size	Parameter	Size	Parameter
0	$P_{CCHP}^{min}$ (kW)	0	$H_{AB}^{min}$ (kW)	1500	${Cost}_{Capital}^{WT}$ (USD/kW)
633	$P_{CCHP}^{max}$ (kW)	350	$H_{AB}^{max}$ (kW)	400	$P_{Capacity}^{WT}$ (kW)
0	$H_{CCHP}^{min}$ (kW)	0.8	${\eta }_{the}$ (%)	0.13	$Gr$ (Percent/year)
700	$H_{CCHP}^{max}$ (kW)	BESS		0.2	${CF}_{WT}$ (%)
0.09	${\beta }_{gas}$ (m3/kWh)	Size	Parameter	20	$T_{Life}$ (year)
0.15	${\rho }_{gas}$ (USD/m3)	1775	${Cost}_{Capital}^{BESS}$ (USD/kW)	0.05	${Cost}_{WT}^{O\&M}$(USD/kWh)
0.3	${\eta }_{ele}$ (%)	1200	$P_{Capacity}^{BESS}$ (kW)	400	$P_{rated}$ (kW)
3674	${Cost}_{Capital}^{CCHP}$ (USD/kW)	200	$P_{BESS}^{ch-max}$ (kW)	12	$v_r$ (m/s)
633	$P_{Capacity}^{CCHP}$ (kW)	200	$P_{BESS}^{dis-max}$ (kW)	25	$v_{co}$ (m/s)
0.13	$Gr$ (Percent/year)	0.25	${CF}_{BESS}$ (%)	3.5	$v_{ct}$ (m/s)
0.2	${CF}_{CCHP}$ (%)	120	$E_{BESS}^{min}$ (kWh)	PV panel
10	$T_{Life}$ (year)	1200	$E_{BESS}^{max}$ (kWh)	Size	Parameter
0.0039	${Cost}_{CCHP}^{Fuel}$ (USD/kWh)	0.85	${\eta }_{BESS}^{ch}$ (%)	6675	${Cost}_{Capital}^{PV}$ (USD/kW)
0.9	${\eta }_{EH}$ (%)	0.95	${\eta }_{BESS}^{dis}$ (%)	400	$P_{Capacity}^{PV}$(kW)
14.45	ER (kg/kWh)	25	$T_{Life}$ (year)	0.13	$Gr$ (Percent/year)
TESS		0.13	$Gr$ (Percent/year)	0.25	${CF}_{PV}$ (%)
		0.05	${Cost}_{BESS}^{O\&M}$ (USD/kWh)	20	$T_{Life}$ (year)
Size	Parameter	CB		0.05	${Cost}_{PV}^{O\&M}$ (USD/kWh)
1800	${Cost}_{Capital}^{TESS}$ (USD/kW)	Size	Parameter	18.6	${\eta }^{PV}$ (%)
1200	$P_{Capacity}^{TESS}$ (kW)	9	${Cost}_{Capital}^{CB}$ (USD/kW)	40	$S^{PV}$ (m2)
0.95	${\eta }_{TESS}^{ch}$ (%)	400	$P_{Capacity}^{CB}$ (kW)	Grid
0.95	${\eta }_{TESS}^{dis}$ (%)	0.13	$Gr$ (Percent/year)	Size	Parameter
120	$E_{TESS}^{min}$ (kWh)	0.2	${CF}_{CB}$ (%)	0.95	$V_{min}$ (v)
1200	$E_{TESS}^{max}$ (kWh)	25	$T_{Life}$ (year)	1.05	$V_{max}$ (v)
0.25	${CF}_{TESS}$ (%)	0.05	${Cost}_{CB}^{O\&M}$ (USD/kWh)	5.46	${ER}_{Grid}$ (kg/kWh)
200	$H_{TESS}^{ch-max}$ (kW)
200	$H_{TESS}^{dis-max}$ (kW)
25	$T_{Life}$ (year)
0.05	${Cost}_{TESS}^{O\&M}$ (USD/kWh)
0.13	$Gr$ (Percent/year)

and

Table 2. GA parameters.

Size	Parameter	Size	Parameter
60	Number of members	1	Probability of crossover
100	Number of iterations	0.04	Probability of mutation

.

4.2. Numerical Results

To investigate the problem addressed in the paper, two cases are conducted; the former is tackled irrespective of the resilience, while the later takes the resilience into account. The results are contrasted to highlight the significance of resilient operations.

4.2.1. Case 1: Optimum Management of MG Regardless of Resiliency

In this section, the MG operator minimizes its objective function without considering the cost of outage and resilience. Consequently, it may experience outages during emergencies. In this scenario, although the MG owner’s perspective does not consider outage costs, the imposed costs are still calculated.

Table 3. Case 1: simulation results.

Hour Mean Values of Variables					Hour
Objective Function (USD)	Power Loss (kW)	Pollution (kg)	Outage Cost (USD)	Operating Cost (USD)
1482.7	22.46	1.89	1229.4	200.8	1
1483.83	23.12	1.85	1229.4	200.8	2
1516.7	22.12	1.85	1259.4	205.7	3
1516.7	22.12	1.85	1259.4	205.7	4
1788.3	29.63	1.95	1482.3	242.1	5
1747	29.87	1.97	1446.3	236.2	6
1692.4	34.99	2.32	1389.7	226.9	7
1678	34.84	2.30	1377.7	225	8
2116.9	45.37	2.65	1258.6	763.7	9
2448.5	45.37	2.65	1114.6	1239.3	10
2645.7	52.12	2.85	1208.1	1330.9	11
2611	52.65	2.87	1190.1	1313.2	12
2535.8	66.89	3.12	1515	889.3	13
3004	66.79	3.04	1377	1496.4	14
1749.9	39.94	2.33	919.2	747.5	15
1829.3	39.94	2.33	967.2	778.9	16
1470.3	30.10	2.01	948.3	456.7	17
1481.3	30.43	2.02	1002.3	413.1	18
2160.6	45.11	2.52	1558.6	509.1	19
1878.1	45.19	2.54	1234.6	550.3	20
1511.6	31.12	2.12	870.3	573.4	21
1533.7	33.87	2.21	990.3	470.4	22
1637.1	30.63	2.04	1350.3	220.5	23
1634.3	29.63	1.91	1350.3	220.5	24
300–150–220			Capacity of Capacitors (kVAr)

provides the simulation results, including the reactive power production by the three capacitive banks and the objective function values. Furthermore, the resource outputs are illustrated in Figure 8 and Figure 9. The remaining energy requirements are supplied by the upstream system. Here, the value of the objective function equals 45,154.4. The convergence curve for case 1 is shown in Figure 10.

Table 4. Evaluating the change in GA parameters on the results of case 1.

Iteration of Convergence	Objective Function	Condition Change	Sample
44	45,154.4	-	Base
25	45,154.4	Increase the members to 120	1
44	45,154.4	Increase the iterations to 200	2
51	45,157.2	Increase in mutation probability to 0.08	3
34	45,155.9	Decrease in crossover probability to 0.9	4

shows the effect of changing the GA parameters on the results of the simulations in case 1. As can be seen, with the increase in the number of members of the initial population in sample 1, the value of the objective function does not change, but the simulation converges earlier due to the increase in the number of members of the population. In sample 2, only the number of iterations increased, which did not affect the value of the objective function and the speed of convergence. In sample 3, the probability of the mutation operator increased, which caused the deterioration of the objective function. Also, in sample 4, the probability of the crossover operator decreased, which has also caused the deterioration of the objective function.

4.2.2. Case 2: Optimal Management of Power in an MG with a CCHP, Taking Resilience into Account

The minimization of the resilience cost is considered in this section in addition to the primary objectives of minimizing operational costs, environmental pollution, and power losses. Hence, the operator manages its energy by taking into account the occurrence of emergencies to achieve resilience cost reduction. The high outage cost compels the MG operator to make every effort to meet the demand of consumers during emergencies.

Table 5. The results of simulation in case 2.

Hour Variables’ Mean Values					Hour
Objective Function (USD)	Power Loss (kW)	Pollution (kg)	Outage Cost (USD)	Operating Cost (USD)
1075.5	22.22	2.34	749.4	269.4	1
1075.6	23.01	2.23	749.4	269.4	2
1112	22.02	2.53	779.4	274.3	3
1108.8	22.02	2.21	779.4	274.3	4
1377.3	29.43	2.02	1002.3	310.7	5
1405.4	29.55	2.02	1026.3	314.6	6
1423.6	34.67	2.67	1029.7	315.1	7
1410.2	34.45	2.76	1017.7	313.2	8
2117.9	45.21	2.77	1258.6	763.7	9
2683.7	45.17	2.44	1234.6	1356.9	10
2876.9	52.02	2.23	1328.1	1448.5	11
2841.9	52.43	2.23	1310.1	1430.8	12
2526.4	66.67	2.21	1515	889.3	13
3231.2	66.66	2.02	1497	1614	14
1945.8	39.77	2.11	1039.2	825.9	15
1926.2	39.76	2.13	1027.2	818.1	16
1315.9	30.01	2.51	828.3	417.5	17
1348.1	30.21	2.51	852.3	425.4	18
1829.8	45.02	2.51	1198.6	538.5	19
1875.9	45.04	2.34	1234.6	550.3	20
1696.6	31.01	2.76	990.3	632.2	21
1535.7	33.65	2.45	990.3	470.4	22
1368.9	30.33	2.44	990.3	308.7	23
1366.4	29.33	2.34	990.3	308.7	24
300–150–220			Capacity of Capacitors (kVAr)

presents the simulation results for the studied case, including the reactive power produced by the three capacitor banks, the costs, and the objective values. The resource outputs are demonstrated in Figure 11 and Figure 12. In this case, the objective value is 42,476.8, representing a 5.93% improvement compared to the previous case. The convergence curve for case 2 can be observed in Figure 13.

Table 6. Evaluating the change in GA parameters on the results of case 2.

Iteration of Convergence	Objective Function	Condition Change	Sample
39	42,476.8	-	Base
31	42,476.8	Increase the members to 120	1
39	42,476.8	Increase the iterations to 200	2
47	42,479.2	Increase in mutation probability to 0.08	3
53	42,481.5	Decrease in crossover probability to 0.9	4

shows the effect of changing the GA parameters on the results of the simulations in case 2. As can be seen in this table, changes similar to Table 4 occurred. As can be seen, with the increase in the number of members of the initial population in sample 1, the value of the objective function did not change, but the simulation converged earlier due to the increase in the number of members of the population. In sample 2, only the number of iterations increased, which did not affect the value of the objective function and the speed of convergence. In sample 3, the probability of the mutation operator increased, which caused the deterioration of the objective function. Also, in sample 4, the probability of the crossover operator decreased, which caused the deterioration of the objective function.

4.2.3. Case 3: Multi-Objective Optimal MG Management, Taking the Resilience into Account

In the previous sections, a single objective is considered to address the problem. In other words, the objective functions, including operation and outage costs, environmental pollution, and power losses, are defined as objective functions by considering the weighting factors. However, considering the estimated weighting factors and converting the technical objectives into economic ones can reduce the accuracy of the results. Therefore, the optimization problem is considered here as a three-objective problem using a GA based on the Pareto front. When solving the optimization problem using Pareto front-based methods, a set of points are identified as optimal solutions. Each of these points has three values representing the objective functions, including operation and outage costs, power losses, and environmental pollution. As not all objective functions expressed in points are superior to those expressed in another point, all of these points are regarded as optimal solutions and are available to the MG’s operator as options. Now, the MG operator can make a decision about how to operate the MG according to the objective function values associated with each of these points. These optimal points are given in Figure 14 and detailed in

Table 7. Case 3: simulation results.

Power Loss (kW)	Pollution (kg)	Operating and Outage Cost (USD)	Point Number
898.55	56.93	40,552.34	1
899.08	56.81	40,554.76	2
899.66	56.78	40,558.12	3
899.98	56.43	40,561.31	4
900.43	56.12	40,563.22	5

.

It is evident that the optimal point, number 3, equals the solution for problem case 2. It can be observed that the three-objective optimization method based on the Pareto front outperforms case 2 in terms of objective function values. Specifically, this method attains a value of 40,552.34 for the operation and outage cost function at point 1. Also, at point 1, a value of 898.55 is reached for the power loss objective function. Furthermore, in point 5, a value of 56.12 is obtained for the objective function of environmental pollution, and all of these values are better than their corresponding values in case 2.

4.2.4. Results Analysis

According to Table 3 and Table 5, it is observed that, in both cases, the costs increase during peak hours. To clarify, as the MG load increases, operational and outage costs, power losses, and environmental pollution also rise. Moreover, according to Figure 8 and Figure 11, the electrical energy is stored during cheaper hours and discharged during more expensive hours. The WT and PV resources generate energy when wind and solar energy are available. It is obvious from Figure 9 and Figure 12 that the heat is stored during less expensive hours and discharged during expensive hours to mitigate costs. The CCHP also generates electricity and heat simultaneously. The total energy generated by the CCHP is depicted in Figure 9 and Figure 12. In the second case, there is a higher generation of this resource to reduce outages in emergency situations compared to the first case.

By comparing the results of these two cases, the effect of considering outages in MG operation is clarified. According to the data presented in Table 3, the overall operational and outage costs for case 1 are 13,716 and 29,528, respectively. Similarly, according to the results found in Table 5, the overall operating and outage costs for case 2 are 15,140 and 25,418, respectively. As observed, considering the resilience concept in the problem discussed in this paper leads to a 6.31% increase in the operational costs, while there is a 13.91% decrease in outage costs. Consequently, this results in a 5.93% reduction in the problem’s objective function. This cost reduction is achieved through increased CCHP generation and reduced outages during an emergency condition. Furthermore, by considering the concept of resilience in the problem, power is decreased by 0.5%, while environmental pollution is increased by 2.8%.

Finally, based on the results from Table 7, the optimal values for the objective functions were obtained in case 3 through the three-objective optimization method utilizing the Pareto front. This method achieves a value of 40,552.34 for the operational and outage cost functions, 898.55 for power loss, and 56.12 for the environmental pollution objective function. These values indicate an improvement of 0.02%, 0.12%, and 1.16% compared to case 2, respectively.

5. Conclusions

The present paper discussed the stochastic multi-objective optimal management of power in a CCHP-based MG with an economic and resilience enhancement approach. Three case studies were undertaken to conduct simulations, and the results are presented as follows:

• Considering the resilience improvement approach in the MG’s operation reduced the MG operator’s reliance on the main grid. This approach led to increased CCHP generation, resulting in higher operational costs due to the resource’s higher cost compared to purchasing from the main grid. Regarding the results of this study, shown in Figure 9 and Figure 12, the amount of energy generated via CCHP doubled in the studied time, taking into account the issue of resilience in the operation of the MG. The reason for this is the desire of the MG operator to rely less on purchasing from the upstream network due to the possibility of natural disasters and the impossibility of purchasing from the upstream network,
• The increased CCHP generation resulted in fewer outages during MG emergencies, leading to a reduction in outage costs for the MG. As the reduction in outage costs outweighed the increase in operational costs, the MG operator’s final objective improved. As indicated in Table 3 and Table 5, considering the issue of resilience in MG operation, the total cost of MG operation increased by 10.38%. The reason for this increase in costs was the use of more resources at the disposal of the MG, which cost more than buying from the upstream network. On the other hand, by using the resources at the disposal of the MG, the MG suffered fewer outages during the occurrence of natural disasters and, as a result, fewer outage costs, so its resilience costs were reduced by 13.92%. Therefore, applying the approach of resilient operation of the MG reduced the total cost of operation and resilience of the MG by 6.2%,
• Solving the problem using a multi-objective optimization approach yielded better values for the objective functions. This approach improved the objective function values related to operation and outage costs, losses, and environmental pollution by 0.02%, 0.12%, and 1.16%, respectively.

According to the results obtained from this study, it can be recommended to MGs similar to the MG studied in this paper that if there is a possibility of disconnection to the upstream network due to the occurrence of various incidents, always have available resources ready to work; even if their use causes a slight increase in costs, use them at the minimum amount of generation, so that when accidents occur, the source in question can be used quickly and with the least interruption to reduce the amount of outages and avoid additional costs. Naturally, a compromise must be made between the use of internal resources and reliance on the use of the upstream network, which requires study and experience in connection with the studied MGs.

Since energy management within MGs remains a highly practical subject, there is still a need for further studies in this area. Therefore, to facilitate future research in this field, the following topics are recommended:

• In this study, power management was conducted for a single MG. Given the numerous advantages associated with networked MGs, it is suggested that the energy management of networked MGs will be explored by considering the resilience concept.
• Another critical aspect of interest in energy management is the consideration of resource flexibility. Given its paramount importance, especially in the context of operating renewable resources, it is suggested that the problem addressed in this study should be investigated by considering resource flexibility. Considering the flexibility of resources improves generation expansion planning and resource operation. It is possible that in normal or contingency conditions, sufficient resources will be available in terms of capacity, but these resources do not have a sufficient ramp rate to increase or decrease their generation in the limited time they have. This issue will increase outages in the network and increase total costs. If the flexibility of the resources is considered in studies related to the planning, operation, and resilience of the network, the available resources will have the appropriate ramp rate at the time of need, and with the appropriate response at the considered time, network outages will be prevented and will reduce the total network costs.
• In this study, the issue of resilience is raised in a general way. In other words, the issue of resilience can be raised in order to prevent outages and crises during natural disasters such as floods, earthquakes, and storms, or during human sabotage such as cyber-attacks. Naturally, with the growth of the use of automation, telecommunications, and especially the Internet in power grids and MGs, as well as the development of digital loads, the possibility of cyber-attacks and outages and crises arising from them increases. Therefore, it is very valuable that, in the continuation of this study, the issue of energy management in MGs in order to improve resilience against cyber-attacks is also studied and measures are taken to reduce outages.