Optimization of Generation Cost, Environmental Impact, and Reliability of a Microgrid Using Non-dominated Sorting Genetic Algorithm-II

The world is facing the ever-increasing risk of the depletion of non-renewable energy sources at an alarming rate. Additionally, the heavy dependency on fossil-fuel-based energy generation is causing excessive environmental pollution. Therefore, the transition from conventional fossil-fuel-based energy to renewable energy resources is a must to ensure a sustainable future. Microgrids are one of the key tools of this sustainable energy transition process since it provides significant advantages such as lower transmission losses and efficient renewable energy source (RES) integration capability. However, the delivered power through the microgrids needs to be cost-effective and at the same time, the environmental impacts need to be checked. Additionally, the grid-system has to be highly reliable, and this is a very big concern especially for the islanding microgrids. To meet the power demand with the least possible cost, lowest possible environmental impact and higher system reliability, an optimal dispatch system must be modeled based on estimated trade-offs.

Microgrids can ameliorate the quality, reliability, and efficiency of a power system; and it can also create an opportunity to provide grid-independence. According to the U.S. department of energy (DOE), a microgrid can be defined as “a batch of distributed energy resources (DERs) and coordinated loads which maintains all the properties of an electrical system that resembles a controllable single unit with respect to the main power grid”. A microgrid has the ability to connect and disconnect from the main grid enabling it to operate in both grid-connected and islanded mode [1]. To deal with the economic dispatch problem, an efficient energy management system (EMS) is required. The EMS optimizes, monitors and co-ordinates the energy sources connected to the microgrid. In the past decades, several approaches have been proposed regarding EMS as well as the co-ordination between different renewable energy sources connected to microgrids. On the other hand, a number of efforts have also been made for effective generation scheduling in order to minimize the cost and pollution caused by the generator for achieving the optimal eco-friendly operation for the microgrids.

Nikmehr and Najafi-Ravadanegh [2] have reported optimal power dispatching technique based on probabilistic analysis considering technical constraints and economic aspects, and it is solved by applying a heuristic algorithm called imperialist competitive algorithm. Monte carlo simulation (MCS) is also applied to compare the obtained results. Bayat et al. [3] have optimized the operation cost using multi objective particle swarm optimization (MOPSO) in grid connected mode considering the voltage constraints. For intelligent energy generation scheduling, a cost minimization optimization problem is formulated and solved by Wang et al. [4] using robust optimization and chance constraints approximations approaches. Hosseinnezhad et al. [5] have proposed an optimal power scheduling technique in order to minimize the operation cost and emission from the generators by applying species-based quantum particle swarm optimization (SQPSO). Gupta and Gupta [6] have proposed a robust optimization approach for optimal management of a microgrid taking wind power uncertainty into consideration. Askarzadeh [7] has used memory-based genetic algorithm (MGA) to solve the best economic DERs dispatching problem as a single objective optimization and compared his result with Genetic Algorithm (GA), and two variants of Particle Swarm Optimization: PSO with inertia weight (PSO_w), PSO with constriction factor (PSO_cf).

Multi-objective PSO (MOPSO) technique can be used to develop a solution for the optimal power flow (OPF) problem of medium electrical systems [8]. Improved harmony search (EHS) algorithm can also be applied to resolve the economic dispatch problem [9]. Sundaram et al. [10] have implemented a synergistic hybrid metaheuristic algorithm merging the Nondominated sorting genetic algorithm-II (NSGA- II) and multi-objective PSO (MOPSO) algorithm in order to solve the combined heat and power economic emission dispatch problem. Zhao et al. [11] presented a dynamic economic dispatch model of a microgrid which uses a variant of NSGA-II algorithm with an external penalty function for dealing with the constraint conditions. An improved NSGA-II method is adopted for planning the production of thermal unit for minimizing the cost and emissions [12].

Akram et al. [13] have proposed a methodology to find an optimal combination of hybrid renewable power generation system and energy storage for grid connected microgrids. The modeled hybrid generation system is comprised of solar photovoltaic (PV) and wind turbine (WT), while the hybrid energy storage system (HESS) includes the battery energy storage (BES) system and supercapacitor (SC) technology [13]. To minimize the generation cost of the microgrid, a reduced gradient algorithm (RGA) is applied on three generating units considering load demand as a constraint [14]. An algorithm based on dynamic programming (DP) has been proposed to minimize the cost and emission of CO₂ as a form of cost [15]. Liu et al. [16] proposed that an optimal day-ahead bidding strategy can be used for finding the lowest possible net expected cost based on a hybrid robust or stochastic algorithm. Based on a robust optimization technique taking islanding capability as a constraint and cost minimization as a single objective, an economic dispatching model for a microgrid can be implemented [17].

For dealing with the reliability analysis, probabilistic techniques are extensively used. Billinton et al. [18] have presented composite DC power system adequacy assessment with monte carlo simulation (MCS). Billinton and Jonnavithula [19] have presented two variance-reduction techniques to improve the performance of sequential monte carlo simulation. The enhanced model is then applied for the adequacy analysis of a six-bus test system and a 24-bus system. It was suggested that system adequacy assessment of a composite system can also be evaluated by using a procedure called contingency evaluation instead of monte carlo simulation (MCS) [20]. A system reliability assessment based on spot pricing has been propose by Goel et al. [21]. Billinton and Wenyuan have evaluated the reliability of bulk composite system with the help of monte carlo simulation [22]. Henneaux et al. [23] have estimated the confidence intervals (CI) of reliability indices using monte carlo simulation method. 

In this paper, we have applied the non-dominated sorting genetic algorithm-II (NSGA-II) for multi-objective optimization by adding a nonlinear and non-convex objective for green-house-gas (GHG) emission, generation cost and system reliability. The proposed non-dominated sorting genetic algorithm-II (NSGA-II) is designed using MATLAB simulation software. The designed algorithm is then applied for multi-objective optimization purposes to optimize cost, emission and reliability by taking them in pairs, considering double objective functions at a time. The simulation results provide trade-offs between fuel cost Vs emission; fuel cost Vs reliability; and emission Vs reliability. In order to compare and verify the effectiveness of our proposed model, the simulation results of the energy dispatch, cost and emission quantities are compared with relevant parameters of another model which uses the strength pareto evolutionary algorithm (SPEA). The comparison between NSGA- II and SPEA model, verifies that the NSGA- II performes efficiently. After verifiying the performance of the proposed model, NSGA- II algorithm is applied to find out the optimized values of cost and emission but at this time the reliability factor is considered as a constraint.

The structure of this paper is organized as follows: section 2 presents the system description, section 3 demonstrates the problem statement, section 4 presents the solution methodology, section 5 shows the simulation results and lastly the conclusion of this study is presented in section 6.

3.1 Problem objective

An economic, environment friendly and reliable power dispatching problem has three objective functions, one is minimizing fuel cost, second one is minimizing the emission from the generators and the third one is minimizing the unserved energy (EENS) or maximizing the reliability.

The electricity generation schemes largely depend on its cost. For minimizing the cost of the studied microgrid, a common quadratic cost function has been considered. This cost function FC(P_g) can be expressed as follows [24]:

$F C\left(P_{g}\right)=C_{g i}\left(P_{g i}\right)=\alpha_{g i} P_{g i}^{2}+\beta_{g i} P_{g i}+\gamma_{g i}$       (1)

In Eq. (1), C denotes the cost per hour in USD($), gi implies the i th distributed generator, P denotes the real power generated by a distributed generator in MW, and α, β and γ are the cost coefficients which are listed in Table I.

The environmental pollutant gases emitted from fossil-fuel-run conventional generators have been modeled separately in [25]. The emission can be expressed in ton/hr to detect the amount of the pollutant gases emitted from the generators. The total emission function E(P_g) can be written as follows [25],

$E\left(P_{g}\right)=\sum_{i=1}^{N G}\left[10^{-2}\left(a_{i}+b_{i} P_{g i}+c_{i} P_{g i}^{2}\right)+d_{i} \exp \left(e_{i} P_{g i}\right)\right]$    (2)

where, NG is the number of generators; P_g refers to the real power output vector consists of all the generator outputs such that $P_{g}=\left[P_{g 1, P_{g 2}, P_{g 3, \ldots \ldots,}} P_{g N G}\right]$; $P_{g i}$ represents the real power generated from i-th generator; and $a_{i}, b, c i, d i, e_{i}$ are the emission coefficients of the i-th generator listed in Table 1 [25].

The reliability evaluation algorithm calculates the total curtailed power with the help of power flow algorithm and finds the total amount of unserved power of the system. Finally, it calculates the expected energy not served (EENS) by using the simplified equation of the reliability evaluation function R(P_g) [26],

$R\left(P_{g}\right)=E E N S\left(P_{g}\right)=\sum_{i \in S} C_{i} F_{i} D_{i}=\sum_{i \in S} 8760 C_{i} p_{i}$

$E D N S=\sum_{i \in S} C_{i} p_{i}$      (3)

where, S is the set of all system states associated with loss of load; C_i is the curtailment of load for ith system state; F_i represents the frequency of departing system state i, p_i is the duration of system state i; and p_i is the probability of the ith system state. In Eq. (3) EDNS stands for expected demand not supplied (MW).

3.2 Objective constraints

Constraint of power balance: The power balance constraint is a very important constraint for the minimization cost problem. That is because if the dispatching algorithm does not meet this constraint, it can generate power more than the demanded power which will end in higher fuel cost and higher emission of toxic gas. Likewise, it can also generate power that is lower than the demand if this constraint is not satisfied. Both these conditions inrrupts the economic power dispatch process. That is why, the total power generation must be equal to the total power load demand P_d and the real power loss P_Loss, for the power transmission. P_Loss is assumed to be zero here. Therefore, the resulting power balance equation is shown in Eq. (4),

$\sum_{i=1}^{N G} P_{g i}-P_{d}-P_{\text {Loss}}=0$      (4)

Constraint of generation capacity: The real power output from each generator is confined by a lower limit and an upper limit for the stable operation of the generators as follows:

$P_{g i}^{\max } \geq P_{g i} \geq P_{g i}^{\min }, i=1,2, \ldots \ldots, N G$      (5)

Reliability constraint: The expected energy not served (EENS) calculated by the Eq. (3) is confined by an upper limit for the reliable operation of the generators. This limitation is implemented by Eq. (6) as follows,

$R\left(P_{g)} \leq E E N S_{P g}^{\max }\right.$     (6)

3.3 Problem statement

Combining all the objectives and constraints, the multi-objective optimization problem can be mathematically formulated as a non-linear constrained problem as follows:

Minimize$P_{g}$$\left[\begin{array}{c}F C\left(P_{g}\right), E\left(P_{g}\right) ; F C\left(P_{g}\right), R\left(P_{g}\right) ;\\ E\left(P_{g}\right), R\left(P_{g}\right)\end{array}\right]$    (7)

A penalty factor has been used as a preference controller for the constraint given in Eq. (4) when combined with Eq. (1) during optimization. The value of this penalty factor is 10 in our case which is directly taken from [7].

3.4 Monte Carlo simulation for reliability analysis

The reliability of a system is represented by a numerical value between 0 and 1 or 0 to 100%. A power system is normally very complex as huge amount of equipments are connected with each other, and it is very difficult to evaluate the reliability value for the whole system. Therefore, some of the system adequacy parameters such as EENS, EDNS etc. are used to define the reliability of a system. So, to define the reliability of a system shown in Figure 1, evaluation of EENS is considered as a reliability function. The EENS is calculated as a function of generated/ dispatched power from each generator. Though the dispatched power is not directly involved in the calculation of EENS, a dispatching sample is passed from the optimization algorithm to the EENS evaluation algorithm. EENS evaluation algorithm then evaluates the amount of load curtailed that means the power shortage between the load and the generated power due to the outage of a generator and line failures [26]. The outage of a generator or a line failure is calculated by the use of Monte Carlo Simulation Technique.

For the reliability analysis, Monte Carlo Simulation has been implemented to find the system adequacy parameters. For the calculation of reliability, some indices are used in order to assess the system adequacy. Since the reliability is a numerical parameter that can vary between zero and one mathematically, the values of these indices will determine the system adequacy. Some of these indices are loss of load probability (LOLP), expected demand not served (EDNS), expected energy not served (EENS), system minutes (SM) etc. EENS calculation will be used for this paper as reliability function.

We have focused on EENS (MWh/yr). It is one of the major indices of system adequacy assessment. All most all literature on system adequacy assessment, calculates the EENS [27, 28].

Different papers have used different formula to calculate the EENS. In our calculation, we have used the formula for the implementation in MATLAB stated below,

$E E N S=\frac{1}{\text {simulation}} \sum_{i=1}^{\text {simulation}}\left(\frac{1}{\text {sample}} \sum_{j=1}^{\text {sample}}\right.$discont.load$)$     (8)

$F O R=\frac{\lambda[f / y e a r] * r[h]}{(\lambda[f / y e a r] * r[h])+8760}$     (9)

where, discont. load is an abbreviation of disconnected load, FOR refers to forced outage rates, λ is outage rate per year and 'r' represents outage duration. 8760 shows the number of hours per year. Eq. (8) has been used in Monte Carlo Simulation to calculate the outage rate of all the generators and lines. According to their outage, the loads are disconnected which is used in Eq. (7) in MATLAB formulation for calculating EENS. All the reliability parameters for our system is given in Table 2.

The proposed non-dominated sorting genetic algorithm-II (NSGA- II) models are designed using MATLAB simulation software. The proposed algorithm is designed to optimize two objective functions at a time, and for this reason, pairwise objective functions among fuel cost, emission, and reliability (expected energy not served -EENS) are considered. The proposed algorithms have been run 30 times and the best performance of each run of each algorithm is reported. The parameter setting of the proposed algorithms are as follows: Population size = 100, maximum number of generations =500, distribution index for crossover=20, distribution index for mutation/mutation constant=100 and Mutation Probability=0.16667. For multi-objective optimization, two different cases have been considered.

Case 1: the power balance equation given in Eq. (4) is taken as constraint to optimize:

a) Eq. (1) and Eq. (2) to find the optimized values or trade-offs between fule cost and emission

b) Eq. (1) and Eq. (3) to find the optimized values or trade-offs between fule cost and reliability (expected energy not served -EENS)

c) Eq. (2) and Eq. (3) to find the optimized values and dispatch parameters for emission and reliability (expected energy not served -EENS)

The capacity of the fossil-fuel-run generators and the cost coefficients of emission function are necessary for analyzing the cost and emission function, and the co-efficient values are listed in Table 1 [24, 25]. In Table 1. α, β, γ represents the cost coefficients and a,b,c,d,e are the emission coefficients.

Table 1. Cost and emission coefficients of the distributed generators [24, 25]

For case 1, at first, the fuel cost of generators and emissions from each generator are minimized with the proposed multi-objective optimization algorithm for the microgrid considered in Figure 1. The best economic fuel cost and the lowest emission objectives have been optimized separately by the proposed approach in order to find the lowest possible cost and emission points in solution space. The diversity of the solution space can also be evaluated by this method. The results of the best fuel cost and lowest emission are given in Table 3. The paprameters G1, G2, ..., G6 represents the power dispatch in mega-watts for generators 1, 2, ..., 6 respectively.

In order to validate the accuracy of the applied NSGA- II, the results are compared with another multi-objective optimization model based on the strength pareto evolutionary algorithm (SPEA). The comparison between NSGA- II and SPEA is demonstrated in Table 4. The Pareto-optimal front of solutions for cost and emission is shown in Figure 3. From Table 4 and Figure 3, it can be stated that the proposed NSGA- II algorithm is a good choice for minimizing the fuel cost and emission. The results are very close to each other with small differences in values in comparison with SPEA but pareto optimal front (shown in Figure 3) proves that the proposed algorithm can also perform the multi-objective optimization very effectively. Therefore, the designed algorithm is capable of performing optimization for other multi-objective optimization processes as well.

Table 2. Modified reliability parameters

Table 3. Fuel cost and emission optimized for case 1

Table 4. Optimized results for fuel cost and emission for case1

In the next stage, the cost function and EENS function (as a measurement of reliability) are considered for double objective optimization. Both the fuel cost and EENS are set as a minimization function in this scenario. The best results obtained after performing the optimization with the proposed algorithm are presented in Table 5.

Table 5. Optimized results for fuel cost and reliability for case 1

In Table 5, EDNS refers to the expected demand not supplied. From Table 5, it can be seen that in order to have the most economic dispatch for the microgrid, the generation system needs to reserve 3.1 more MW generation capacity than the most reliable dispatch. On the other hand, for maintaining the most reliable dispatch for the microgrid, 11.60 more USD needs to be spent on the fuel every hour.

Lastly for case 1, the emission objective function of the generators in equation Eq. (2) is considered along with the EENS function shown in Eq. (3). The proposed algorithm has been tried to minimize both of them remaining in the boundary of power balance or subjecting to the constraint of power balance alone. The simulation results have been presented in Table 6. Table 6, presents the dispatch values of all the six generators for achieving the least emission and least EENS (in other words highest reliability).

Table 6. Optimized results for emission and reliability objectives for case 1

Case 2: The fuel cost function in Eq. (1) and the emission function in Eq. (2) have been considered for a multi-objective optimization process taking the power balance, generation capacity, and reliability constraints into account. In case 1, reliability is considered as an objective in terms of EENS. But, in this case, the EENS factor is applied as a constraint in order to optimize cost and emission. The maximum EENS allowed in this case, is 1.165% (50000 MWhr/yr) of the total energy supplying capacity of the generators. The best results for cost and emission are presented in Table 7.

From Table 7, it can be observed that by keeping the EENS within a certain limit to ensure higher system reliability, saving fuel cost will cause more environmental pollution, while saving the environment by keeping the system highly reliable will cost 27.1 USD more per hour.

Table 7. Optimized results for cost and emission for achieving highest reliability for case 2