—In this paper, a new approach is proposed to solve a multi-objective economic-emission scheduling problem in microgrids (MGs) by simultaneously minimizing the energy and emission costs of the MG with various distributed energy resources (DERs). The proposed approach is an extension of a computationally effective multiobjective optimization technique, Pareto concavity elimination transformation (PaCcET). The proposed approach, referred to as Fuzziﬁed-PaCcET, employs a fuzzy logic controller to dynamically revise crossover and mutation rates in the original PaCcET leading to the faster convergence of the solution. The proposed approach ﬁnds the best Pareto front, also referred to as a Non-dominated set (NDS) of solutions, instead of ﬁnding a single optimal solution. In order to ﬁnd the solutions on concave areas of the Pareto front, an iterative objective space transformation is performed in the PaCcET algorithm to allow a linear combination of objective functions (in the transformed objective space). The proposed Fuzziﬁed-PaCcET-based scheduling is implemented on a MG with various dispatchable and non-dispatchable DERs to ﬁnd the set of optimal solutions according to the total fuel cost of DERs, as well as the most optimum environmental cost. In order to extract the best compromise solution (BCS) among NDS of solutions, a fuzzy-based method is implemented. The comparison of the simulation results of the Fuzziﬁed-PaCcET with that of PaCcET shows that Fuzziﬁed-PaCcET can generate better solution with less computational burden. Index Terms —Economic-emission scheduling, fuzzy logic con- troller, multi-objective optimization, PaCcET. I. I NTRODUCTION Microgrids (MGs) are small-scale power systems consisting of various distributed energy resources (DERs) such as solar photovoltaics (PVs), small wind turbines, fuel cells, micro- tubines, energy storage devices, and other controllable loads. MG can also be regarded as a single controllable system providing both power and heat to a certain speciﬁc area, and it can be operated in both grid-connected and isolated modes [1]. This enables MGs to meet its continuously varying demands with the help of power imported from the main grid if it is not economical or insufﬁcient to supply only using its own DERs. Therefore, the economic dispatch is needed for a grid connected MG for its optimal operation [2]. Different techniques have been reported in the literature in order to solve the economic emission dispatch (EED) prob- lems in MGs and power systems. In [3], the multi-objective economic load dispatch problem has been solved by reducing it to a single objective problem by treating the emissionas a constraint with a permissible limit, which does not consider the tradeoff between generation cost and emissions. A stochastic EED has been formulated in [4], where uncertainties in the system production cost and random nature of the load demand have been considered. A dynamic programming based approach has been presented in [5] to solve EED problem in real-time. In [6], a dynamic non-dominated sorting multi- objective biogeography-based optimization technique has been proposed to solve multi-objective dynamic EED problem considering charging of plug-in electric vehicles (PEVs). An interior search algorithm has been applied in [7] to solve multi- objective EED problem. In [8], an exchange market algorithm, inspired by the method of exchange of shares by stockholders, has been proposed to solve EED and reliability problem in case of thermal power plants. In [9], a normal boundary intersection method has been proposed to solve multiobjective EED problem considering combined heat and power. The multi-objective optimization problems including EED have been solved using different types of evolutionary algo- rithms including Non-dominated Sorting Genetic Algorithm-II (NSGA-II), Multi-objective Fireworks Algorithm (MOFWA), Multi-objective Particle Swarm Optimization (MOPSO), and Pareto Concavity Elimination Transformation (PaCcET), to name a few. NSGA-II has been implemented for dynamic EED problem in [10], where a nonlinear constrained multi- objective optimization problem has been formulated. In [11], a-dominance based evolutionary algorithm ( -DEA) has been proposed to solve combined heat and power EED prob- lem. A time-varying acceleration particle swarm optimization (TV AC-PSO) has been presented in [12] for stochastic multi- objective optimization of combined heat and power economic- emission dispatch. In [13], a NSGA-II based approach has been proposed to determine optimal or near-optimal sizes and locations of multi-purpose utility-scale shared energy storage in distribution systems. In [14], a Paired Bacteria Optimization (PBO) algorithm has been been proposed to solve Security Constrained Optimal Power Flow (SCOPF) with distributed load variations and uncertain wind power. In [2], the multi- objective economic emission dispatch problem has been solved using PaCcET. This paper proposes the Fuzziﬁed-PaCcET, an extension to PaCcET, to solve a multi-objective Economic Emission scheduling problem that minimizes the emission apart from minimizing generation cost of various distributed energy re- sources (DERs). The Fuzziﬁed-PaCcET employs a fuzzy logic controller to dynamically revise crossover and mutation rates in the original PaCcET leading to the faster convergence of the solution. The Fuzziﬁed-PaCcET is implemented on a MG with various DERs to obtain the most economic operating condition not only by minimizing the total fuel cost but also by ﬁnding the most environmentally friendly solutions. To extract the best compromise solution (BCS) from NDS of solutions, an approach based on fuzzy set theory is implemented. The results show that the Fuzziﬁed-PaCcET can generate better solution than PaCcET with less computational time. The rest of the paper is organized as follows. The method of formulation of economic-emission scheduling problem with the description of objective functions and constraints are pre- sented in Section II. Section III describes the proposed multi- objective optimization technique, which is the combination of the original PaCcET and a fuzzy logic controller. Also, the proposed solution methodology is explained in Section III. Section IV presents the simulation results and discussions, along with the comparison of the proposed Fuzziﬁed-PaCcET technique with PaCcET in terms of convergence and the dom- inated hypervolume. Section V provides concluding remarks. II. E CONOMIC -EMISSION SCHEDULING PROBLEM FORMULATION FOR MG S Since the objective of this paper is to minimize both economic and emission costs, a multi-objective economic- emission scheduling problem is to be formulated. The two objectives for the economic-emission scheduling problem un- der consideration are as follows [2]. 1) Economic cost minimization minfFg=TX t=1 qtPt grid+NgX i=1(aiPi;t DER2+biPi;t DER +ci) g, (1) 2) Fuel emission minimization minfEg=TX t=1NgX i=1(ki ePi;t DER ) g, (2) whereqtandPt grid are electricity rate and the amount of power exchanged between the main grid and the MG at time t, respectively. The total scheduling time Tis divided into subintervals t.Pt gridis positive if power is being bought from the main grid and negative if power is being sold to the main grid. The generation cost comprising of power generation and operation and maintenance costs of a DER is a quadratic function ofPi;t DER , the active power of DER iat timet. In (1), ai,bi, andcirepresent the cost coefﬁcients of each DER and Ngrepresents the total number of dispatchable DERs. Non- dispatchable resources such as solar photovoltaics (PVs) and wind-turbines are considered as negative loads. Equation (2) is the other objective function that represents the cost related to emission as an index for environmental conservation, where ki eis the emission cost coefﬁcient of ith DER.The constraints of the optimization problem are as follows. NgX i=1Pi;t DER +NndX j=1Pj;t nd+Pt gridPt load = 0, (3) Pi min<Pi;t DER<Pi max, (4) where (3) represents active power balance equation, Pj;t ndis the active power generated by jth non-dispatchable resource at timet,Pt load is the total load of MG at time t, andNnd is the total number of non-dispatchable resources. Equation (4) represents the active power limit constraint of dispatch- able DER,Pi min andPi max respectively being minimum and maximum limit of ith DER. III. P ROPOSED MULTI -OBJECTIVE OPTIMIZATION TECHNIQUE A. Background The proposed Fuzziﬁed-PaCcET technique draws from many distinct concepts from within multi-objective research, some of which are brieﬂy described as follows [15]. Dominance: A solution dominates another solution if it scores lower on all criteria. A solution is said to weakly dom- inate another solution if it scores equal on some objectives, but less on other ones. Pareto front: The outcome of a multi-objective optimization problem is a set of solutions referred to as Pareto front. Each point (or solution) in the Pareto front cannot be dominated by other points in feasible solutions. Utopian and nadir points: The best possible solution ob- tained by optimizing each objective function individually is referred to as utopian point. This point is an infeasible and unattainable solution due to the conﬂict between the objec- tives. Nadir point is the upper bound of the Pareto front. It is the solution resulting from the worst value for each objective function in the Pareto front. It is to be noted that the nadir point is a concept that is different from the worst feasible point in the complete set of solutions. Dominated hypervolume: The dominated hypervolume refers to the feasible space of multi-objective optimization problem that is enclosed by nadir and Pareto points. This con- cept is widely used as a criterion to compare the accuracy and performance of different types of multi-objective optimization techniques. B. Fuzziﬁed-PacCET Fuzziﬁed-PaCcET, an extension to PaCcET, is a new itera- tive multi-objective transformation. It transforms the objective space in such a manner as to make the Pareto Front convex, and a single user-deﬁned parameter is sufﬁcient for it [15]. Fuzziﬁed-PaCcET is composed to two main components, viz, the original PaCcET and a fuzzy logic controller. r unadir,norm uo,normλBnorm pI*,norm pI*,τ Obj. Fcn. 1 (norm or τ)Fig. 1. Visualization of PaCcET. It transforms the green cross (a normalized point) to red cross (a transformed point) and green dots on the normalized border (norm B) to red ones on the transformed border ( B) [15]. 1) PaCcET: PaCcET searches the points in the Pareto front in multi-objective optimization problem based on the linear combination of multiple objective functions in the transformed space using optimizers with single objective function. The PaCcET starts by calculating the utopian uoand nadirunadir points by minimizing each objective function individually. In the ﬁrst step, all other points are normalized and mapped from multi-objective space ( ) to a normalized space ( norm) using (5). vnorm(i) =v(i)uo(i) unadir (i)uo(i), (5) wherevnorm(i)is theith element of normalized point v. In the second step, the normalized space normis mapped to a transformed space (  ) using (6)-(11) . r=vnorm jvnormj, (6) L1:kvnormk1=X ivnorm i , (7) L2:kvkB=min( )3 rp I, (8) L3=kvkhp= 3X i ri= (m1), (9) d =L3L1 L2=kvkhpkvnorm 1k kvkB, (10) v =d r, (11) whereL1is linear combination or Manhattan distance from uo;normtovnorm,L2is distance from uo;normto the nor- malized dominated border norm B alongr,L3is distance from uo;normto the normalized utopian hyperplane norm B alongr, mis the number of objective functions, and p Iis the Pareto front atIth iteration. Figure 1 shows all the variables employed in the PaCcET. μ (x1) x1 1.0 0.5 0 0 0.4 0.8 μ (pc) pc 1.0 0.5 0 0.6 0.8 0.99μ (x2) x2 1.0 0.5 0 0 0.008 0.024L M H L M H L M Hμ (pm) pm 1.0 0.5 0 0.06 0.10 0.12L M H EL 0.08(a) (b) (c) (d)Fig. 2. The membership function of inputs and outputs of fuzzy logic controller. In the last step a function, referred to as PaCcET Linear Combination (PLC), is determined which is a linear combina- tion of the elements of v . PLC =X iv i. (12) PaCcET ﬁnds the points with PLC less than 1 in each iter- ation using single-objective optimizers. In this paper, genetic algorithm (GA) is used as single objective optimizer. 2) Fuzzy logic controller: Since GA is used as single objective optimizer, normally crossover and mutation rates are constants. Contrary to this, the proposed Fuzziﬁed-PaCcET technique utilizes a fuzzy logic controller to dynamically up- date crossover and mutation rates. The crossover and mutation rates are updated in such a manner so as to enhance diversity of the population in the early stage of evolution, improve global search ability in the medium stage of evolution, and strengthen the convergence of the algorithm in the late stage of evolution [16]. The inputs of the fuzzy logic controller are x1andx2 and its outputs are crossover rate pcand mutation rate pm. The fuzzy logic controller consists of three parts: fuzzi- ﬁcation, fuzzy reasoning, and defuzziﬁcation. In fuzziﬁcation process, the crisp quantities are converted into fuzzy quantities. The crisp quantities x1(n)andx2(n)fornth generation, given respectively by (13) and (14), are converted into fuzzy quantities using membership functions shown in Figure 2(a) and 2(b). x1(n) =jFmax(n)Favg(n)j jFmax(n)j, (13) x2(n) =jFavg(n)Favg(n1)j jFmax(n)j, (14) wherex1(n)denotes the normalized Euclidean distance be- tween maximum objective vector Fmaxand average objective vectorFavgboth atnth generation, x2(n)denotes the nor- malized euclidean distance between average objective vector atnth generation and that at (n1)th generation. The membership functions for inputs and outputs of fuzzy logic controller are shown in Figure 2. Each membership function, lying on the interval [0,1], expresses the degree of membership that the speciﬁc combination of parameters has in a particular fuzzy set. The inputs x1andx2have three fuzzy sets, viz, L (low value), M (medium value), and H (high value). Crossover rate pchas three fuzzy sets, viz, L, M, and H as shown in Figure 2(c), and mutation rate pmhas four fuzzy sets, viz, EL (extra low value), L, M, and H as shown in Figure 2(d). In the fuzzy reasoning part, also referred to as fuzzy inference system, the fuzzy values of the outputs are obtained on the basis of fuzzy rules. For the rule-based fuzzy reasoning, linguistic variables are used as antecedents and consequents. The antecedents are used to express an inference, which when satisﬁed result in certain consequents [17]. Table I shows the IF-THEN fuzzy rules in the early, medium, and late stages of evolution [16]. The fuzzy rule-based system adopts IF-THEN fuzzy rules, given by, IF antecedent and THEN consequent. For example, in the early stage of evolution, if x1is M and x2is L, then membership function M is used as fuzzy set for crossover rate pcand membership function H is used as fuzzy set for mutation rate pm. The degree of membership for crossover and mutation rates are determined as follows. (pc) =(pm) =minf(x1);(x2)g. (15) In the defuzziﬁcation part of the fuzzy logic controller, the fuzzy quantities (pc)and(pm)are converted to crisp quantitiespcandpm, respectively. The crisp value of an output is determined using the centroid method. The centroid method, also referred to as center of area or center of gravity, is the most prevalent and is regarded as physically appealing of all the defuzziﬁcation methods [18] and is given by the following algebraic expression. z=R (z):zdzR (z)dz, (16) wherezany quantity to be defuzziﬁed, i.e., zispcorpm. TABLE I FUZZY RULES IN THE EARLY /MEDIUM /LATESTAGE OF EVOLUTION Fuzzy value of crossover rate pc (pc)x1 L M H x2L L/M/L M/M/L H/H/L M L/M/L M/M/L M/H/L H L/M/M M/H/M M/M/M Fuzzy value of mutation rate pc (pc)x1 L M H x2L H/H/L H/M/L M/M/EL M H/L/L H/L/EL M/L/EL H M/L/EL M/L/EL M/L/EL In this way, the fuzzy logic controller is used to dynamically update crossover and mutation rates over generations. Figure 3 shows the ﬂowchart of Fuzziﬁed-PaCcET technique to ﬁnd Pareto front of the multi-objective optimization problem. Start Read system data Generate initial populationPerform crossover and mutation Generate children population using elitist -selection StopFind utopia & nadir pointsGenerate first Pareto front Sort combined population based on PLC Update Pareto front with the best populationCompute crisp quantities, x 1(n) and x2(n) Fuzzify input crisp quantities using fuzzificationGenerate output fuzzy set using fuzzy reasoning Defuzzify the output set using centroid methodMaximum generation reached? YesNoFig. 3. Flowchart of the proposed economic-emission optimization technique IV. C ASE STUDIES AND DISCUSSIONS A. Case Study Parameters In this paper, the proposed Fuzziﬁed-PaCcET is applied to economic-emission multi-objective scheduling of microgrid (MG) in either grid-connected or islanded mode. The MG consists of three dispatchable DERs, one non-dispatchable PV generation, and time-varying loads. In this paper, the economic-emission scheduling of dispatchable DERs is per- formed according to the most-probable PV generations and loads during each scheduling period. The cost coefﬁcients of the dispatchable DERs are shown in Table II and load, PV generation, and the electricity price for different scheduling hours are presented in Table III. The day-ahead electricity price schedule for distribution network retail customers is adopted from the ComEd utility [19]. TABLE II COST COEFFICIENTS DATA OF DISPATCHABLE DER S TitlesPmax (kW)Pmin (kW)c ()b (/kWh)a (/kWh2)ke (/kWh) DER1 200 40 954 63.6 0.0018 5 DER2 240 48 813 61.1 0.0011 6 DER3 400 80 1054 45.4 0.0005 8 B. Comparison The superiority of PaCcET for solving multi-objective op- timization problems in power systems, including economic- TABLE III HOURLY LOAD, PV G ENERATION AND ELECTRICITY PRICE DATA HourPloadPpvqtHourPloadPpvqt (kW) (kW) (/kWh) (kW) (kW) (/kWh) 1 308 0 56 13 693 114 79 2 341 0 55 14 770 132 70 3 308 0 51 15 825 111 77 4 319 0 51 16 781 97.5 77 5 341 0 50 17 814 82.5 83 6 407 3 51 18 792 61.5 83 7 517 13.5 54 19 759 40.5 77 8 649 27 56 20 693 18 76 9 704 51 66 21 649 4.5 76 10 814 72 78 22 594 0 74 11 737 102 85 23 561 0 72 12 748 108 83 24 429 0 68 Fig. 4. Comparison of dominated hypervolume for Fuzziﬁed-PaCcET and PaCcET with respect to number of GA iterations reliability scheduling or V olt-Var control in distribution net- works, is previously established in [20], [21]. For compari- son of Fuzziﬁed-PaCcET with the original PaCcET, different criteria can be used. According to our prior results, the con- vergence and diversity of the best Pareto points are the most important ones. Various metrics such as distance, diversity, and hypervolume metrics can be used to compare the results of different multi-objective optimization techniques. Dominated hypervolume is a metric that measures the area enclosed by the Pareto front and the nadir point. It identiﬁes the closeness between the Pareto front to the feasible area boundary. Figure 4 shows the percentage of dominated hypervolume as a function of number of GA genrations for PaCcET and Fuzziﬁed- PaCcET. It can be seen from the ﬁgure that Fuzziﬁed-PaCcET proceeds faster than PaCcET towards the Pareto front. Figure 5 and Figure 6 show the comparison of Pareto fronts obtained using Fuzziﬁed-PaCcET and PaCcET after 25and 50GA generations, respectively. In each of the scenarios, the population size is 40. From these two ﬁgures, we can see that Fuzziﬁed-PaCcET converges to ﬁnal solution faster than PaCcET. But after large number of iterations, both the methods may converge to same solutions. Fig. 5. Pareto fronts obtained using Fuzziﬁed-PaCcET and PaCcET after 25 GA generations Fig. 6. Pareto fronts obtained using Fuzziﬁed-PaCcET and PaCcET after 50 GA generations C. Best Compromise Solution Once the ﬁnal Pareto front including non-dominated so- lutions is obtained by Fuzziﬁed-PaCcET technique, the best compromise solution (BCS) is chosen from non-dominated solutions using a fuzzy membership approach. The following function is used to calculate the membership value of each point on the ﬁnal Pareto front [20]. i(k) =8 >< >:1 iffi(k)<fm i fM ifi(k) fM ifm iiffm i<fi(k)<fM i 0 iffM i<fi(k)(17) wherei(k)is the membership value of the ith objective func- tion ofkth Pareto point. For each Pareto point, the normalized membership value, norm(k), is computed as follows. norm(k) =PNo i=1i(k) PNP j=1PNo i=1i(j), (18) whereNoandNPare the number of objective functions and Pareto points, respectively. The BCS is the solution with the maximum value of norm. Fig. 7. Hourly loads, scheduled power generation of DERs, and power ex- change between main grid and MG based on Fuzziﬁed-PaCcET optimization Using the BCS obtained using (17) and (18), the values ofPi;t DER andPt grid over the entire scheduling horizon can be computed. Figure 7 shows the hourly loads, scheduled power generation of DERs, and the power exchange between the main grid and the microgrid when the Fuzziﬁed-PaCcET algorithm is run for 100 generations with the population size of 40. In the ﬁgure, transparent bars with red border represent the power transfer Pt grid. When the transparent bar is above the load line, the power is sold to the main grid and when it is below the load line, the power is bought from the main grid. V. C ONCLUSION In this paper, a Fuzziﬁed-PaCcET technique has been proposed for economic-emission scheduling of microgrids with dispatchable DERs. The proposed Fuzziﬁed-PaCcET is composed of a fuzzy logic controller in addition to the original PaCcET. The PaCcET has been used to iteratively transform multi-objective space to allow a linear combination of objective functions to ﬁnd the solutions on concave areas of the Pareto front. The fuzzy logic controller has been employed to dynamically update crossover and mutation rates at different stages of evolution of the Pareto solution. The comparison of the dominated hypervolume of the Pareto solutions between PaCcET and Fuzziﬁed-PaCcET has shown that Fuzziﬁed- PaCcET can generate superior solution with less number of iteration and computational time. This feature makes it more attractive for practical applications with resource-constrained computational devices. ACKNOWLEDGEMENT This work was supported in Part by the U.S. Department of Energy (DOE) under Grant DE-EE0009022. This work was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Gov- ernment nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, orusefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights.