An NSGA-III algorithm for solving multi-objective economic/environmental dispatch problem

The main ambition of utility is to provide continuous reliable supply to customers, satisfying power balance, transmission loss while generators are allowed to be operated within rated limits. Meanwhile, achieving this from fossil fuel fired power plant emission value and fuel cost should be as less as possible. An allowable deviation in fuel cost and feasible tolerance in fuel cost has been additively called as multi objective combined economic emission dispatch (MOCEED) problem. MOCEED problem is applied to newly proposed non dominated sorting genetic algorithm-III (NSGA-III). NSGA-III method is really powerful to handle problems with non-linear characteristics as well as having many objectives. The proposed NSGA-III is firstly applied to unconstraint/constraints multi-objective test functions then applied to solve MOCEED problem with 6-generation unit, IEEE 118 bus 14 generating unit system with a smooth quadratic fuel/emission objective functions and 10-unit with non-smooth/valve point loading effect test system. Statistical results of MOCEED problem obtained by NSGA-III is compared with other well-known techniques proposed in recent literature, validates the effectiveness of proposed approach. Subjects: Artificial Intelligence; Evolutionary Computing; Power Engineering; Engineering Economics


Introduction
The main ambition of fossil fuel fired power plant management is to optimal scheduling of active power to committed units so as to achieve least possible generation fuel cast as well as emission value. Primarily economic load dispatch problem (ELDP) (Trefny & Lee, 1981) is main objective of electricity generation utilities but with large consciousness towards environment protection and

PUBLIC INTEREST STATEMENT
In these days, every person or every country suffer from due to increased load demand and emission gases like as CO 2 , SO 2 , and others. These gases increase the pollution that directly affects human health. So in this research, we have used a new method to solve this problem that prevents pollution, saves human life, and fulfills load demand that reduced human problems.
passage of the clean air act amendments have forced generation utilities to reduce emission (Le et al., 1995;Talaq, El-Hawary, & El-Hawary, 1994) to a certain level.
Evolutionary techniques are capable to overcome the difficulties associated with classical methods such as multiple run. The target of multi-objective optimization technique is not only to steer the search towards the pareto optimal front but also to preserve population diversity in the set of nondominated solutions.
Newly proposed NSGA-III  is powerful technique to eliminate the drawbacks of NSGA-II such as lack of uniform diversity and absence of lateral diversity preserving operator among the current best non dominated solutions.
In this paper term used emission constrained economic dispatch (ECED) problem is similar to term combined economic emission dispatch (CEED) problem. Various conventional linear optimization methods (Wood, Wollenberg, & Sheble, 2013) were used to solve the ELD problem, (a) lambda-iteration method, (b) gradient-method, (c) linear-programming-method and (c) Newton's method. Linear programming techniques are fast and reliable but these methods are failed to obtain the optimal solution for solving highly complex non-linear objective functions.
The multi-objective power system dispatch problem can be transformed into single objective by Secularization Methods (Priori Approach) using these techniques: • Price penalty factor technique and • Weighted sum method (WSM) • Goal Attainment method • Lexicographic method etc.
The ECED problem consists of either single objective or multi-objective is solved using various algorithms such as: After Scalarization technique applied ECED problem can be classified into two forms with and without considering valve point loading effect of generators further classified into the equation used either quadratic and cubic equation to evaluate fuel cost and emission value. ECED problem is solved without considering valve point effect and with price penalty factors based approach is solved with various computational techniques. ECED problem solved using "Max-Max" price penalty factor approach by various artificial intelligence (AI) techniques (Jacob Raglend, Veeravalli, Sailaja, Sudheera, & Kothari, 2010) consisting of genetic algorithm (GA), Evolutionary Programming (EP), Particle swarm optimizer (PSO) and Differential evolution (DE) is applied on IEEE-30 Bus system. "Max-Max" price penalty factor is also used to solve CEED/ECED problem with Gravitational Search Algorithm (Güvenç, Sönmez, Duman, & Yörükeren, 2012), Parallelized PSO (PPSO) (Hamedi, 2013), Evolutionary programming (EP), GA and Micro GA (MGA) (Venkatesh, Gnanadass, & Padhy, 2003), Assessment of available transfer capability for practical power system with CEED problem for IEEE-30bus system with 6 generating units and Indian utility system 62-Bus (IUS-62) with nineteen generators (Gnanadass, Padhy, & Manivannan, 2004). Analytical solution for CEED problem with IUS-62 with six generators (Palanichamy & Babu, 2008), comparative study (Krishnamurthy & Tzoneva, 2012b) with "Min-Max" price penalty factor using PSO and Lagrange's algorithm (LA), with LA (Krishnamurthy & Tzoneva, 2011a) and PSO (Krishnamurthy & Tzoneva, 2012a) taking "Min-Max" and "Max-Max" price penalty factors approach ECED problem is solved. Lagrange's algorithm is used to solve ECED problem with four penalty factors (Krishnamurthy & Tzoneva, 2012d) with the quadratic equation is considered for evaluating fuel cost and emission value, six penalty factors with cubic equation (Krishnamurthy & Tzoneva, 2012c) used for calculation of ECED problem. Scenario based dynamic economic emission dispatch problem is solved by Fuzzy adaptive improved PSO (FAIPSO) (Aghaei, Niknam, Azizipanah-Abarghooee, & Arroyo, 2013).
Combined Economic Emission Dispatch (CEED) problem is also known as Emission Constrained Economic Dispatch (ECED) problem, Combined Economic and Environmental Dispatch, Environmental/ Economic dispatch (EED), Multiobjective CEED (MOCEED) and Constraint Environment Dispatch (CED) problem.
Recently proposed nature based and evolutionary algorithms with various applications like energy management of RES in a Microgrid using Cuckoo Search (CS) Algorithm , constrained engineering design problem by Moth-Flame optimizer (MFO) , Voltage stability improvement by BAT optimization algorithm , Seyedali Mirjalili et al. algorithms such as Grey wolf optimizer (GWO) (Mirjalili, Mirjalili, & Lewis, 2014), Whale optimizer (WOA) (Mirjalili & Lewis, 2016), Moth-flame optimizer (MFO) (Mirjalili, 2015), swarm based Dragonfly Algorithm (DA) (Mirjalili, 2016), Gai-Ge Wang et al. elephant herding optimization (EHO) (Wang & Suash Deb, in press) and evolution based NSGA-III. Among these algorithms NSGA-III is chosen because it has comparatively better capability of handling many objectives (up to 50 objectives) and it also has a uniform diversity to obtained Pareto optimal front in a set of non-dominated solutions. Overview of this paper is shown in Figure 1

With valve-point loading effect
where, F c = Total fuel cost $/h.
where, E T = Total emission value kg/h.

Non-dominated sorting algorithm-III
NSGA-III   At a generation t, complete population Pt convert in non-dominated solutions in the same way of NSGA-II algorithm Sorting mechanism, after that Pt produces new offspring population Q t with the help of mutations and recombination operators in which everyone population member associated with each reference point& any selection operator will allow a competition to be set among different reference points. A combined population R t = P t ∪ Q t is then formed. So we have get first non-dominated solution P t+1 until every solution cannot be included from whole fronts. Suppose we have denoted some front that cannot be associated to select F L after that P t+1 and F L perform niching and normalized mechanism after that each member associated with a specific reference point based on shortest perpendicular distance (d()) of each population member with a reference line created by joining the origin with a supplied reference point. At finally niching mechanism choose the F L member that is linked with minimum reference points in P t+1 .
The whole process is then expected to find one population member corresponding to each supplied reference point close to the Pareto-optimal front, based on crossover, mutation and recombination operators that are used to develop uniform solutions. The use of a well-spread reference points ensures a well-distributed set of trade-off points at the end.
A better advantage of NSGA-III  is that have not required additional parameter compare to NSGA-II. NSGA-III Algorithm step by step representation shown in Figure  2. Main difference of selection mechanism of both NSGA-II & NSGA-III algorithms given below: (1) NSGA-III algorithm cannot have required another selection operator for P t to create new operator q t . On the other hand, NSGA-II's selection operator uses non-dominated rank and a crowding distance value to choose a winner between two feasible individuals from p t . It is worth noting however that, NSGA-III performs selection if and only if at least one of the two individuals being compared is infeasible. In that case NSGA-III prefers feasible over infeasible, and less violating over more violating individuals.
(2) To maintain better Coverage of pareto solutions NSGA-III uses reference point mechanism and other side NSGA-II uses crowding distance operator to maintain uniform coverage. (3) NSGA-III  uses to pre-allocated reference set mechanism to choose better diverse solutions in the size of population in free space, whereas NSGA-II algorithm does not require any pre-allocated methods on the objective space. So, more time taken to generate first solution in spaces, NSGA-III have easily generated first solution so NSGA-III better than NSGA-II algorithm for solving many objective problems. Figure 2 shows step by step procedure of NSGA-III Algorithm has been explained in pseudo code named step 2, step 3 and step 4 respectively.

Application and results
The meta-heuristic techniques are implemented to resolve the MOCEED problem for standard test system and for a number of cases with dissimilar objective functions. The software program is inscribed in MATLAB 2014b and applied on a 2.60 GHz i5 PC having 4 GB RAM.

Test systems
NSGA-III technique is applied to three different test systems. In order to represent the effectiveness of proposed algorithm statistical results are compared with NSGA-II (Dhillon et al., 1994), Strength pareto evolutionary algorithm 2 (SPEA 2) (Dhillon et al., 1994), pareto differential evolution (PDE) (Dhillon et al., 1994). NSGA-III results are also provided with single objective such as minimum fuel cost and minimum emission value. For each test system internal parameter like population/search agent, maximum/termination count and maximum archive size are 100,200 and 100 respectively. In the way to check performance of NSGA-III algorithm, it has been tested on: the IEEE 30-bus, the 39bus New England system network and the IEEE 118-bus test systems. All test systems are used to check the performance of NSGA-III algorithm compare to other algorithms.

Test system 1
In IEEE 30-bus test system contain 30-Buses, 41-Branches, 4-Transformers, 9-Shunt Var Compensators and 6-Genrators. Further details, the Cost and emission coefficients, generator data and the minimum and maximum limits for the control variables are given in Table B1. In test system consists of six operational generating unit with simply a quadratic fuel and emission objective function for a power demand of 1,200 MW (Basu, 2011). Input data for operational generating unit loading limits and loss parameters are given in Table B1 of Appendix B. Single line diagram of 6-unit system is shown in Figure 4.    Table 3 that with the objective of least cost objective minimum fuel cost is 6.41E+04 $ and emission value is 1,346 lb. But fuel cost increases to 6.599E+04 $ and emission value reduced to a numeric value 1,241 lb with the objective of emission minimization. Compromise point or true operating point obtained by NSGA-III for MOCEED problem is as fuel cost is 6.4830E+04 $ that is higher than minimum fuel cost 6.41E+04 $ and lower than 6.599E+04 $ obtained during least cost and emission value objectives respectively. So as with emission value for true operating point is 1,285 lb that is lower than 1,346 lb and higher than 1,241 lb obtained during least cost and emission value objectives respectively. Statistical value obtained for compromise point is compared with other techniques solves same MOCEED problem like SPEA2, NSGA-II and PDE in Table 3. Figure 5 shows 100 non-dominated solutions as true pareto front for 6-opertaional generating for PD = 1,200 MW.

Test system 2
In 39-bus New England test system, contain 39-Buses, 46-Branches, and 10-Genrators. Further details, the Cost and emission coefficients, generator data and the minimum and maximum limits for the control variables are given in Table B2. This test system consists of ten operational generating unit with a non-smooth/non-convex fuel and emission objective function for a power demand of  2,000 MW (Basu, 2011). Input data for operational generating unit loading limits and loss parameters are given in Table B2 of Appendix B. Single line diagram of 10-unit system is shown in Figure 6.
It is represented in Table 4 that with the objective of least cost objective minimum fuel cost is 1.115E+05 $ and emission value is 4,562 lb. But fuel cost increases to 1.164E+05 $ and emission value reduced to a numeric value 3,932 lb with the objective of emission minimization. Compromise point or true operating point obtained by NSGA-III for MOCEED problem is as fuel cost is 1.1340E+05 $ that is higher than minimum fuel cost 1.115E+05 $ and lower than 1.164E+05 $ obtained during least cost and emission value objectives respectively. So as with emission value for true operating point is  4,105 lb that is lower than 4,562 lb and higher than 3,932 lb obtained during least cost and emission value objectives respectively. Statistical value obtained for compromise point is compared with other techniques solves same MOCEED problem like SPEA2, NSGA-II and PDE in Table 4. Figure 7 shows 100 non-dominated solutions as true Pareto front for 6-opertaional generating for PD = 2,000 MW.

Test system 3
In IEEE 118-bus test system contain 118-Buses, 186-Branches, 9-Transformers, 14-Shunt Var Compensators and 14-Genrators. Further details, the Cost and emission coefficients, generator data and the minimum and maximum limits for the control variables are given in Table B3. The IEEE 118bus 14-operational generating unit test system with a smooth quadratic fuel and emission objective function neglecting transmission losses for a power demand of 950 MW. Input data for operational generating unit loading limits and loss parameters are given in Table B3 of Appendix B. Single line diagram of 14-unit system is shown in Figure 8.   Table 5 that with the objective of least cost objective minimum fuel cost is 4,265 $/h and emission value is 446.5 ton/h. But fuel cost increases to 4,485 $/h and emission value reduced to a numeric value 24.09 ton/h with the objective of emission minimization. Compromise point or true operating point obtained by NSGA-III for MOCEED problem is as fuel cost is 4,335.9 $/h that is higher than minimum fuel cost 4,265 $/h and lower than 4,485 $/h obtained during least cost and emission value objectives respectively. So as with emission value for true operating point is 124.6384 ton/h that is lower than 446.5 ton/h and higher than value 24.09 ton/h obtained during least cost and emission value objectives respectively. Statistical value obtained for compromise point is compared with other techniques solves same MOCEED problem like NSGA-II in Table 5. Figure 9 shows 100 non-dominated solutions as true pareto front for 14-opertaional generating for PD = 950 MW.

It is represented in
In order to check the robustness of the NSGA-III algorithm for solving Multi-Objective Economic/ Environmental dispatch problem, different standard benchmark functions and the IEEE 30-bus, the 39-bus New England system network and the IEEE 118-bus test systems used for experimental study. Tables 1 and 2 represents the statistical results on benchmark functions achieved by the NSGA-III, NSGA-II and SPEA-2 algorithms for unconstraint and constraint problem in terms of IGD and HV metrics. From these table clear that NSGA-III algorithm have better value compare to NSGA-II and SPEA-2 in terms of SD and mean value. Tables 3-5 represents the statistical results on the IEEE 30-bus, the 39-bus New England system network and the IEEE 118-bus test systems achieved by the NSGA-III, NSGA-II, MODE, PDE and SPEA-2 algorithms for Economic/Environmental dispatch problem in terms of compromise solution point. From these tables clear that NSGA-III algorithm have better value compare to other algorithm reported on literature survey.

Performance evaluation study of NSGA-III algorithm
In this Section describe why NSGA-III algorithm Comparison with other published techniques to demonstrate the accuracy and the validity of the NSGA-III technique should be presented in details. (b) NSGA-IIIalgorithm are used to many objective means easily implemented more than 10 objectives that future make more power full NSGA-III algorithm compare to other algorithm.  (c) Results of NSGA-III algorithm in term of IGD and HV metrics in term of SD and mean value on standard unconstrained/constraint benchmark functions shown in Tables 1 and 2 represents NSGA-III algorithm represent better solution quality compare to NSGA-II and SPEA-2 algorithms.
(d) Results of NSGA-III algorithm in terms of best value on economic constraint emission dispatch (ECED) problem with the IEEE 30-bus, the 39-bus New England system network and the IEEE 118-bus test systems represents the better effectiveness of NSGA-III algorithm compare to other algorithm shown in Tables 3-5.

Conclusion
In this paper, a new NSGA-III algorithm, an improved version of most popular multi-objective algorithm NSGA-II, is successfully applied to standard benchmark functions and economic constraint emission dispatch (ECED) problem on three test systems, such as IEEE 30-Bus, New England 39-Bus, and IEEE 118-Bus, with different fuel cost curve characteristics such as simple quadratic fuel and non-smooth/non-convex fuel with emission value and various constraints. NSGA-III algorithm removes drawbacks of NSGA-II algorithm such as disability to maintaining the diversity among population members that is avoided by supplying and adaptively updating a number of well-spread reference points in NSGA-III. The obtained results using NSGA-III algorithm have been compared to well recognize NSGA-II, SPEA-2 and other multiobjective techniques reported in literatures. The comparative study among that algorithms represent the solution quality in terms of SD and mean value, superiority in terms of IGD and HV metrics and effectiveness in terms of best values of NSGA-III algorithm over other algorithms on standard unconstraint/constraint test functions and economic constraint emission dispatch (ECED) problem.
In future direction of this research work is to improve the coverage and convergence characteristics of NSGA-III algorithm considering integrated with different oppositional strategy, cross over and mutation schemes. and, .sin(10 f 1 ) and,