Hopfield lagrange network based method for economic emission dispatch problem of fixed-head hydro thermal systems

POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE Hopfield Lagrange Network Based Method for Economic Emission Dispatch Problem of Fixed-Head Hydro Thermal Systems Thang NGUYEN TRUNG 1, Dieu VO NGOC 2 1Department of Electrical Engineering, Faculty of Electrical and Electronics Engineering, Ton Duc Thang University, 19 Nguyen Huu Tho Street, Ho Chi Minh City, Vietnam 2Department of Power Systems, Faculty of Electrical and Electronics Engineering, Ho Chi Minh City University of Technology, 268 Ly Thuong Kiet Street, Ho Chi Minh City, Vietnam [email protected], [email protected] DOI: 10.15598/aeee.v14i2.1543 Abstract. This paper proposes a Hopfield Lagrange Network (HLN) based method (HLNM) for economic emission dispatch of fixed head hydrothermal systems. HLN is a combination of Lagrange function and continuous Hopfield neural network where the Lagrange function is directly used as the energy function for the continuous Hopfield neural network. In the proposed method, HLN is used to find a set of non-dominated solutions and a fuzzy based mechanism is then exploited to determine the best compromise solution among the obtained ones. The proposed method has been tested on four hydrothermal systems and the obtained results in terms of total fuel cost, emission, and computational time have been compared to those other methods in the literature. The result comparisons have indicated that the proposed method is favorable for solving the economic emission dispatch problem of fixed-head hydrothermal systems. Keywords Economic emission dispatch, fixed head, hopfield Lagrange network, hydrothermal systems. 1. Introduction The short term hydro-thermal scheduling (HTS) problem is to determine the power generation among the available thermal and hydro power plants so that the total fuel cost of thermal units is minimized over a scheduled time of a single day or a week while satisfying both equality and inequality constraints including power balance, available water, and generation limits of both thermal and hydro plants [1]. In practical systems, thermal power generating stations are the sources of carbon dioxide (CO2), sulfur dioxide (SO2), and nitrogen oxides (NOx) causing atmospheric pollution [2]. Therefore, the optimal scheduling of generation in a hydrothermal system involves the allocation of generation among the hydro and thermal plants to simultaneously minimize the fuel cost and emission level of thermal plants satisfying the various constraints on the hydraulic and system network becomes a practical requirement. In the past decades, several conventional methods have been used to solve the HTS problem neglecting environmental aspects such as lambdagamma iteration method (LGM) [1], an effective conventional method (ECM) based on Lagrange multiplier theory [3], dynamic programming (DP) [4], Lagrange relaxation (LR) method [5], and decomposition and coordination method [6]. Among these methods, Lagrange multiplier theory based method does not find out optimal solution and it must be used together with other optimization techniques [7] whilethe DP and LR methods are more popular ones. However, the computational and dimensional requirements of the DP method increase drastically with large-scale system planning horizon, which is not appropriate for dealing with large-scale problems [8]. On the contrary, the LR method is more efficient and can deal with largescale problems. However, the solution quality of the LR for optimization problems depends on its duality gap which is a result of the dual problem formulation and might oscillate, leading to divergence for some problems with operation limits and non-convexity of incremental heat rate curves of the generators. Besides, the other methods require simplifications to solve the original model which may yield sub-optimal solutions [2]. Several optimization techniques have been proposed to c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 113 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE deal with the economic emission dispatch problems. A particle swarm optimization and gamma based method (γ-PSO) has been suggested in [1] to solve the problem. Similar to LGM [1], the coordination equations are used in the iterative algorithm to obtain optimal solution in the γ-PSO method. Unlike existing PSO [9], each particle in the method is represented with respect to gamma, leading to easier convergence. Two novel search methods have been presented in [10] for dealing with the problem. Those are hybrid algorithm and heuristic searches with genetic algorithm (GA). Both techniques can achieve convergence with a smallermaximum number of generations. However, the computational time of the heuristic searches with GA is slower than the one of the hybrid algorithm. An improved bacterial foraging algorithm (BFA) has been applied to solve the short-term HTS problem considering the environmental aspects given in [11]. A non-dominated sorting genetic algorithm-II (NSGA II) method [12] has been applied to economic environmental dispatch of fixed head hydrothermal scheduling problem with both convex and non-convex fuel cost and emission functions. Another method based on integration of predator-prey optimization and Powell search method (PPO-PS) [13] has been implemented for solving economic emission dispatch for fixed-head hydrothermal systems. The PPO-PS is a powerful method for solving the problem,however, there are many control parameters in this method and an appropriate selection of penalty parameters for a good performance is really a difficult work This paper proposes a Hopfield Lagrange network (HLN) based method (HLNM) for solving the economic emission dispatch of fixed-head hydrothermal systems. The proposedHLN method is a combination of Lagrange function and continuous Hopfield neural network where the Lagrange function is directly used as the energy function for the continuous Hopfield neural network. In addition, the HLN is developed by applying the augmented Hopfield terms;therefore, HLN can tackle oscillation of conventional Hopfield network and get faster convergence as well as obtain higher quality solutions. There is a fact that the proposed HLN is a family of deterministic algorithms, so it also copes with the limited applicability to objective function not to be differentiable. Consequently, the HLN cannot deal with systems where fuel cost and emission functions are represented as nonconvex curves.In the proposed method, HLN is used to find a set of nondominated solutions and a fuzzy based mechanism is then exploited to determine the best compromise solution among the obtained ones. The proposed method has been tested on four hydrothermal systems and the obtained results in terms of total fuel cost, emission, and computational time have been compared to those other methods in the literature. 2. Problem Formulation Consider an electric power system having N1 thermal plants and N2 hydro plants. The problem is to find the active power generation of each plant in the system so as the total generation cost and emission of thermal plants is minimized over an M-schedule period time satisfying power balance, water availability constraint, and generation limits. 2.1. Fuel Cost Objective The fuel cost function F1 for all thermal units is approximated by a quadratic function as follows [12]: F1= M X k=1 N1 X i=1 tkafsi +bfsiPsik +cfsiP2 sik,(1) where afsi,bfsi, cfsiarefuel cost coefficients of thermal plant i;Psik is power output of thermal unit iat subinterval k;tkis the duration of subinterval k. 2.2. Emission Objective The atmospheric pollutants such as sulphur oxides (SOx) and nitrogen oxides (NOx) caused by fossilfueled thermal generator can be modeled separately. Each gaseous emission is represented by quadratic function as follows [2]: NOsik =α1si +β1siPsik +γ1siP2 sik,(2) SOsik =α2si +β2siPsik +γ2siP2 sik,(3) COsik =α3si +β3siPsik +γ3siP2 sik,(4) and then the total emission can be calculatedas follows [2]: F2=w1NOsik +w2SOsik +w3COsik,(5) where w1,w2, and w3are positive weighting factors of the individualgaseous emission contribution to the emission objective; α1si,β1si, and γ1si are emission coefficients for NOx;α2si,β2si, and γ2si are emission coefficients for SOx; and α3si,β3si, and γ3si are emission coefficients for CO2. 1) Load Demand Equality Constraint The total power generation from thermal and hydro plants satisfies the total power demand of the system and transmission losses: N1 P i=1 Psik + N2 P j=1 Phjk −PLK −PDK = 0, k= 1,2, . . . , M, (6) c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 114 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE where the power losses in transmission lines are calculated as follows: PLK = N1+N2 P i=1 N1+N2 P j=1 PikBij Pjk+ + N1+N2 P i=1 B0iPik +B00, (7) where PDk,PLk are load demand, transmission loss during subinterval k, in MW; Phjk is generation output of hydro unit jduring subinterval k, in MW; Bij,B0i, and B00 are loss formula coefficients of transmission system. 2) Water Availability Constraints The total water discharge for each hydro plant during the schedule time is fixed: M X k=1 tkqjk =Wj, j = 1,2, . . . , N2,(8) where Wjis volume of water available for generation by hydro plant jduring the scheduled period, and the water discharge qjk for hydro unit jat subinterval kis determined by: qjk =ahj +bhj Phjk +cjP2 hjk,(9) where ahj,bhj,chj are water discharge coefficients of hydro unit j. 3) Generator Operating Limits The power output of thermal and hydro plants should be limited between their upper and lower boundaries: Psi min ≤Psik ≤Psi max, i= 1,2, . . . , N1, k = 1,2, . . . , M. (10) Phj min ≤Phjk ≤Phj max, i= 1,2, . . . , N2, k = 1,2, . . . , M. (11) where Psi max,Psi min are maximum and minimum power output of thermal unit i, respectively; and Phj max,Phj min are maximum and minimum power output of hydro plant j, respectively. 3. HLN for the Problem The Lagrange function Lof the problem is formulated as follows: L= M P k=1 N1 P i=1 tkasi +bsiPsik +csiP2 sik + M P k=1 λk PLk +PDk − N1 P i=1 Psik − N2 P j=1 Phjk! + N2 P j=1 γhj M P k=1 (tkqjk −Wj). (12) In Eq. (12) λk,γhj are Lagrangian multipliers associated with power balance and water constraint, respectively. Further: asi =ψafsi +(1−ψ)(w1α1si +w2α2si +w3α3si),(13) bsi =ψbfsi + (1 −ψ)(w1β1si +w2β2si +w3β3si),(14) csi =ψcfsi + (1 −ψ)(w1γ1si +w2γ2si +w3γ3si),(15) 0≤ψ≤1,(16) where ψis weighting factor for combination of objectives [14]. The energy function Eof the problem is described in terms of neurons is determined in Eq. (17), E= M P k=1 N1 P i=1 tkasi +bsiVsik +csiV2 sik + M P k=1 Vλk PLk +PDk − N1 P i=1 Vsik − N2 P j=1 Vhjk! + N2 P j=1 Vγhj M P k=1 tkqjk −Wj (17) + M P k=1 N1 P i=1 RVsik 0g−1(V)dV + N2 P j=1 RVhjk 0g−1(V)dV !, where Vλk and Vγhj are outputs of the multiplier neurons associated with power balance and water constraint, respectively; Vhjk, Vsik are output of continuous neuron hjk,sik representing Phjk,Psik, respectively. The dynamics of the model for updating neuron inputs are defined as follows: dUsik dt =∂E ∂Vsik =−(tk(bsi + 2csiVsik) +Vλk ∂PLk ∂Vsik −1+Usik)(18) dUhjk dt =∂E ∂Vhjk =−   Vλk ∂PLk ∂Vhjk −1 +Vγhj tk ∂qjk ∂Vhjk +Uhjk   (19) dUλk dt = + ∂E ∂Vλk =PDk +PLk − N1 X i=1 Vsik − N2 X j=1 Vhjk (20) dUγhj dt = + ∂E ∂Vγhj = M X k=1 tkqjk −Wj.(21) The inputs of neurons at step nare updated: U(n) sik =U(n−1) sik −αsi ∂E ∂Vsik ,(22) U(n) hjk =U(n−1) hjk −αhj ∂E ∂Vhjk ,(23) U(n) λk =U(n−1) λk +αλk ∂E ∂Vλk ,(24) U(n) γhj =U(n−1) γhj +αγhj ∂E ∂Vγhj ,(25) c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 115 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE where Uλk,Uγhj are inputs of the multiplier neurons; Usik and Uhjk are inputs of the neurons sik and hjk respectively; αλk,αγh are step sizes for updating of multiplier neurons; αsi,αhj are step sizes for updating of continuous neurons. The outputs of continuous neurons and multiplier neurons: Vsik =g(Usik) = (Psi max −Psi min)1+tanh(σUsik) 2+Psi min,(26) Vhjk =g(Uhjk) = (Phj max −Phj min)1+tanh(σUhjk) 2+Phj min,(27) where σis slope of the sigmoid functionwhich determines the shape of the sigmoid function. The outputs of multiplier neurons are determined using a transfer function: Vλk =Uλk,(28) Vγhj =Uγhj,(29) 3.1. Initialization The initial outputs of continuous neurons are set at their middle limits and the multiplier neurons are set as follows: V(0) λk =1 N1 N1 X i=1 tkbsi + 2csiV(0) sik /1−∂PLk ∂Vsik ,(30) V(0) γhj =1 M M X k=1 V(0) λk 1−∂PLk ∂Vhjk /tk ∂qjk ∂Vhjk .(31) 3.2. Stopping Criteria The algorithm will be terminated when either the maximum error Errmax is lower than a predefined threshold or maximum number of iterations Nmax is reached. 4. Best Compromise Solution by Fuzzy-Based Mechanism The economic emission dispatch of hydrothermal system is a very complex problem due to many variables and objectives. Moreover, three cases of dispatch for each system consisting of economic dispatch, emission dispatch and economic emission dispatch are carried out. For economic dispatch, only fuel cost is minimized while emission is neglected and for emission dispatch, only emission is minimized whereas the fuel cost is neglected. On the contrary, for economic emission dispatch, both fuel cost and emission are considered and the compromise solution for the economic emission dispatch must satisfy both fuel cost and emission objectives. However, the determination of the compromise is not simple since there is a conflict between the two objectives for an optimal solution. In fact, if a solution tends to have good fuel cost, its emission will become worse and vice versa. Consequently, the Fuzzy-Based Mechanism is carried out to determine the best compromise. In the technique, two weight factors associate with fuel objective and emission objective are employed to determine a set of non-dominated solutions and then the cardinal priority of each non-dominated solution is calculated. As a result, solution with the highest value of cardinal priority is chosen as a compromise solution. On the other hand, the set of non-dominated solutions has a significant impact on the determination of the compromise solution. If the number of non-dominated solutions is low, a good compromise can be skipped and if a large number of non-dominated solutions is calculated, the task for obtaining the solution is time consuming. Therefore, the determination of the best compromise is not simple and must be carefully carried out. In this paper, the best compromise solution for the problem is determined using the fuzzy satisfying method [14]. The fuzzy goal is represented in linear membership function as follows [14]: µ(Fj) =        1if Fj≤Fjmin, Fjmax −Fj Fjmax −Fjmin if Fjmin < Fj< Fjmax, 0if Fj≥Fjmax, (32) where Fjis the value of objective j;Fjmax and Fjmin are maximum and minimum values of objective j, respectively. For each knon-dominated solution, the membership function is normalized as follows [15]: µk D= Nobj X i=1 µ(Fk i)/ Np X k=1 Nobj X i=1 µ(Fk i),(33) where µk Dis the cardinal priority of k−th nondominated solution, µ(Fj)is membership function of objective j,Nobj is number of objective functions, and Npis number of Pareto-optimal solutions. The solution that attains the maximum membership µk Din the fuzzy set is chosen as the ’best’ solution based on cardinal priority ranking [16]: Max µk D:k= 1,2, . . . , Np.(34) 5. Numerical Results The proposed method has been tested on four systems where the first system has one thermal and one hydro power plant, the second one consists of one thermal and two hydropower plants, the third and last ones c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 116 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE Tab. 1: Result comparison for the economic dispatch for the first three systems (ψ= 1, w1=w2=w3= 0). System Method Fuel cost ($) Emission (kg) CPU time (s)NOxSO2CO2 1 LGM [2] 96 024.42 14 829.94 44 111.89 247 838.53 - EPSO [2] 96 024.61 14 830.00 44 111.98 247 839.50 - γ-PSO [2] 96 024.40 14 829.93 44 111.88 247 838.43 - HLN 96 024.37 14 834.48 44 112.91 247 696.31 0.92 2 LGM [2] 848.241 575.402 4 986.16 2 951.46 - EPSO [2] 848.204 575.513 4 986.00 2 952.00 - γ-PSO [2] 847.908 575.477 4 985.74 2 951.65 - HLN 848.349 575.261 4 986.42 2 950.19 0.4 3 LGM [2] 53 053.79 28 199.21 74 867.81 454 063.64 - EPSO [2] 53 053.79 28 199.21 74 867.80 454 063.56 - γ-PSO [2] 53 053.79 28 199.21 74 867.80 454 063.63 - HLN 53 051.61 28 556.53 74 954.09 458 621.31 0.32 Tab. 2: Result comparison for the emission dispatch for the first three problems (ψ= 0, w1=w2=w3= 1/3). System Method Fuel cost ($) Emission (kg) CPU time (s)NOxSO2CO2NOx+SO2+CO2 1 LGM [2] 96488.08 14376.32 44202.36 242406.08 300984.76 - EPSO [2] 96488.38 14376.41 44202.51 242407.42 300986.33 - γ-PSO [2] 96488.08 14376.32 44202.36 242406.08 300984.76 - HLN 96809.80 14267.87 44312.40 241263.61 299843.87 0.49 2 LGM [2] 851.98 571.99 4993.75 2922.82 8488.56 - EPSO [2] 853.15 571.73 4995.19 2922.14 8489.06 - γ-PSO [2] 851.98 571.99 4993.75 2922.82 8488.56 - HLN 851.91 572.00 4993.66 2922.81 8488.47 1.8 3 LGM [2] 54359.64 21739.27 74131.82 373122.57 468993.66 - EPSO [2] 54359.66 21739.27 74131.82 373122.57 468993.66 - γ-PSO [2] 54359.53 21739.19 74131.68 373121.27 468992.14 - HLN 55392.75 19986.58 73824.88 350972.26 444783.71 0.12 have two thermal and two hydropower plants. The data for the thermal and hydro plants in the first three systems are from [3] whereas emission data are from [16]. The data for the last one are from [12]. The proposed method is coded in Matlab 7.2 programming language and run on an Intel 1.8 GHz with 4GB of RAM PC. 5.1. The First Three Systems The objectives of the test systems in this section include one fuel cost and three emissions of NOx, SO2 and CO2scheduled in 24 subintervals with one hour for each. For each system, three cases of dispatches are considered including economic dispatch (ψ= 1, w1= w2=w3= 0), emission dispatch (ψ= 0, w1= w2=w3= 1/3), and economic emission dispatch (ψ= 0.5, w1=w2=w3= 1/3). The obtained results from the proposed method for three dispatch cases including economic dispatch, emission dispatch, and economic emission dispatch for the three test systems are compared to those from other methods including LGM, EPSO, and γ-PSO in [2] as given in Tab. 1, Tab. 2, and Tab. 3. For the economic dispatch, the proposed HLN can obtain better total costs than the others except for the System 2 where the cost is slightly higher than for the others. For the emission dispatch, the proposed HLN can obtain less total emission than the others for all test systems. In the economic emission dispatch, there is a trade-off between total cost and emission objectives and the obtained solutions from the methods are non-dominated as in Tab. 3. The total computational time for each system for the three cases is given in Tab. 4. The study in [2] has not reported computer processor and we fail to compare the processor. However, as indicated in Tab. 4 in the paper, HLN is very fast compared to LGM [2], EPSO [2], γ-PSO [2] since HLN has gotten optimal solutions with 1.51 seconds for System 1, 3 seconds for System 2 and 0.740 second for System 3 whereas that time from LGM is 10 seconds higher, from EPSO is about 100 seconds and from γ-PSO is about 40 seconds. Clearly, these methods are time consuming and it is very slow for convergence as compared to HLN. Convergence characteristics obtained by HLN in terms of maximum error and number of iterations for economic dispatch of System 1, System 2 and System 3 are depicted in Fig. 1, Fig. 2 and Fig. 3. Clearly, HLN has obtained the optimal solution with the lowest number of iterations at System 3, 2594 iterations and with the highest number of iterations at System 1, 6263 iterations. Consequently, the convergence time for economic dispatch of the System 1 is the longest meanwhile this time for System 3 is the fastest and they are respectively 0.92 and 0.32 as reported in Tab. 1. The optimized control variables for test System 1 is given in table Tab. A in Appendix section. c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 117 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE Tab. 3: Result comparison for the economic emission dispatch of the first three system (ψ= 0.5, w1=w2=w3= 1/3). System Method Fuel cost ($) Emission (kg) CPU time (s)NOxSO2CO2NOx+SO2+CO2 1 LGM [2] 96421.702 14384.101 44176.312 242456.004 300984.76 - EPSO [2] 96421.725 14384.108 44176.324 242456.109 300986.33 - γ-PSO [2] 96421.46 14384.03 44176.195 242454.92 300984.762 - HLN 96465.712 14328.17 44181.95 241776.424 300286.544 0.1 2 LGM [2] 851.208 572.235 4992.707 2923.986 8488.928 - EPSO [2] 851.079 572.264 4992.547 2923.061 8487.872 - γ-PSO [2] 852.388 571.97 4994.167 2923.301 8489.438 - HLN 850.065 572.723 4991.026 2927.027 8490.776 0.8 3 LGM [2] 54337.014 21745.127 74144.989 373165.02 469025.136 - EPSO [2] 54337.027 21745.138 74115.007 373165.186 469025.331 - γ-PSO [2] 54336.888 21745.021 74114.821 373163.42 469023.262 - HLN 55158.62 20031.652 73731.958 351363.758 445127.368 0.3 1000 2000 3000 4000 5000 6000 7000 0 0.5 1 1.5 2 2.5 x 104 Number of iterations = 6263 Maximum error Fig. 1: Convergence characteristic obtained by HLN for economic dispatch of System 1. 1000 1500 2000 2500 3000 3500 0 100 200 300 400 500 600 Number of iterations = 3499 Maximum error Fig. 2: Convergence characteristic obtained by HLN for economic dispatch of System 2. 5.2. The Fourth System The test system in this case includes one total cost function and one emission function scheduled in three subintervals with eight hours for each [12]. The proposed HLN method is applied for obtaining the opti500 1000 1500 2000 2500 3000 0 0.5 1 1.5 2 2.5 3 3.5 4 Number of iterations = 2594 Maximum error Fig. 3: Convergence characteristic obtained by HLN for economic dispatch of System 3. Tab. 4: Computational time comparison for the first three systems. Method System 1 System 2 System 3 LGM [2] 14.83 11.46 12.26 EPSO [2] 95.36 83.73 105.0 γ-PSO [2] 43.44 39.27 49.01 HLN 1.51 3 0.740 mal solutions for the economic, emission and economic emission dispatches. The values of w1,w2and w3in Eq. (13), Eq. (14), Eq. (15) are fixed at 1, 0 and 0, respectively. The value of ψin Eq. (16) is set to one and zero for the economic and emission dispatches, respectively. For the case of economic emission dispatch, we have determined 11 non-dominated solutions to form Pareto optimal front with the change of weight factor ψfrom 0 to 1. The best compromise solution from the obtained 11 nondominated solutions is determined by the fuzzy based mechanism in Section 4. The obtained results in terms of fuel cost, emission and computational time for the three cases from the proposed method are compared to those from PSO, PSO with penalty method (PSO-PM), predator-prey optimization (PPO), PPO with penalty c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 118 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE Tab. 5: Result comparison for the three cases of dispatch of the fourth system. Method Economic dispatch Emission dispatch Compromise dispatch Cost ($) CPU (s) Emis. (lb) CPU (s) Cost ($) Emis. (lb) CPU (s) PSO-PM [13] 65741 18.25 585.67 18 65821 620.78 18.98 PSO [13] 65241 18.32 579.56 18.31 65731 618.78 19.31 PPO-PM [13] 64873 16.14 572.71 15.93 65426 612.34 16.53 PPO [13] 64718 15.99 569.73 15.18 65104 601.16 16.34 PPO-PS-PM[13] 64689 15.98 568.78 15.92 65089 600.24 16.15 PPO-PS [13] 64614 15.89 564.92 15.45 65058 594.18 16.74 HLN 64576 0.3 579.12 0.68 64807 617.64 0.74 method (PPO-PM), PPO-PS with penalty method (PPO-PS-PM), and PPO-PS in [13] as given in Tab. 5. As observed from the table, the proposed method can obtain better cost than other methods for the two cases of economic and combined economic emission dispatch. However, HLN gets lower emission than PSO-PM and PSO only and higher emission than rest of methods for emission dispatch and economic emission dispatch. Furthermore, as seen in Tab. 5 HLN has been run under one second for each dispatch case while it has taken from 15 to 20 seconds for other methods. Obviously, HLN is much faster than these methods although no computer has been reported for the methods in [13] and computer processor comparison has not been performed. Figure 4 shows the convergence characteristic obtained by HLN for economic dispatch of the system. Obviously, the applied HLN method can obtain the optimal solution for the case with fewer number ofiterations than that for three systems above and therefore the execution time for the system is shorter than that for the three systems. 200 400 600 800 1000 1200 1400 0 0.5 1 1.5 2 x 105 Number of iterations = 1338 Maximum error Fig. 4: Convergence characteristic obtained by HLN for economic dispatch of System 4. 6. Conclusions In this paper, a Hopfield Lagrange network based method has been efficiently implemented for solving the economic emission short-term hydrothermal scheduling problem. The proposed method is a combination of Lagrange function and continuous Hopfield neural network for solving optimal single-objective dispatch problem and a fuzzy based mechanism for obtaining the best compromise solution among several non-dominated solutions. The Hopfield Lagrange network is an improvement of the continuous Hopfield neural network by using the Lagrange function as its energy function. The advantages of the Hopfield Lagrange network are that it is simple, fast, and efficient for solving optimization problems. The proposed method has been tested on four systems with different number of objectives and the obtained results have been compared to those from other methods in the literature. The result comparisons have indicated that the proposed method can obtain better solution than many other methods with shortercomputational time. Therefore, the proposed method can be very favored for solving economic emission dispatch of short-term fixed-head hydrothermal problems. 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DOI: 10.1049/ip-gtd:20020176. About Authors Thang NGUYEN TRUNG was born in 6th August 1985. He received his M.Sc. from university of technical education Ho Chi Minh City in 2011. His research interests include optimization of power system, power system operation and control and Renewable Energy. Dieu VONGOC received his B.Sc. and M.Sc. degrees in Electrical Engineering from Ho Chi Minh City University of Technology, Ho Chi Minh city, Vietnam, in 1995 and 2000, respectively and his Ph.D. degree in Energy from Asian Institute of Technology (AIT), Pathumthani, Thailand in 2007. He is currently a Research Associate at Energy Field of Study, AIT and a lecturer at Department of Power Systems Engineering, Faculty of Electrical and Electronic Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh City, Vietnam. His research interests are applications of AI in power system optimization, power system operation and control, power system analysis, and power systems under deregulation. c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 120 POWER ENGINEERING AND ELECTRICAL ENGINEERING VOLUME: 14 |NUMBER: 2 |2016 |JUNE Appendix Tab. A: Control variables for System 1 with four objective function. Subinterval Economic dispatch Emission dispatch Economic emission dispatch Vsk (MW) Vhk (MW) Vsk (MW) Vhk (MW) Vsk (MW) Vhk (MW) 1 231.8904 235.1858 273.9742 191.3175 262.9423 202.7468 2 203.7237 232.3999 255.0545 178.9122 241.6737 192.7512 3 194.3511 231.4743 248.7678 174.7781 234.6012 189.4216 4 186.8589 230.735 243.7456 171.4712 228.9492 186.7587 5 180.3075 230.0889 239.3563 168.578 224.0081 184.4292 6 199.0364 231.9369 251.91 176.8451 238.1364 191.0863 7 262.0162 238.1735 294.2544 204.5551 285.7133 213.4201 8 372.883 249.2464 369.2893 252.9993 369.7242 252.5449 9 431.1394 255.1173 408.9719 278.2835 414.0031 273.0073 10 440.7196 256.0864 415.5146 282.4302 421.2937 276.3661 11 459.9057 258.0303 428.6318 290.7249 435.9021 283.0872 12 469.5116 259.0052 435.2064 294.873 443.2198 286.4496 13 350.0483 246.9553 353.7829 243.0565 352.3935 244.5063 14 373.8355 249.3421 369.9367 253.4137 370.4474 252.88 15 384.3185 250.3959 377.0649 257.9718 378.4086 256.5668 16 419.6542 253.9568 401.1346 273.3081 405.2662 268.9784 17 484.8987 260.569 445.7479 301.5109 454.9472 291.8319 18 503.1996 262.4327 458.3018 309.395 468.9038 298.2275 19 464.7076 258.5175 431.9177 292.7989 439.5598 284.7682 20 443.5954 256.3775 417.4794 283.6742 423.4827 277.3739 21 397.6752 251.7403 386.1555 263.774 388.5566 261.2612 22 354.8016 247.4318 357.0085 245.1277 355.9999 246.1804 23 312.0972 243.1598 328.071 226.4914 323.6217 231.1235 24 277.1106 239.6738 304.4332 211.1759 297.1317 218.7611 c 2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING 121