Hopfield Lagrange Network Based Method for Economic Emission Dispatch Problem of Fixed-Head Hydro Thermal Systems

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.


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 lim-its of both thermal and hydro plants [1].In practical systems, thermal power generating stations are the sources of carbon dioxide (CO 2 ), sulfur dioxide (SO 2 ), and nitrogen oxides (NO x ) 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 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.

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.

Fuel Cost Objective
The fuel cost function F1 for all thermal units is approximated by a quadratic function as follows [12]: where a f si , b f si , cfsiarefuel cost coefficients of thermal plant i; P sik is power output of thermal unit i at subinterval k; t k is the duration of subinterval k.

Emission Objective
The atmospheric pollutants such as sulphur oxides (SO x ) and nitrogen oxides (NO x ) caused by fossilfueled thermal generator can be modeled separately.Each gaseous emission is represented by quadratic function as follows [2]: and then the total emission can be calculatedas follows [2]: where w 1 , w 2 , and w 3 are positive weighting factors of the individualgaseous emission contribution to the emission objective; α 1si , β 1si , and γ 1si are emission coefficients for NO x ; α 2si , β 2si , and γ 2si are emission coefficients for SO x ; and α 3si , β 3si , and γ 3si are emission coefficients for CO 2 .

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: where the power losses in transmission lines are calculated as follows: where P Dk , P Lk are load demand, transmission loss during subinterval k, in MW; P hjk is generation output of hydro unit j during subinterval k, in MW; B ij , B 0i , and B 00 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: where W j is volume of water available for generation by hydro plant j during the scheduled period, and the water discharge q jk for hydro unit j at subinterval k is determined by: where a hj , b hj , c hj 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: where P si max , P si min are maximum and minimum power output of thermal unit i, respectively; and P hj max , P hj min are maximum and minimum power output of hydro plant j, respectively.

HLN for the Problem
The Lagrange function L of the problem is formulated as follows: In Eq. ( 12) λ k , γ hj are Lagrangian multipliers associated with power balance and water constraint, respectively.Further: where ψ is weighting factor for combination of objectives [14].
The energy function E of the problem is described in terms of neurons is determined in Eq. ( 17), where V λk and V γhj are outputs of the multiplier neurons associated with power balance and water constraint, respectively; V hjk , Vsik are output of continuous neuron hjk, sik representing P hjk , P sik , respectively.The dynamics of the model for updating neuron inputs are defined as follows: V hjk (20) The inputs of neurons at step n are updated: hjk where U λk , U γhj are inputs of the multiplier neurons; U sik and U hjk are inputs of the neurons sik and hjk respectively; α λk , α γh are step sizes for updating of multiplier neurons; α si , α hj are step sizes for updating of continuous neurons.
The outputs of continuous neurons and multiplier neurons: V hjk = g(U hjk ) = (P hj max − P hj min ) 1+tanh(σU hjk ) 2 where σ is slope of the sigmoid functionwhich determines the shape of the sigmoid function.The outputs of multiplier neurons are determined using a transfer function:

Initialization
The initial outputs of continuous neurons are set at their middle limits and the multiplier neurons are set as follows:

Stopping Criteria
The algorithm will be terminated when either the maximum error Err max is lower than a predefined threshold or maximum number of iterations N max is reached.

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]: where F j is the value of objective j; F j max and F j min are maximum and minimum values of objective j, respectively.For each k non-dominated solution, the membership function is normalized as follows [15]: where µ k D is the cardinal priority of k − th nondominated solution, µ(F j ) is membership function of objective j, N obj is number of objective functions, and N p is number of Pareto-optimal solutions.The solution that attains the maximum membership µ k D in the fuzzy set is chosen as the 'best' solution based on cardinal priority ranking [16]:

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 Tab. 1: Result comparison for the economic dispatch for the first three systems (ψ = 1, w 1 = w 2 = w 3 = 0).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.

The First Three Systems
The objectives of the test systems in this section include one fuel cost and three emissions of NO x , SO 2 and CO 2 scheduled in 24 subintervals with one hour for each.For each system, three cases of dispatches are considered including economic dispatch (ψ = 1, w 1 = w 2 = w 3 = 0), emission dispatch (ψ = 0, w 1 = w 2 = w 3 = 1/3), and economic emission dispatch (ψ = 0.5, w 1 = w 2 = w 3 = 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.

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 opti- Tab.4: Computational time comparison for the first three systems.The values of w 1 , w 2 and w 3 in Eq. ( 13), Eq. ( 14), Eq. ( 15) are fixed at 1, 0 and 0, respectively.The value of ψ in 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 ψ from 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 Tab.5: Result comparison for the three cases of dispatch of the fourth system.method (PPO-PM), PPO-PS with penalty method (PPO-PS-PM), and PPO-PS in [13] as given in Tab. 5.

Method
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.

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.

About Authors
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2016 ADVANCES IN ELECTRICAL AND ELECTRONIC ENGINEERING

Fig. 1 :
Fig. 1: Convergence characteristic obtained by HLN for economic dispatch of System 1.

Fig. 2 :
Fig. 2: Convergence characteristic obtained by HLN for economic dispatch of System 2.

Fig. 3 :
Fig. 3: Convergence characteristic obtained by HLN for economic dispatch of System 3.

Fig. 4 :
Fig. 4: Convergence characteristic obtained by HLN for economic dispatch of System 4.
NGUYEN TRUNG was born in 6 th 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.