This paper proposes an augmented Lagrange Hopfield network (ALHN) based method for solving multiobjective short term fixed head hydrothermal scheduling problem. The main objective of the problem is to minimize both total power generation cost and emissions of NO
_{x}
, SO
_{2}
, and CO
_{2}
over a scheduling period of one day while satisfying power balance, hydraulic, and generator operating limits constraints. The ALHN method is a combination of augmented Lagrange relaxation and continuous Hopfield neural network where the augmented Lagrange function is directly used as the energy function of the network. For implementation of the ALHN based method for solving the problem, ALHN is implemented for obtaining nondominated solutions and fuzzy set theory is applied for obtaining the best compromise solution. The proposed method has been tested on different systems with different analyses and the obtained results have been compared to those from other methods available in the literature. The result comparisons have indicated that the proposed method is very efficient for solving the problem with good optimal solution and fast computational time. Therefore, the proposed ALHN can be a very favorable method for solving the multiobjective short term fixed head hydrothermal scheduling problems.
1. Introduction
The short term hydrothermal scheduling (HTS) problem is to determine power generation among the available thermal and hydro power plants so that the fuel cost of thermal units is minimized over a schedule time of a single day or a week while satisfying both hydraulic and electrical operational constraints such as the quantity of available water, limits on generation, and power balance
[1]
. However, the major amount electric power in power systems is produced by thermal plants using fossil fuel such as oil, coal, and natural gases
[2]
. In fact, the process of electricity generation from fossil fuel releases several contaminants such as nitrogen oxides (NO
_{x}
), sulphur dioxide (SO
_{2}
), and carbon dioxide (CO
_{2}
) into the atmosphere
[3]
. Therefore, the HTS problem can be extended to minimize the gaseous emission as a result of the recent environmental requirements in addition to the minimization the fuel cost of thermal power plants, forming the multiobjective HTS problem. The multiobjective HTS problem is more complex than the HTS problem since it needs to find several obtained nondominated solutions to determine the best compromise solution which leads to time consuming. Therefore, the solution methods for the multiobjective HTS have to be efficient and effective for obtaining optimal solutions.
In the past decades, several conventional methods have been used to solve the classical HTS problem neglecting environment aspects such as dynamic programming (DP)
[4]
, network flow programming (NFP)
[5]
, Lagrange relaxation (LR)
[6]
, and Benders decomposition
[7]
methods. Among these methods, the DP and LR methods are more popular ones. However, the computational and dimensional requirements of the DP method increase drastically with largescale system planning horizon which is not appropriate for dealing with largescale problems. 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 results from the dual problem formulation and might oscillate, leading to divergence for some problems with operation limits and nonconvexity of incremental heat rate curves of generators. The Benders decomposition method is usually used to reduce the dimension of the problem into subproblems which can be solved by DP, Newton’s, or LR method. In addition to the conventional methods, several artificial intelligence based methods have been also implemented for solving the HTS problem such as simulated annealing (SA)
[8]
, evolutionary programming (EP)
[9]
, genetic algorithm (GA)
[10]
, differential evolution (DE)
[11]
, and particle swarm optimization (PSO)
[12]
. These methods can find a near optimum solution for a complex problem. However, these metaheuristic search methods are based on a population for searching an optimal solution, leading to time consuming for largescale problems. More, these methods need to be run several times to obtain an optimal solution which is not appropriate for obtaining several nondominated solution for a multiobjective optimization problem. Recently, neural networks have been implemented for solving optimization problem in hydrothermal systems such as twophase neural network
[13]
, combined Hopfield neural network and Lagrange function (HLN)
[14]
, and combined augmented Lagrange function with Hopfield neural network
[15

17]
. The advantage of the neural networks is fast computation using parallel processing. Moreover, the Hopfield neural network based on the Lagrange function can also overcome other drawbacks of the conventional Hopfield network in finding optimal solutions for optimization problems such as easy implementation and global solution. Therefore, the neural networks are more appropriate for solving multiobjective optimization problems with several solutions determined for each problem.
In this paper, an augmented Lagrange Hopfield network (ALHN) based method is proposed for solving multiobjective short term fixed head HTS problem. The main objective of the problem is to minimize both total power generation cost and emissions of NO
_{x}
, SO
_{2}
, and CO
_{2}
over a scheduling period of one day while satisfying power balance, hydraulic, and generator operating limits constraints. The ALHN method is a combination of augmented Lagrange relaxation and continuous Hopfield neural network where the augmented Lagrange function is directly used as the energy function of the network. For implementation of the ALHN based method for solving the problem, ALHN is implemented for obtaining nondominated solutions and fuzzy set theory is applied for obtaining the best compromise solution. The proposed method has been tested on different systems with different analyses and the obtained results have been compared to those from other methods available in the literature including λγ iteration method (LGM), existing PSObased HTS (EPSO), and PSO based method (PM) in
[3]
and bacterial foraging algorithm (BFA)
[2]
.
The organization of this paper is as follows. Section 2 addresses the multiobjective HTS problem formulation. The proposed ALHN based method is described in Section 3. Numerical results are presented in Section 4. Finally, the conclusion is given.
2. Problem Formulation
The main objective of the economic emission dispatch for the HTS problem is to minimize the total fuel cost and emissions of all thermal plants while satisfying all hydraulic, system, and unit constraints. Mathematically, the fixedhead shortterm hydrothermal scheduling problem including
N_{1}
thermal plants and
N_{2}
hydro plants scheduled in
M
subintervals is formulated as follows:
where
F_{1sk}
is fuel cost function;
F_{2sk}
,
F_{3sk}
and
F_{4sk}
are emission function of NO
_{x}
, SO
_{2}
, and CO
_{2}
of sth thermal plant at
k
th subinterval scheduling, respectively;
w_{i}
(
i
= 1, …, 4) are weights corresponding to the objectives.
subject to:
Power balance constraints:
where
B_{ij}, B_{0i}
, and
B_{00}
are loss formula coefficients of transmission system
Water availability constraints:
Generator operating limits:
3. ALHN based Method for the Problem
 3.1 ALHN for optimal solutions
For implementation of the proposed ALHN for finding optimal solution of the problem, the augmented Lagrange function is firstly formulated and then this function is used as the energy function of conventional Hopfield neural network. The model of ALHN is solved using gradient method.
The augmented Lagrange function
L
of the problem is formulated as follows:
where
λ_{k}
and
γ_{h}
are Lagrangian multipliers associated with power balance and water constraints, respectively;
β_{k}
,
β_{h}
are penalty factors associated with power balance and water constraints, respectively; and
The energy function
E
of the problem is described in terms of neurons as follows:
where
V_{λk}
and
V_{γh}
are the outputs of the multiplier neurons associated with power balance and water constraints, respectively;
V_{hk}
and
V_{sk}
are the outputs of continuous neurons
hk, sk
representing
P_{hk}, P_{hk}
, respectively.
The dynamics of the model for updating inputs of neurons are defined as follows:
where
where
B_{hj}
and
B_{si}
are the loss coefficients related to hydro and thermal plants, respectively;
B_{sh}
and
B_{hs}
are the loss coefficients between thermal and hydro plants and
B_{sh}
=
B_{hs}^{T}
.
The algorithm for updating the inputs of neurons at step
n
is as follows:
where
U_{λk}
and
U_{γh}
are the inputs of the multiplier neurons;
U_{sk}
and
U_{hk}
are the inputs of the neurons
sk
and
hk
, respectively;
α_{λk}
and
α_{γh}
are step sizes for updating the inputs of multiplier neurons; and
α_{sk}
and
α_{hk}
are step sizes for updating the inputs of continuous neurons.
The outputs of continuous neurons representing power output of units are calculated by a sigmoid function:
where σ is slope of sigmoid function that determines the shape of the sigmoid function
[15]
.
The outputs of multiplier neurons are determined based on the transfer function as follows:
The proof of convergence for ALHN is given in
[15]
.
3.1.1 Initialization
The algorithm of ALHN requires initial conditions for the inputs and outputs of all neurons. For the continuous neurons, their initial outputs are set to middle points between the limits:
where
V_{hk}^{(0)}
and
V_{sk}^{(0)}
are the initial output of continuous neurons
hk
and
sk
, respectively.
The initial outputs of the multiplier neurons are set to:
The initial inputs of continuous neurons are calculated based on the obtained initial outputs of neurons via the inverse of the sigmoid function for the continuous neurons or the transfer function for the multiplier neurons.
3.1.2 Selection of parameters
By experiment, the value of σ is fixed at 100 for all test systems. The other parameters will vary depending on the data of the considered systems. For simplicity, the pairs of
α_{sk}
and
α_{hk}
as well as
β_{k}
and
β_{h}
can be equally chosen.
3.1.3 Termination criteria
The algorithm of ALHN will be terminated when either maximum error
Err_{max}
is lower than a predefined threshold
ε
or maximum number of iterations
N_{max}
is reached.
3.1.4 Overall procedure
The overall algorithm of the ALHN for finding an optimal solution for the HTS problem is as follows.

Step 1: Select parameters for the model in Section 3.1.2.

Step 2: Initialize inputs and outputs of all neurons using (33)(36) as in Section 3.1.1.

Step 3: Setn= 1.

Step 4: Calculate dynamics of neurons using (18)(21).

Step 5: Update inputs of neurons using (25)(28).

Step 6: Calculate output of neurons using (29)(32).

Step 7: Calculate errors as in section 3.1.3.

Step 8: IfErrmax> ε andn
 3.2 Best compromise solution by fuzzybased mechanism
In a multiobjective problem, there often exists a conflict among the objectives. Therefore, finding the best compromise solution for a multiobjective problem is a very important task. To deal with this issue, a set of optimal nondominated solutions known as Paretooptimal solutions is found instead of only one optimal solution. The Pareto optimal front of a multiobjective problem provides decision makers several options for making decision. The best compromise solution will be determined from the obtained nondominated optimal solution. In this paper, the best compromise solution from the Paretooptimal front is found using fuzzy satisfying method
[18]
. The fuzzy goal is represented in linear membership function as follows:
where
F_{j}
is the value of objective
j
and
F_{jmax}
and
F_{jmin}
are maximum and minimum values of objective
j
, respectively.
For each
k
nondominated solution, the membership function is normalized as follows
[19]
:
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 Paretooptimal 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:
4. Numerical Results
The proposed ALHN based method has been tested on four hydrothermal systems. The algorithm of ALHN is implemented in Matlab 7.2 programming language and executed on an Intel 2.0 GHz PC. For termination criteria, the maximum tolerance
ε
is set to 10
^{−5}
for economic dispatch and emission dispatches and to 5×10
^{−5}
for determination of the best compromise solution.
 4.1 Economic and emission dispatches
In this section, the proposed ALHN is tested on four systems. There are one thermal and one hydro power plants for the first system, one thermal and two hydropower plants for the second system, two thermal and two hydropower plants for the third systems, and two thermal and two hydropower plants for the fourth system. The data for the first three systems are from
[1]
and emission data from
[20]
. The data for the fourth system is from
[2]
.
4.1.1 Case 1: The first three systems
For each system, the proposed ALHN is implemented to obtain the optimal solution for the cases of economic dispatch (
w_{1}
= 1,
w_{2}
=
w_{3}
=
w_{4}
= 0), emission dispatch (
w_{1}
=0,
w_{2}
=
w_{3}
=
w_{4}
=1/3), and the compromise case (
w_{1}
= 0.5,
w_{2}
=
w_{3}
=
w_{4}
= 0.5/3). The result comparisons for the three cases from the proposed ALHN with other methods including LGM, EPSO, and PM in
[3]
are given in
Tables 1
,
2
, and
3
. For the economic dispatch, the proposed ALHN can obtain better total costs than the other except for the system 2 which is slightly higher than the others. For the emission dispatch, the proposed ALHN can obtain less total emission than the others for all systems. In the compromise case, there is a tradeoff between total cost and emission and the obtained solutions from the methods are nondominated as in
Table 3
. The total computational times for economic dispatch, emission dispatch, and compromise case of the three systems from the proposed ALHN are compared to those from LGM, EPSO, and PM methods in
[3]
. As observed from the table, the proposed method is faster than the others for obtaining optimal solution. There is no computer reported for the methods in
[3]
.
Result comparison for the economic dispatch for first three systems (w1= 1,w2=w3=w4= 0)
Result comparison for the economic dispatch for first three systems (w_{1} = 1, w_{2} = w_{3} = w_{4} = 0)
Result comparison for the economic dispatch for first three systems (w1= 0,w2=w3=w4= 1/3)
Result comparison for the economic dispatch for first three systems (w_{1} = 0, w_{2} = w_{3} = w_{4} = 1/3)
Result comparison for the compromise case of three first systems (w1= 0.5,w2=w3=w4= 0.5/3)
Result comparison for the compromise case of three first systems (w_{1} = 0.5, w_{2} = w_{3} = w_{4} = 0.5/3)
4.1.2 Case 2: The fourth system
For this system, each of the four objectives is individually optimized. The results obtained by the proposed ALHN for each case is given in
Table 4
. The minimum total cost and emission from the proposed ALHN is compared to those from BFA
[2]
in
Table 5
. In all cases, the proposed ALHN method can obtain better solution than BFA except for the case of CO
_{2}
emission individual optimization.
Total cost and emission for each individual objective minimization
Total cost and emission for each individual objective minimization
Result comparison for individual minimization of each objective
Result comparison for individual minimization of each objective
 4.2 Determination of the best compromise solution
In this section, the best compromise solution is determined for the first system in Section 4.1. For obtaining the best compromise solution for the system, three following cases are considered.
4.2.1 Case 1: Best compromise for two objectives
The best compromise solution for two objectives among the four objectives of this system is determined. The two objectives include the fuel cost and another emission objective while the other emission objectives are neglected. Therefore, there are three subcases for this combination including fuel cost and NO
_{x}
, fuel cost and SO
_{2}
, and fuel cost and CO
_{2}
. For each subcase, 21 nondominated solutions are obtained by ALHN to form a Paretooptimal front and the best compromise solution is determined by the fuzzy based mechanism. The best compromise solution for each subcase is given in
Table 7
. In this table, the best compromise solution for each subcase is determined via the value of the membership function
μ_{D}
and the weight associated with each objective function is determined accordingly. For Subcase 1, the best compromise solution is found at
w_{1}
= 0.35 and w
_{2}
= 0.65 corresponding to
μ_{D}
= 0.0547 at the solution number 14 among the 21 nondominated solutions. The total fuel cost for this subcase is $96,293.5771 with the total emission of 14,397.5374 kg NO
_{x}
. The Paretooptimal front for this subcase is given in
Fig. 1
.
Fig. 2
depicts the methodology to determine the best compromise solution based on the relationship between membership function and the weight of objective. Similarly, the best compromise solution for Subcase 2 and Subcase 3 is determined in the same manner of Subcase 1.
Computational time comparison for the first three systems
Computational time comparison for the first three systems
The best compromise solutions for Case 1 with two objectives
The best compromise solutions for Case 1 with two objectives
Paretooptimal front for fuel cost and NO_{x} emission in Subcase 1 of Case 1
Variation of membership functions against weight w_{2} = 1− w_{1}, w_{3} = w_{4} = 0 in Subcase 1 of Case 1
4.2.2 Case 2: Best compromise for three objectives
The best compromise solution for three objectives among the four objectives is determined. The three objectives include the fuel cost and two other emission objectives among NO
_{x}
, SO
_{2}
, and CO
_{2}
. Therefore, there are three subcases considered for this case.
Table 8
shows the best compromise solution for each subcase with three objective functions with corresponding weight factors. For each subcase, the best compromise solution is obtained based on the value of the membership function from different 43 nondominated solutions.
The best compromise solutions for Case 2 with three objectives
The best compromise solutions for Case 2 with three objectives
4.2.3 Case 3: Best compromise for four objectives
The best compromise for all four objectives is considered in this section. The best compromise solution for this case is obtained from 284 nondominated solutions based on the value of membership function
μ_{D}
given in
Table 9
.
The best compromise solutions for Case 3 with four objectives
The best compromise solutions for Case 3 with four objectives
The total computational times for the three cases above are given in
Table 10
. The total computational time here is the total time for calculation of all nondominated solutions and determination of the best compromise solution. The total computational time for Case 1, Case 2, and Case 3 includes 21, 43, and 284 nondominated solutions, respectively. Obviously, the computational time increases with the number of objective functions.
Computational time for all test cases
Computational time for all test cases
5. Conclusion
In this paper, the proposed ALHN based method is effectively implemented for solving the multiobjective shortterm fixed head hydrothermal scheduling problem. ALHN is a continuous Hopfield neural network with its energy function based on augmented Lagrange function. The ALHN method can find an optimal solution for an optimization in a very fast manner. In the proposed method for solving the problem, the ALHN method is implemented for obtaining the optimal solutions for different cases and a fuzzy based mechanism is implemented for obtaining the best compromise solution. The effectiveness of the proposed method has been verified through four test systems with the obtained results compared to those from other methods. The result comparison has indicated that the proposed method can obtain better optimal solutions than other methods. Moreover, the proposed method has also implemented to determine the best compromise solutions for different cases. Therefore, the proposed ALHN method is an efficient solution method for solving multiobjective shortterm fixed head hydrothermal scheduling problem.
Nomenclature
a_{1s}, b_{1s}, c_{1s}Cost coefficients for thermal unit s, a_{h}, b_{h}, c_{h}Water discharge coefficients for hydro unit h, d_{1s}, e_{1s}, f_{1s}NO_{x} emission coefficients, d_{2s}, e_{2s}, f_{2s}SO_{2} emission coefficients, d_{3s}, e_{3s}, f_{3s}CO_{2} emission coefficients, P_{Dk}Load demand of the system during subinterval k, in MW, P_{hk}Generation output of hydro unit h during subinterval k, in MW, P_{h}^{min}, P_{h}^{max}Lower and upper generation limits of hydro unit h, in MW, P_{Lk}Transmission loss of the system during subinterval k, in MW, P_{sk}Generation output of thermal unit s during subinterval k, in MW, P_{s}^{min}, P_{s}^{max}Lower and upper generation limits of thermal unit s, in MW, q_{hk}Rate of water flow from hydro unit h in interval k, in acreft per hour or MCF per hour, r_{hk}Reservoir inflow for hydro unit h in interval k, in acreft per hour or MCF per hour, t_{k}Duration of subinterval k, in hours, W_{h}Volume of water available for generation by hydro unit h during the scheduling period.
BIO
Thang Trung Nguyen received his B.Eng. and M.Eng. degrees in Electrical Engineering from University of Technical education Ho Chi Minh City (UTE), Ho Chi Minh city, Vietnam in 2008 and 2010, respectively. Now, he is teaching at department of electrical and electronics engineering, Ton Duc Thang university and pursuing D.Eng. Degree at UTE, Ho Chi Minh city, Vietnam. His research interests include optimization of power system, power system operation and control and Renewable Energy.
Dieu Ngoc Vo received his B.Eng. and M.Eng. degrees in electrical engineering from Ho Chi Minh City University of Technology, Ho Chi Minh city, Vietnam, in 1995 and 2000, respectively and his D.Eng. degree in energy from Asian Institute of Technology (AIT), Pathumthani, Thailand in 2007. He is Research Associate at Energy Field of Study, AIT and lecturer at Department of Power Systems, Faculty of Electrical and Electronic Engineering, Ho Chi Minh City University of Technology, Ho Chi Minh city, Vietnam. His interests are applications of AI in power system optimization, power system operation and control, power system analysis, and power systems under deregulation.
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