Different versions of Harmony Search Algorithm to Solve Dynamic EconomicEnvironmental Dispatch
^{1}Department of Computer Science
U.S.T.O, B.P 1505,
^{2}Department of Electrical
Engineering,U.S.T.O, B.P 1505,
Email(s): jbenasla@yahoo.fr; belmadan@univusto.dz; rahlim@yahoo.fr
Abstract
This paper presents an application of different versions of Harmony Search algorithm (Harmony Search Algorithm (HSA), Improved Harmony Search Algorithm (IHSA), Global Harmony Search Algorithm (GHSA) and Fast Harmony Search algorithm (FHSA)) to solve the Dynamic EconomicEnvironmental Dispatch (DEED) problem under some equality and inequality constraints. The equality constraints reflect a real power balance, and the inequality constraint reflects the limits of real generation. The voltage levels and security are assumed to be constant. Dynamic EconomicEnvironmental Dispatch problem is obtained by considering both the economy and emission objectives. This biobjective problem is converted into a single objective function using a price penalty factor.
In this paper different versions of HSA are tested on six generators system and its results are compared with the previously published solutions. The results are quite encouraging and useful in the economic emission environment.
Keywords
Dynamic EconomicEnvironmental Dispatch (DEED); Pollution Emission; Fuel cost; Harmony Search Algorithm (HSA); Improved Harmony Search Algorithm (IHS); Global Harmony Search Algorithm (GHSA); Fast Harmony Search Algorithm (FHSA).
Introduction
Evolutionary algorithms are generalpurpose stochastic search methods simulating natural selection and biological evolution. They differ from other optimization methods in the fact maintaining a population of potential solutions to a problem, and not just one solution. Generally, these algorithms work as follows: a population of individuals is randomly initialized where each individual represents a potential solution to the problem. The quality of each solution is evaluated using a fitness function. A selection process is applied during each iteration in order to form a new solution population. This procedure is repeated until convergence is reached. The best solution found is expected to be a nearoptimum solution.
HSA that was proposed by Greem and al. [2] is an evolutionary algorithm imitating the improvisation process of musicians. This process is constituted of three steps, in the original HSA, with a fourth step added in the improved version [3]. In order to improve the fine–tuning characteristic of HSA, Mahdavi and al. [4] developed an Improved Harmony Search Algorithm (IHSA) that differs from original HSA in the fact that some parameters (pitch adjusting rate “PAR” and bandwidth “bw”) are adjusted during the improvisation process. Omran and al [5] proposed another version of HSA named Globalbest Harmony Search Algorithm (GHSA), which borrows concepts from swarm intelligence to enhance the performance of HSA. GHSA is an IHSA version with the pitchadjustment modified such that the new harmony can mimic the best harmony in the Harmony Memory (HM). Our version of HSA (FHSA), is inspired by the concept of reactive search [6] where parameter tuning, which is usually performed offline by the researcher, becomes an integral part of the search algorithm, ensuring flexibility without human intervention. The "learning" component is implemented as a reactive feedback scheme that uses the past history of the search to increase its efficiency and efficacy.
Economic dispatch is one of the most important optimization problems in power system operation and forms the basis of many application programs. The main objective of economic load dispatch of electric power generation is to schedule the committed generating unit outputs so as to meet the load demand at minimum operating cost while satisfying all unit and system constraints. One of those constraints which are always taken into account is the environmental constraint. That is minimization of pollution emission (NO_{x}, CO_{2}, SO_{2}, small quantities of toxic metals, etc) in case of power plants. However, the rise of interests in environment problem in the recent years, it become necessary for power utilities to count this constraint as one of the main objectives, which should be solved together with the cost problem [7, 8]. Thus, we are facing with a biobjective optimization problem to deal with. This problem is converted into monoobjective optimisation problem by introducing price penalty factor.
In this paper, different versions of HSA are applied to solve DEED problem. A six generators system is considered to investigate the effectiveness of the proposed versions.
Problem Formulation
Economic Dispatch
Economic dispatch is the important component of power system optimization. It is defined as the minimization of the combination of the power generation, which minimizes the total cost while satisfying the power balance relation. The problem of economic dispatch can be formulated as minimization of the cost function subjected to the equality and inequality constraints [9].
In power stations, every generator has its input/output curve. It has the fuel input as a function of the power output. But if the ordinates are multiplied by the cost of $/Btu, the result gives the fuel cost per hour as a function of power output [10].
In the practical cases, the fuel cost of generator i may be represented as a polynomial function of real power generation:
_{} (1)
where is the total fuel cost of the system at time t, _{}is real power output at time t, _{}is the number of generators including the slack bus,_{}, _{}and _{}are the cost coefficients of the i^{th} unit.
The total cost of active power generation may be expressed by:
_{} (2)
The Economic Dispatch Problem can be mathematically represented as:
Minimize _{} (3)
Emission Dispatch
The emission function can be expressed as the sum of all types of emission considered, such as NO_{x}, SO_{2}, CO_{2}, particles and thermal emissions, ect, with suitable pricing of weighting on each pollutant emitted [11].
In the present study, only emission of NO_{x} is taken into account. The emissions function of the unit i can be expressed as polynomial function of real power generation of the unit. Therefore, the objective function is:
Minimize NO_{x} emission
_{} (4)
Where _{}is NOx emission at time t and _{}, _{}and _{}are the emission coefficients of the ith unit
Constraints
Equality Constraints
The total power generation must cover the total demand _{} and the real power losses of the system_{}. Hence,
_{} (5)
The real power losses are a function of real power injection_{}and reactive power _{}and voltage nodes. Their expression is given by [12]:
_{}
where_{}, _{}, _{} and _{}, _{} is the total bus number, _{}is the active power demand at bus i, _{}is the reactive demand at bus i, _{}are the real components of bus impedance matrix.
The voltages nodes _{} (in module _{} and phase_{}) and the reactive power injection are assumed constant. In this case, the power transmission losses can be expressed in terms of active power generations by assuming that the demand for power remains constant during dispatch period. This expression is given by:
_{}
where _{}
Inequality Constraints
Generation power should be within the minimum output _{} and the maximum output_{}.
_{} (6)
Combined Economic and Emission Dispatch
Aggregating equations (1) to (6), the power dispatch problem is expressed as a biobjective optimization problem as follows:
_{} (7)
The biobjective combined economic emission dispatch problem is converted into single optimization problem by introducing price penalty factor _{} as follows:
_{} (8)
Subject to the power constraints given by (5) and (6).
The price penalty factor _{} is the ratio between the maximum fuel cost and maximum emission of corresponding generator [13, 14]
_{} (9)
The steps to determine the price penalty factor for a particular load demand are:
1. Find the ratio between maximum fuel cost and maximum emission of each generator.
2. Arrange _{} in ascending order.
3. Add the maximum capacity of each unit_{} one at a time, starting from the smallest _{} until_{}.
4. In this stage, _{} associated with the last unit in the process is the price penalty factor of the given load.
Once the value of _{} is known, then equation (8) can be rewritten in terms of known coefficients and the unknown output of the generators.
_{} (10)
where _{}, _{}and_{},
The Harmony Search Algorithms
Harmony Search Algorithm
The Harmony Search Algorithm (HSA) is inspired from the musical process of searching for a perfect state of harmony. All Harmony Search versions consider the optimization problem defined as:
_{}
Subject to: _{}
The optimization process is directed by four parameters [2, 15]:
1. Harmony Memory Size (HMS) is the number of solution vectors stored in HM.
2. Harmony Memory Considering Rate (HMCR) is the probability of choosing one value from HM and (1HMCR) is the probability of randomly choosing one new feasible value.
3. Pitch Adjusting Rate (PAR) is the probability of choosing a neighbouring value of that chosen from HM.
4. Distance bandwidth (bw) defines the neighbourhood of a value as [xj ± bw·U(0,1)]. U (0,1) is a uniform distribution between 0 and 1.
Another intuitively important parameter is the Number of Iterations (NI) which is the stop criterion of the three previous versions of HSA.
HSA works as follows:
Step 1: Initialize the problem and HSA parameters.
Step 2: Initialize HM by randomly generated (improvised) harmonies.
Step 3: Improvise a new harmony as follows:
for j=1 to p do
if _{} then_{}
else (*Memory consideration*)
begin
_{}
if _{} then (*pitch adjustment*)
begin _{}
endif
endif
done
Step 4: If the New Harmony (NH) is better than the Worst Harmony (WH) in HM then replace WH by NH.
Step 5: Reiter Steps 3 and 4 until satisfaction of the stop criterion.
Improved Harmony Search Algorithm
The IHSA dynamically updates PAR and bw in improvisation step (Step 3). These two parameters change dynamically with generation number as follows: _{}
where: _{} : minimum pitch adjusting rate; _{} : maximum pitch adjusting rate; _{}: number of iterations; _{} : generation number; _{} : minimum bandwidth; _{}: maximum bandwidth.
Global Harmony Search Algorithm
The GHSA modifies the pitch adjustment step of the IHSA as follows:
if U(0,1) ≤ PAR then (*pitch adjustment*)
begin _{}
endif
where best is the index of the best harmony in the HM and k ≈ U(1,p). This pitch adjustment is inspired by the concept of swarm intelligence in Particle Swarm Optimization. The position of a particle is influenced by the best position visited by itself and the best particle in the swarm.
Fast Harmony Search Algorithm
FHSA [16, 17] introduces a prohibition step between step 4 and step 5. It consists in defining a permanent prohibition of the search space (bounds adjustment) to prevent the system from going back on its track and is performed as follows:
for j=1 to p do
if _{}then _{}
else if _{}then _{}
else _{}
endif
done
The stop criterion becomes:
if _{} then STOP endif
where_{}is a real number “small enough” and_{}is the precision term.
Application
All applications have
been performed using ERELEC2 (Etude des Réseaux ELECtriques V2) software. The
software has been developed by LORE (Laboratoire d’Optimisation des Réseaux
Electriques) laboratory at the
Figure 1. IEEE 30bus system in ERELEC2
The generator fuel cost, and NO_{x} emissions function are given in Tables 1 and 2.
Bus 
Minimum output _{}(MW) 
Maximum output _{}(MW) 
Fuel cost _{} ($/h) 

Cost coefficient _{}($/MW².h) 
Cost coefficient _{}($/MW.h) 
Cost coefficient _{}($/h) 

1 
50 
200 
0.00375 
2.0 
0 
2 
20 
80 
0.00175 
1.5 
0 
5 
15 
50 
0.06250 
1.8 
0 
8 
10 
40 
0.00834 
2.0 
0 
11 
10 
30 
0.02500 
1.5 
0 
13 
12 
40 
0.02500 
1.8 
0 
BUS 
Emission function _{} (ton/h) 

Emission coefficient _{}(ton/MW².h) 
Emission coefficient _{}(ton/MW.h) 
Emission coefficient _{}(ton/h) 

1 
6.490 e6 
5.554 e4 
4.091e2 
2 
5.638 e6 
6.047 e4 
2.543 e2 
5 
4.586 e6 
5.094 e4 
4.258 e2 
8 
3.380 e6 
3.550 e4 
5.326 e2 
11 
4.586 e6 
5.094 e4 
4.258 e2 
13 
5.151 e6 
5.555 e4 
6.131 e2 
The daily load demands of IEEE 30bus system are shown in Figure 2.
Figure 2. Daily load curve of IEEE 30bus system in ERELEC2
The total system load of IEEE 30bus system is 283.4 MW. The corresponding coefficients losses are:
_{}, _{}and K = 0.00364.
The NO_{x} price penalty factors are given in Table 3
Dispatch periods 
Price penalty factor _{}(ton/h) 
Dispatch periods 
Price penalty factor _{}(ton/h) 
1 
3380.4055 
13 
7217.8478 
2 
3380.4055 
14 
19182.1697 
3 
3579.9067 
15 
19182.1697 
4 
3579.9067 
16 
19182.1697 
5 
7217.8478 
17 
19182.1697 
6 
7217.8478 
18 
19182.1697 
7 
19182.1697 
19 
19182.1697 
8 
19182.1697 
20 
7217.8478 
9 
19182.1697 
21 
7217.8478 
10 
7217.8478 
22 
3579.9067 
11 
7217.8478 
23 
3380.4055 
12 
3579.9067 
24 
3380.4055 
The parameters of the different versions of HSA are resumed in Table 4.
Version 
Parameters 
Harmony Search Algorithm (HSA) 

Improved Harmony Search Algorithm (IHSA) 

Global Harmony Search Algorithm (GHSA) 

Fast Harmony Search Algorithm (FHSA) 
_{} 
HSA, IHSA and GHSA were allowed to run for 50,000 iterations with these common parameters: HMCR = 0.9; HMS = 14.
Figure 3 shows the result of the DED problem (cost optimization). The FHSA outperforms the other versions with a daily cost of 24 099.184 $/day, which corresponds to a gain of at least 50.71 $/day and more than 18 000 $/year.
The results of the four optimization methods are compared, for the third hour (load=283.4 MW) with those obtained in [1]. Table 5 shows that all Harmony Search based algorithms outperforms the Genetic Algorithm in the case of ED and FHSA is gives the best solution of all.
Algorithm 
Generation cost ($/hr) 
Power losses (MW) 
Genetic Algorithm [1] 
803.1060 
9.9308 
HSA 
798.884 
9.021 
IHSA 
800.584 
9.085 
GHSA 
799.571 
8.958 
FHSA 
788.887 
8.608 
The DEED results for the different versions of HSA are depicted in figures 4 to 6.
These graphs clearly indicate that total fuel cost, transmission losses and NOx emissions increase by increasing the power demand.
The daily results show that FHSA finds a better solution than the other versions of HSA. Indeed, it achieves a minimum daily gain of 187 $/day which is more than 68 000 $/year. Besides, the FHSA results reduce the transmission losses (a daily gain of approximately 70 MW) and the NOx emissions.
In order to demonstrate the performance of the proposed algorithms, a comparison for the power demand of 283.4 MW emerges between Harmony Search based algorithms and Genetic Algorithm [1], as is shown in table 6.

Fuel cost ($/hr) 
Power loss (MW) 
NOx emission (ton/hr) 
Genetic Algorithm[1] 
824.9884 
6.9661 
0.2659 
HSA 
881.131 
3.913 
0.204 
IHSA 
883.329 
3.958 
0.204 
GHSA 
885.757 
3.812 
0.203 
FHSA 
877.110 
2.635 
0.204 
From the results, it is obvious that the Harmony Search based algorithms are performing well in the solution of combined economic emission dispatch. Moreover, they obtain better compromises between cost optimization and emission minimization.
Conclusion
In this paper the Dynamic EconomicEnvironmental Dispatch problem is solved by different versions of Harmony Search Algorithm to minimize the fuel cost and NOx emissions.
Four harmony search based algorithms are implemented in ERELEC2 software that allows the user to resolve different types of problems one can meet in the field of power network optimization.
Satisfactory results are obtained by adapting harmony search based algorithms to six generators system and found that those results outperform some results in the literature.
The advantage of using ERELEC2 software to resolve the power network optimization problem is its simplicity in modelling and visualising all the results of economic and/or environmental dispatch.
Thus, the proposed software is a simple tool for the power industry to aid in curbing pollution and protecting our environment.
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