A Comparison of Harmony Search Algorithm, Improved Harmony Search Algorithm and Biogeography-Based Optimization Algorithm for Solving Constrained Economic Load Dispatch Problems

Abstract

This paper presents a comparison of Harmony Search (HS) algorithm, Improved Harmony Search (IHS) algorithm and Biogeography-Based Optimization (BBO) algorithm for solving constrained economic load dispatch problems in the power system. In the IHS algorithm, multiple harmony memory consideration rates and dynamic pitch adjusting rate are used to generate new solution vector. The proposed algorithms have been successfully tested in the test system which consists of 20 generating units with ramp rate limits and valve point loading constraint. The results obtained through the simulation results reveal that IHS algorithm has minimum total fuel cost and has good convergence characteristics when compared to both HSA and BBO algorithm.

Headnote

This paper presents a comparison of Harmony Search (HS) algorithm, Improved Harmony Search (IHS) algorithm and Biogeography-Based Optimization (BBO) algorithm for solving constrained economic load dispatch problems in the power system. In the IHS algorithm, multiple harmony memory consideration rates and dynamic pitch adjusting rate are used to generate new solution vector. The proposed algorithms have been successfully tested in the test system which consists of 20 generating units with ramp rate limits and valve point loading constraint. The results obtained through the simulation results reveal that IHS algorithm has minimum total fuel cost and has good convergence characteristics when compared to both HSA and BBO algorithm.

Keywords: Improved Harmony Search (IHS) algorithm, Harmony Search (HS) algorithm, Biogeography-Based Optimization (BBO) algorithm, Economic load dispatch, Ramp rate limits, Valve point loading constraints

(ProQuest: ... denotes formulae omitted.)

Introduction

Most of the electrical power utilities in the world are required to ensure that electrical energy requirement from the customer is served smoothly in accordance with the respective policy of the country. Despite serving the power demands of the country, the power utility has also to ensure that the electrical power is generated within minimal cost. Thus, the total demand must be appropriately shared among the generating units with an objective to minimizing the total generation cost for the system in order to satisfy the economic operation of the system. Economic dispatch is a procedure to determine the electrical power to be generated by the committed generating units in a power system so that the total generation cost of the system is minimized, while satisfying the load demand simultaneously.

Economic dispatch can be defined as "the operation of generation facilities to produce energy at the lowest cost to reliably serve consumers, recognizing any operational limits of generation and transmission facilities". Economic Load Dispatch (ELD) is an important optimization task in power system operation for allocating generation among the committed units such that the constraints imposed are satisfied and the energy requirement in terms of $/h is minimized. A simplified input-output curve of a thermal unit known as heat rate curve is shown in Figure 1(a). Converting the ordinate of heat-rate curve from Btu/h to $/h results in the fuel cost curve shown in Figure 1(b).

Earlier, a number of derivatives-based approach like lambda iteration, gradient method, Lagrangian multiplier method and dynamic programing method have been applied to solve ELD problems. But in reality, input-output characteristics of modern generators are nonlinear because of valve point effect, ramp rate constraints and so on. Recently, heuristics optimization techniques such as Genetic Algorithm (GA), Particle Swarm Optimization (PSO) and Artificial Bee Colony (ABC) have been successfully applied to solve ELD problem with non-smooth cost functions. One such recent technique is Harmony Search (HS) algorithm. HS algorithm has been recently developed in an analogy with improvisation process where musicians always try to polish their pitches to obtain better harmony. Music improvisation process is similar to the optimum design process which seeks to find optimum solution. The pitch of each musical instrument determines the certain quality of harmony, just like the objective function assigned to the set of variables. An improved version of HS algorithm is called Improved Harmony Search (IHS) algorithm (Coelho and Mariani, 2009). This paper deals with IHS algorithm used to solve ELD problem with inclusion of ramp rate limit (Mahdavi et al., 2007).

Problem Formulation

The main objective of ELD problem is to minimize:

...(1)

where Fi is the total fuel cost for the generator unity i (in $/h), which is defined by the equation:

...(2)

where ai, bi,and ci are cost coefficients of generator i. Two constraints are considered in this problem, i.e., the generation capacity of each generator and the power balance of the entire power system.

Constraint 1: This constraint is an inequality constraint for each generator. For normal system operations, real power output of each generator is within its lower and upper bounds and is known as generation capacity constraint given by:

...(3)

Constraint 2: This constraint is an equality constraint, in which the equilibrium is met when the total power generation equals the total demand PD and the real power loss in transmission lines PL. This is known as power balance constraint and can be expressed as given in:

...(4)

Ramp Rate Limit Constraint

The power generated, Pi0 by the ith generator in certain interval may not exceed that of previous interval Pi0 by more than a certain amount URi, the up-ramp rate limit, and neither may it be less than that of the previous interval by more than some amount DRi, the down ramp rate limit of the generator .These give rise to the following constraints:

As generation increases

...

As generation decreases

and

...(5)

Valve Point Loading Constraint

The valve point loading is taken into consideration by adding a sine component to the cost of the generating units. Typically, the fuel cost function of the generating units with valve point loadings is represented as in the following equation:

...

The input-output curve under valve point loading coefficients is shown below:

Harmony Search Algorithm

The HS algorithm, proposed by Geem, is a nature-inspired algorithm, mimicking the improvisation of music players (Lee and Geem, 2005). The harmony in music is analogous to the optimization solution vector, and the musician's improvisations are analogous to the local and global search schemes in optimization techniques. The HS algorithm uses a stochastic random search, instead of a gradient search. This algorithm uses Harmony Memory Considering Rate (HMCR) and Pitch Adjustment Rate (PAR) for finding the solution vector in the search space. The HS algorithm uses the concept, how aesthetic estimation helps to find the perfect state of harmony, to determine the optimum value of the objective function. The HS algorithm is simple in concept, few in parameters and easy in implementation. It has been successfully applied to various optimization problems. The optimization procedure of the HS algorithm (Hadi Saadat, 1999; and Kothari and Dhillon, 2006) is as follows:

* Initialize the optimization problem and algorithm parameters;

* Initialize the harmony memory;

* Improvisation of a new harmony memory;

* Update the harmony memory; and

* Check for stopping criteria. Otherwise, repeat step 3 to 4.

The detailed descriptions of the above steps are given in the following sections.

Initialization of Problem and HS Algorithm Parameters

In this step, the optimization problem is expressed in Equations (1) and (2) and the decision variable is taken as x and HS algorithm parameters are the Harmony Memory Size (HMS) or the number of solution vectors in the harmony memory, HMCR, PAR, Bandwidth rate (BW) and the Number of Improvisations (NI) or the stopping condition.

Initialization of Harmony Memory

The Harmony Memory (HM) is a memory location where all the solution vectors (sets of decision variables) are stored. The HM is similar to the number of population in other evolutionary algorithms. The HM matrix is filled with as many randomly generated values between its minimum and maximum limits.

...(6)

Improvisation of a New Harmony from the HM

A new harmony vector, Xc=(x1c, xc2,.....xcN) is generated based on three rules:

* Memory consideration;

* Pitch adjustment; and

* Random selection.

Generating a new harmony is called improvisation. In the memory consideration, the value of decision variables X c for the new vector is selected from (x1-xHMS). The HMCR, which varies between 0 and 1, is the rate of choosing one value from the historical values stored in HM, while (1-HMCR) is the rate of randomly selecting one value from the possible range of values as:

...(7)

where rand is the uniform random number in the range between 0 and 1; and xi the set of possible range of values for each decision variable, that is xi,min £ xi £ xi,max . For example, a HMCR of 0.8 indicates that the HS algorithm will choose the decision variable from historically stored values in the HM with an 80% probability or from the possible range of values with a 20% probability. After the memory consideration, every component is examined to determine whether it should be pitch adjusted. This operation uses the PAR parameter, which is the rate of pitch adjustment as follows:

...(8)

where BW is the arbitrary distance bandwidth. To improve the performance of the HS algorithm, PAR and BW are changed during each generation as follows:

...(9)

where PAR(g) is the pitch adjusting rate of current generation; PARminis the minimum pitch adjusting rate; PARmax is the maximum pitch adjusting rate; g is the current generation number; and NI is the number of improvisations.

...(10)

where BW(g) is the bandwidth rate of current generation; BWminis the minimum bandwidth rate; and BWmax is the maximum bandwidth rate.

Updating the Harmony Memory

Updating the harmony memory in HS algorithm for optimization problem differs from that of basic HS algorithm. The new harmony memory, generated by improvisation process, is combined with the existing harmony memory to form 2 ' HMS solution vectors. Then non-dominated sorting and ranking procedure is performed on the combined harmony memory. Once the ranking is assigned to all the solution vectors in the combined harmony memory, a diversity rank is assigned to the solution vectors, which are in the same nondominated front, using the crowding distance metric. The crowding distance is an indication of the density of the solution vectors surrounding a particular solution vector. The measure of crowding distance is generally based on the average distance of the two solution vectors on either side of a solution vector, along each of the objectives. Finally, the best HMS harmony memory is selected from the combined harmony memory in the order of their ranking for the next improvisation. To choose exactly HMS solution vectors from the last non-dominated front, crowded comparison operator is used to select the best solutions needed to fill the HMS.

Stopping Condition

The HS algorithm is stopped when the NI has been met. Otherwise, the above two sections are repeated.

Implementation of the Proposed Approach

The proposed approach to solve ELD problem is described in the following steps (Geem et al., 2001):

* Input the system parameters, minimum and maximum limits of control variables.

* Choose the HMS, pitch adjusting rate PAR, bandwidth BW and the maximum number of improvisations NI.

* Initialize the harmony memory as explained under the subhead "Initialization of Harmony Memory". While initializing, all the control variables are randomly generated within their limits.

* Start the improvisation.

* For each solution vector in HM, evaluate the objective functions.

* Improvise the new harmony memory as explained in the under the subhead "Improvisation of a New Harmony from the HM".

* Perform the non-dominated sorting and ranking on the combined existing and new harmony memory

* Choose the best harmony memory from the combined solution vectors as given under the subhead "Updating the Harmony Memory" for the next improvisation.

* Check for stopping conditions. If the number of improvisations has been reached, stop the algorithm. Otherwise, go to step 5.

Biogeography-Based Optimization Algorithm

BBO, suggested by Dan Simon in 2008, is a stochastic optimization technique for solving multimodal optimization problems. It is based on the concept of biogeography, which deals with the distribution of species that depend on different factors like rainfall, diversity, etc., (Bhattacharya, 2009).

Parts of BBO

The main parts of BBO algorithm are:

* Migration

* Mutation

Migration

This BBO algorithm is similar to other population-based optimization techniques where population of candidate solutions can be represented as vectors of real numbers. Each real number in the vector is considered as one Suitability Index Variable (SIV). Fitness (in BBO, a term called Habitat Suitability Index (HSI)) of each candidate solution is evaluated using its SIVs.

HSI represents the quality of each candidate solution. High HSI solution represents better quality solution and low HSI solution represents inferior solution in the optimization problem. The emigration and immigration rates of each solution are used to probabilistically share information between habitats. Immigration rate (ls) and emigration rate (ms) can be evaluated as shown in Figure 4. Using habitat modification probability each solution is modified based on other solutions.

Immigration rate, ls of each solution, is used to probabilistically decide whether or not to modify each SIV in that solution. If an SIV in a given solution is selected for modification, emigration rates, ms of other solutions, are used to probabilistically select which of the solutions should migrate a randomly selected SIV to that solution.

Mutation

Due to some natural calamities or other events, HSI of a natural habitat can change suddenly and it may deviate from its equilibrium value. In BBO, this event is represented by the mutation of SIV, and species count probabilities are used to determine mutation rates. The species count probability can be calculated.

Each population member has an associated probability, which indicates the likelihood that it exists as a solution for a given problem. If the probability of a given solution is very low, then that solution is likely to mutate to some other solution. Similarly, if the probability of some other solution is high, then that solution has very little chance to mutate.

Therefore, very high HSI solutions and very low HSI solutions are equally improbable for mutation, i.e., they have less chances to produce more improved SIVs in the later stage. But medium HSI solutions have better chances to create much better solutions after mutation operation. Mutation rate of each set of solution can be calculated in terms of species count probability.

...(11)

where

m(s) = the mutation rate for habitat containing S species;

m = maximum mutation rate; and

P = maximum probability.

BBO Steps

The algorithm steps of BBO are as follows:

Step 1: Initialization of the BBO parameters.

Step 2: The initial position of SIV of each habitat should be randomly selected while satisfying different equality and inequality constraints of ELD problems. Several numbers of habitats depending upon the population size are being generated. Each habitat represents a potential solution to the given problem.

Step 3: Calculate each HSI, i.e., value of objective function for each ith habitat of the population set n for given emigration rate ms, immigration rate ls and species S.

Step 4: Based on the HSI values, some elite habitats are identified.

Step 5: Each non-elite habitat is modified by performing probabilistically immigration and emigration operation.

Step 6: Species count probability of each habitat is updated using Equation 11. Mutation operation is performed on the non-elite habitat, and HSI value of each new habitat is computed.

Step 7: Feasibility of a problem solution is verified, i.e., each SIV should satisfy equality and inequality constraints.

Step 8: Go to step 3 for the next iteration.

Step 9: Stop iterations after a predefined number of iterations.

Simulation Results

Fuel cost coefficients and generation limits for each generating unit of the 20 generator test system are given in Table 1.

The simulation results give the comparison of HS, IHS, BBO algorithms and the results obtained are shown in Table 2.

The results obtained with the inclusion of ramp rate limit constraint are shown in Table 2.

The results obtained with the inclusion of valve point loading constraint are shown in Table 3.

Ramp Rate Constraint

The convergence characteristics obtained for all the three algorithms with the inclusion of ramp rate limit costraint are shown in Figure 4 (Agrawal and Swarnkar, 2012):

The comparison of fuel cost from the table with ramp rate is shown in Figure 5:

Valve Point Loading Constraint

The convergence characteristics obtained for all the three algorithms with the inclusion of valve point loading constraint are shown in Figure 6:

The comparison of fuel cost from the table with valve point loading is shown in Figure 7:

Conclusion

The ELD problem is solved by using IHS, HS, BBO algorithms in the power system with the inclusion of ramp rate limit constraint, valve point constraint for 20 generating units. From the simulation results, the performance of IHS algorithm is found better and gives minimum total fuel cost when compared to both HS and BBO algorithms. The IHS algorithm is better when compared to other algorithms because of two reasons: (i) Total fuel cost is minimum when compared to others; and (ii) Convergence characteristics response is fast when compared to others.9

References

References

1. Agrawal N and Swarnkar K (2012), "Economic Load Dispatch of Thermal Generator with Ramp Rate Limit Using Biogeography Based Optimization", Proc. IJEIT.

2. Bhattacharya A (2009), "Biogeography Based Optimization for Different Economic Load Dispatch Problems", Proc. of IEEE.

3. Coelho L D S and Mariani V C (2009), "An Improved Harmony Search Algorithm for Power Economic Load Dispatch", Energy Conversion and Management , Vol. 50, pp. 2522-2526, May.

4. Geem Z W, Kim J H and Loganathan G V (2001), "A New Heuristic Optimization Algorithm: Harmony Search", Simulation, Vol. 76, pp. 60-68.

5. Hadi Saadat (1999), Power System Analysis, WCB/McGraw-Hill.

6. Kothari D P and Dhillon J S (2006), Power System Optimization, Prentice-Hall of India.

7. Lee K S and Geem Z W (2005), "A New Meta-Heuristic Algorithm for Continuous Engineering Optimization: Harmony Search Theory and Practice", Comput. Methods Appl. Mech. Engr., Vol. 194, pp. 3902-3933.

8. Mahdavi M, Fesanghary M and Damangir E (2007), "An Improved Harmony Search Algorithm for Solvi ng Optimization Problems ", Applied Mathematics and Computation, Vol. 188, pp. 1567-1579.

Reference # 59J-2015-07-04-01