Multiple of Hybrid Lambda Iteration and Simulated Annealing Algorithm to Solve Economic Dispatch Problem with Ramp Rate Limit and Prohibited Operating Zones
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Abstract
Aims
This paper presents a multiple hybrid methods combining the lambda iteration and simulated annealing methods (MHLSA) to solve an economic dispatch (ED) problem with smooth cost function characteristics. The constraints of the economic dispatch were load demand and transmission loss.
Methods
The proposed MHLSA algorithm is a hybrid of the lambda iteration and simulated annealing methods and increases efficiency by adding multiple search mechanisms. It is a sequential execution of an individual hybrid algorithm of the lambda iteration and simulated annealing algorithm (HLSA) using only one personal microcomputer. The MHLSA algorithm introduces additional techniques for improvement of the search processes, such as the hybrid method, scope for multiple adaptive searches, and multiple searches.
Results
To show its effectiveness, the MHLSA is applied to two systems consisting of three and six power generating units. The optimized results from MHLSA are compared with those from HLSA and from conventional approaches, such as bee colony optimization (BCO), hybrid particle swarm optimization (HPSO) and simulated annealing (SA). Results confirm that the proposed MHLSA approach is capable of obtaining the greater speed of convergence and a higher quality solution efficiently.
Conclusions
Thus, it has great potential to be implemented in different types of power system optimization problem.
Keywords
Multiple adaptive searches scope Lambda iteration Simulated annealing Economic dispatchIntroduction
Increasing power demand, which has led to an increase in fuel costs over the years, has resulted in reducing operating costs of power systems becoming an important issue. One of the choices is to use the generator efficiently and economically. Operational economics, involving power generation and delivery, that minimizes power production costs, is called Economic Dispatch (ED). The objective of finding solutions to the ED problem is to control the committed generator’s output to minimize total costs while the power demand and other constraints are satisfied.
Traditionally, the fuel cost function of a generator is represented by a single quadratic function. To solve the problem of ED, there are two approaches including numerical and metaheuristics methods. Recently, many numerical conventional optimization methods have been tested, such as lambda iteration [1], gradient method [2], linear programming [3], interior point [4], lagrangian relaxation [5] and dynamic programming techniques [6]. These methods impose no restrictions on the nature of the cost curves and therefore can solve ED problems with inherently nonlinear and discontinuous costs. However, numerical methods can result in problems with complicated and large power systems where problems of dimensionality or local optimality might exist. Previously, metaheuristics methods have been used to solve the economic dispatch and power electrical system problem, including the genetic algorithm (GA) [7, 8, 9], particle swarm optimization (PSO) [10, 11, 12], ant colony optimization (ACO) [13, 14], bee colony optimization (BCO) [15, 16, 17], the cuckoo search algorithm (CSA) [18], the shuffled frog leaping algorithm (SFLA) [19], the firefly algorithm (FA) [20] and simulated annealing (SA) [21, 22]. These methods often provide fast and reasonable solutions, including global optimization with short time searching. Among them, the SA method is a probabilistic technique for approximating the global optimum of a given function. Specifically, it approximates global optimization in a large search space and has fast convergence. However, the conventional SA algorithm can have a limitation in reaching a global optimum solution in a reasonable computational time when the initial solution is far away from the region where an optimum solution is required. In the SA algorithm, the initial population is randomly generated which is the reason for computation and convergence taking a long time as the population is initiated too early. This problem has been solved by estimating an initializing value and defining the search scope for the random method. Lambda iteration has been used as an initial value instead of the traditional random value, the scope of the search has sometimes been defined with the SA algorithm properties then used to find the best solution. However, this method can have the problem of narrowing and inappropriately restricting the search scope. Determining the proper scope of the search is difficult, so repeated testing can waste time. This problem can be solved by using a “multiple searches” method. Multiple searches help to find the region where the most appropriate global optimum solution exists.
Problem Formulation
The economic dispatch problem, with prohibited operating zones, is approached by allocating system load demand commitments to groups of generators over the scheduled time period so that they have minimal generation costs while satisfying physical constraints and operating requirements.
Objective Function
Constraints
 (a)
Power balance constraint
 (b)
Generator rating constraint
 (c)
Ramprate limits and prohibited operating zone
Hybrid of Lambda Iteration (λ) and Simulated Annealing (HLSA)
HLSA is an initial estimate using the principle of equal cost (incremental cost: λ), defined boundaries around λ values, and performing each procedure according to the SA method. The SA, lambda iteration (λ) and HLSA for solving the ED problem are described below.
Simulated Annealing (SA) [23]
The Simulated Annealing (SA) algorithm is a natureinspired method which is adapted from the process of gradual cooling of metal in nature. In the metallurgical annealing process, a solid is melted at high temperature until all molecules can move about freely and then a cooling process is performed until thermal mobility is lost. SA is a probabilistic technique for approximating the global optimum of a given function. Specifically, it is a metaheuristic approach to approximating global optimization in a large search space. It is often used when the search space is discrete. For problems where finding an approximate global optimum is more important than finding a precise local optimum in a fixed amount of time, simulated annealing may be the preferable method. In general, Simulated Annealing algorithms work as follows. At each time step, the algorithm randomly selects a solution close to the current one, measures its quality, and then decides to move to that solution or to stay with the current solution based on either one of two probabilities.
 (a)
Starting Temperature
 (b)
Final Temperature
 (c)
Temperature Decrement
 (d)
Iterations at Each Temperature
Lambda Iteration (λ)
Hybrid of Lambda Iteration (λ) and Simulated Annealing (HLSA) for Solving the ED Problem
A hybrid of lambda iteration and simulated annealing is used to solve the economic dispatch problem by using the principle of equal cost (incremental cost: λ), an estimate of the initial value to narrow search scope and to find the most appropriate solution as SA properties around the estimates. The HLSA algorithm to solve the economic dispatch is described as follows:
Step 1: Specify the parameters of hybrid lambda iteration and simulated annealing including the initial temperature, the final temperature, cool shed, maximum number of iterations and the maximum number of successes.
Step 2: Calculate the value of λ for the initial configuration of the system using the SA algorithm. In this process, the initial λ is determined from Eq. (14).
The value must satisfy the system requirements by considering the system conditions. Capacity constraints of the generator must be within the minimum and maximum ranges of each generator.
Step 5: Create a random close proximity (S_{j}) and calculate the cost of production to compare with the value obtained from Step 4.
Step 6: If the new answer is better than the original answer, then the new answer will be chosen. But if the new answer is inferior to the original answer, calculate the deviation of cost ΔS = S_{j}− S_{i} and generate random numbers. Then accept the new solution (S_{j}) instead of the original solution (S_{i}).
Step 8: Repeat until all conditions have been met.
Multiple Hybrid of Lambda Iteration (λ) and Simulated Annealing (MHLSA)
Specify Parameter
The parameters of SA, HLSA, and MHLSA
Parameters  Number 

Initial temperature  100 
Final temperature  5 
Cool shed  0.83 
Maximum number of tries  1000 
Maximum number of successes within  50 
Calculate the Value of Lambda (λ)
Lambda iteration is based on the principle of equal cost (incremental cost: λ), and an estimate of the initial value to narrow the search scope and finding the most appropriate solution as a swarm of bees around the estimates. Total fuel cost is lowest when λ is equal among the generators. The value of λ for the initial configuration of the system is calculated from Eq. (14).
Multiple Adaptive Search Scope
In this method, \(rank\) is a multiplier in scoping the answer. It is used to determine the lower and upper power limits of the i^{th} generating unit by defining the scope of the value of λ, which comes from Eqs. (16) and (17). The size of \(rank\) effects search answers. Determining the proper \(rank\) size of the search is difficult, so repeated testing can waste time. Adjusting the size of \(rank\) to multiple values at the same time provides a scalable range for the search. The adaptive search scope is a process for finding nearby locations. This process depends on the size of the \(rank\). \(Rank\) is the variance, which is the key factor in the search. Hence the right size \(rank\) must be chosen. In general, low \(rank\) sizes can be used to reduce computational time due to narrower search spans. On the other hand, a \(rank\) size with a high value provides a more accurate solution but makes the calculation time longer. Nevertheless, the searched result may not reach the global optimum. In this method, the \(rank\) size can be adjusted based on the number iterations of the SA algorithm as defined in the MHLSA algorithm. As a result, the problem can be solved more accurately. Therefore, the adaptive search scope improves the calculation speed and accuracy of the solution. Multiple \(rank\) values in this article are defined as follows: \(rank\) #1 = 0.1, \(rank\) #2 = 0.15, \(rank\) #3 = 0.2 and \(rank\) #4 = 0.25.
Multiple Search
In largescale solutions, it may be necessary to use several computers at the same time to calculate the result. These solutions involve parallel searches. However, the multiple searches implemented in this paper are performed using a single personal computer [24]. The basic idea is to reduce the time spent searching results by improving the accuracy and relevance of each search, as well as the ability to manage the results. Multiple searches use a multitasking algorithm that combines algorithm features and additional search capabilities. The answer may come from one or more algorithms. Multiple searches are a way of taking advantage of the power of many algorithm, that may not be as eloquent as the traditional algorithm, and help to find the promising region where the global optimum solution exists. This concept is used to improve the efficiency of the HLSA algorithm after initial estimation by the lambda iteration and to define the range of multiple searches scope, as described in Sect. 4.3. The search algorithm for better algorithms is used in the SA algorithm step as described in Sect. 3.1. The sequence of execution starts from SA#1 to SA#4 for execution for finding better solutions.
Evaluate the Fitness and Select the Best Value
The fitness function is the specificity of the objective function that is used to summarize similar design approaches to achieve the goal that is set. After each round of testing or simulation, the idea was to bring the worst design solution out and rebuild it from the best design solution. The results of each solution have to be good enough to indicate an approximation of the overall specification, and this is built on using the fitness function to test or simulate the results of the solution. If the algorithm is poorly designed, it will come to an abnormal solution or will have difficulty converging. Moreover, the result must be computed quickly. The whole process involves “evaluating the fitness and selecting the best value”.
Check Stopping Criterion
In general, there are several possible conditions to stop a search. Here, stop search is used if any of the following three conditions are met: firstly, as defined by the limit time. That means that if the calculation time is more than or equal to the time limit, the search process will stop. Secondly, defined by the maximum allowable iterations—if the iteration number is greater than or equal to the maximum allowed number of iterations, the search process will stop. Finally, when the appropriate values meet the objective function within the constraints and conditions of the system.
Case Study
The proposed MHLISA algorithm has been applied to solving the economic dispatch problem with prohibited operating zones in two different test cases for verifying its feasibility. These are a three unit system and a six unit system. Each optimization method was implemented in the MATLAB program which was run on a 2.30 GHz Intel (R) Core (TM) i5 with 8 GB of RAM.
The First Case Study
Generator data in case 1
Unit  a _{ i}  b _{ i}  c _{ i}  \(P_{i}^{\text{min} }\)  \(P_{i}^{\text{max} }\) 

1  0.00525  8.663  328.13  50  250 
2  0.00609  10.04  136.91  5  150 
3  0.00592  9.76  59.16  15  100 
Ram rate limits and prohibited zone in case 1
Unit  \(P_{i}^{0}\)  UR _{ i}  DR _{ i}  Prohibited zone  

Zone 1  Zone 2  
1  215  55  95  [105–117]  [165–177] 
2  72  55  78  [50–60]  [92–102] 
3  98  45  64  [25–32]  [60–67] 
The results of the proposed method are divided into two parts: convergence speed and convergence with BCO, SA and HLSA methods and the best solution compared to EP, PSO and HPSO methods.
The Second Case Study
The test system for this case consisted of six thermal units. The constraints included the generation limits, fuel cost coefficients, ramprate limits and prohibited operating zones of these thermal units, 26 buses and 46 transmission lines. The transmission loss was calculated using a B matrix as given by [26]. The results of the proposed method were divided into two parts: convergence speed and convergence with BCO, SA and HLSA methods and the best solution compared to PSO, HPSO and MSSA methods.
Generator data in case 2
Unit  a _{ i}  b _{ i}  c _{ i}  \(P_{i}^{\text{min} }\)  \(P_{i}^{\text{max} }\) 

1  0.0070  7.00  240  100  500 
2  0.0095  10.0  200  50  200 
3  0.0090  8.50  220  80  300 
4  0.0090  11.0  200  50  150 
5  0.0080  10.5  220  50  200 
6  0.0075  12.0  190  50  120 
Ram rate limits and prohibited zone in case 2
Unit  \(P_{i}^{0}\)  UR _{ i}  DR _{ i}  Prohibited zone  

Zone 1  Zone 2  
1  340  80  120  [210–240]  [350–380] 
2  134  50  90  [90–110]  [140–160] 
3  240  65  100  [150–170]  [210–240] 
4  90  50  90  [80–90]  [110–120] 
5  110  50  90  [90–110]  [140–150] 
6  52  50  90  [75–85]  [100–105] 
Simulation and Results

Population size (n) = 50

Number of selected sites (m) = 20

Number of best sites (e) = 5

Number of bees around best sites (n_{ep}) = 50

Number of bees around other sites (n_{sp}) = 50
Results of MHLSA, HLSA, SA and BCO for three units
Unit output  BCO  SA  HLSA  MHLSA 

P_{1} (MW)  199.57  200.40  191.19  191.65 
P_{2} (MW)  78.97  78.15  85.30  85.12 
P_{3} (MW)  34.00  34.02  34.63  34.00 
P_{T} (MW)  312.54  312.56  311.12  310.76 
T_{C} ($/h)  3631.66  3631.71  3618.14  3614.64 
P _{ Loss}  12.54  12.56  11.12  10.76 
Time (S)  14.10  10.13  10.06  40.34 
Three generators test system: comparison of results
Unit output  EP  PSO  HPSO  MHLSA 

P_{1} (MW)  199.53  190.59  200.18  191.65 
P_{2} (MW)  75.68  85.77  76.26  85.12 
P_{3} (MW)  38.19  34.80  34.40  34.00 
P_{T} (MW)  313.40  311.16  310.84  310.76 
T_{C} ($/h)  3641.70  3631.10  3623.11  3614.64 
P _{ Loss}  13.40  11.16  10.84  10.76 
Results of MHLSA, HLSA, SA and BCO for six generating unit
Unit output  BCO  SA  HLSA  MHLSA 

P_{1} (MW)  452.52  447.26  455.52  450.44 
P_{2} (MW)  171.56  170.47  171.62  170.83 
P_{3} (MW)  256.61  261.92  254.98  253.33 
P_{4} (MW)  138.45  138.53  134.60  137.81 
P_{5} (MW)  163.97  166.46  155.10  160.93 
P_{6} (MW)  92.29  90.55  99.62  98.02 
P_{T} (MW)  1275.10  1275.20  1271.44  1271.36 
T_{C} ($/h)  15,439.61  15,439.80  15,394  15,392 
P _{ Loss}  12.09  12.20  8.44  8.36 
Time (S)  15.56  10.26  10.07  60.02 
Six generators test system: comparison of results
Unit output  MSSA  PSO  HPSO  MHLSA 

P_{1} (MW)  447.50  457.26  462.45  450.44 
P_{2} (MW)  173.32  160.72  184.53  170.83 
P_{3} (MW)  263.46  247.53  246.60  253.33 
P_{4} (MW)  139.07  131.52  108.83  137.81 
P_{5} (MW)  165.43  170.50  171.07  160.93 
P_{6} (MW)  87.13  106.62  98.50  98.02 
P_{T} (MW)  1275.96  1274.15  1271.98  1271.36 
T_{C} ($/h)  15,449.90  15,433  15,404  15,392 
P _{ Loss}  12.96  11.15  8.98  8.36 
Conclusion
In this work, new mechanisms are implemented for solving different ED problems with ramp rate limits and prohibited operating zones. Many sophisticated techniques, such as initial estimation using the lambda iteration, adaptive search scopes, multiple searches, and simulated annealing have been added to the MHLSA, in order to enhance the search potential. The MHLSA algorithm has been proposed to solve the ED problem by initial estimation using the lambda iteration method, using the SA method to find the best solution, and using multiple searches as a mechanism to increase efficiency in finding appropriate solutions. The MHLSA optimization mechanisms outperformed a number of other algorithms. The strength of the algorithm was demonstrated in two case studies involving different ED problems with ramp rate limits and prohibited operating zones. The convergence quality of MHLSA algorithm in these two case studies proved the robustness of the algorithm. Study results from the studies involving either three or six generating units confirmed that the MHLSA was superior to the HLSA, SA and BCO methods in terms of providing a highquality solution, stable convergence characteristics, and good computation efficiency. The results clearly showed that the proposed algorithm yielded better results than the comparison method. Electric system operators can use this algorithm for optimization of their networks.
Notes
Acknowledgements
The authors thank Mahasarakham University, Mahasarakham, Thailand, for the financial supporting of this project and I would like to thank Prof. Ian Warrington for encouragement and insightful comments.
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