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The key object of electric power generation is to deliver required energy to the consumer by optimum economic operation. Power system is a large interconnected system and usage of power also increasing drastically with modernization and the cost of production of electric energy will increase with rapid growth of transmission and distribution infrastructure as losses for new transmission and distribution occurs. So there is a need for optimizing generation of power at different locations of power system. This leads to minimization of cost of input thermal generating units meeting the load demand and transmission losses. This is a challenging task called Economic Dispatch (ED) which includes nonlinear quadratic function and valve point loadings, and requires complex solving. Literature exhibits those convex cost functions of steam thermal generating units when included with valve point loadings leads to nonconvexity resulting in nonconvex problem. Perfect modeling of ED for steam thermal generating units is possible with valve point loadings. Solving nonconvex problem by conventional methods is a difficult issue as sudden changes and discontinuities are possible in incremental cost function. Improved Harmony Search (EHS) algorithm has been introduced in this work to resolve the ED problem. The proposed method is applied to thirteenunit test system at 2different load demands and the results also presented.
non convex, economic load dispatch, harmony search algorithm (HS), enhanced harmony search algorithm (EHS), valve point loading
Operation of power system network at low price is the most precious factor and this can be achieved effectively by distributing the true power demand from available source of generation properly. Running of the system with effective economic point view termed as ED problems. The ED Problem have association to solve 2dissimmer obstacles (Wood & Wollenberg, 1996). Here the pre dispatch obstacle is the first one that is how to meet the requirement of the load by selecting the available unit out of other units optimally for required time and online ED is second obstacle that means to decrease the generation cost how to select the one unit from parallel running units for minute to minute requirements. The effective work of ED problem is by satisfying the load demand requirement produce the real power with low price. So the production cost always depending on the load demand at particular time (Rad & Amjady, 2010).
Due to large interconnection of electrical power system, the power demand is huge and that will continuous hike in the cost of energy, so we must diminishes the operating cost of plant by minimizing the fuel consumption for a particular load. ED is one of the aspects of Optimal Power Flow (OPF) has to decrease overall input generating costs of generating units, objective function and total generation must maintain load demand and transmission line losses, equality constraints (Walters & Sheble, 1993). Traditionally ED is solved by Lagrangian approaches, Lambdaiteration approach, Newton methods, linear programming and quadratic programming and Gradient approaches (Mahor et al., 2009). For higher test systems above mentioned approaches solve the ED problem with lower efficiency.
The generation cost of the power system can be minimized with proper economic load dispatch only. To make economic dispatch as a convex issue, generating units cost function taken in quartic form this can be solved using general existed methods. By one of the conventional method we can solve the ED without considering the transmission losses (Xia & Elaiw, 2010). Another traditional approach solved ED problem by bearing in mind network transmission losses (Li & Wang, 2013), and those approaches are unable to give the exact results due to approximation power balance equations. Hence for improved solution those can be united with an iterative to obtain precise result for convex problem.
ED problem cracking approaches can be divided into three assemblies. The ﬁrst assembly comprises the approaches applied to ED problems in their unique versions. Rare samples are tabu search algorithm (TS) (Boonseng et al., 2012), particle swarm optimization (PSO), differential evolution (DE) (Iba & Noman, 2008). The approaches of the second assembly are the modiﬁed kinds of the ﬁrst assembly including modiﬁed tabu search (MTS) (Li et al., 2002), improved PSO (IPSO) (JoongRin et al., 2010), shufﬂed differential evolution (SDE) (Vaisakh & Srinivasa, 2013). Final assembly comprises of the mixture approaches as the grouping of approaches from the earlier groups mentioned above.
In this work, we recommend an advanced approach for cracking the ED problem by means of an Enhanced harmony search (EHS) algorithm. An ED problem constructed on a 13unit test system (Chattopadhyay et al., 2003) through incremental fuel cost function captivating into account the valvepoint loading effects is employed to validate the presentation of the EHS. The valvepoint loading effects familiarize various minima in the solution space. Numerical outcomes gained with the projected EHS approach were compared with classical HS technique and other optimization outcomes stated in literature.
For newly added units in power station technical fellows got successes to enhance the output of the generating plant apparatus like boilers, generators and turbines without interruption while comparing with the older units. To decrease the cost of generation for any operating condition of power system we should know the contribution of each power station. generation of power can be done by thermal, nuclear, diesel etc. and each and every power station have their own characteristics and various cost of generations at any loads to minimize the operation cost of plant we should go with effective scheduling plants. Every generating plant cost characteristic is nonlinear (Xia et al., 2013). So to achieve the minimum operating cost also a nonlinear problem and difficult.
2.1. Cost function
The major impact on total cost of generation due to useful power only. If we want to meet more true power then we should boost up the speed of prime mover of particular plant and that will hike the fuel cost. Reactive power generation can’t affect the total cost of generation because that can be controlled by field current. Hence any generating station is for production of true power only (Jeddi & Vahidinasab, 2014).
2.1.1. Operating cost of a generator
Fuel cost, labor cost and maintains all together gives the total operating cost of generation. Generally conventional fuel power plants having low out power value that can be increased by proper adjustment of the turbine inlet valve. For amount of incoming fuel the cost of generation of all factors are fixed. Depending on the opening and closing of the turbine valve the efficiency of turbine also changes so we have open or close the valve properly depends on requirement. Fig: 1 represents the total operation involved in generation.
Figure 1. Thermal power generation cycle
The cost of fuel curve in true power generation has represented bellow (He et al., 2015),
${{F}_{i}}\left( {{P}_{i}} \right)={{a}_{i}}p_{i}^{2}+{{b}_{i}}{{p}_{i}}+{{c}_{i}}$Rs/hr (1)
Here,
$F _ { i } \left( P _ { i } \right) \quad$ Generator cost function $j$
$a _ { i } , b _ { i } , c _ { i } \quad$ Generator cost coeffiecnt i
$P _ { i }$ Electrical output of generator $i ,$ MW
i set for all generators
2.1.2. Equality constrains
To reduce Objective function, Total Input Fuel Cost, power constraint must be satisfied; where the power constraint is that overall power generation should maintain overall system demand and transmission line losses.
$\sum\limits_{i=1}^{N}{{{P}_{i}}}({{P}_{D}}+{{P}_{L}})=0$ (2)
Here, P_{D} –overall system demand and P_{L} Transmission line loss. In this work P_{L}_{}is neglected.
2.1.3. Inequality constrains
Generated output power of each unit should be within bounds, which is satisfied the inequality constrains given by (ModiriDelshad et al., 2016)
$P _ { \min , i } \leq P _ { i } \leq P _ { \max , i }$ (3)
Here, $P _ { \text {min} , i } , P _ { \text {max} , i }$ are the bounds of $\mathrm { f }$ it alternator.
Geem et al. (2001) suggested an effective harmony search meetsheuristic algorithm that was impressed by music process for a perfect state of harmony. Optimization technique and music in harmony both are similar and musician’s improvisations are similar to local and global quest optimization methods. Harmony Search (HS) algorithm never asks about initial values and uses a stochastic random quest instead of gradient quest that is based on Harmony Memory Considering Rate (HMCR) and the Pitch Adjusting Rate (PAR).
HS algorithm, musical performances requires an effective state of harmony find from aesthetic estimation, as the optimization algorithms requires a perfect state find from objective function value. It has been effectively applied to different optimization problems in computation and engineering sectors.
The Optimization procedure of the HS algorithm has 15 steps, as bellow (Javadi et al., 2012).
Step1. Frame the optimization problem, objective function with constraints and initialize the HS parameters.
Step 2. Harmony Memory (HM) is initialized.
Step 3. New Harmony from the HM is improved.
Step 4. Modernize the memory of harmony.
Step 5. Continue the loop of steps 3 and 4 until the total iterations are completed, stopping criterion.
3.1. Enhanced harmony search (EHS) algorithm
HS is effective one to give the best performance with in minimum time, but facing problems to local quest for data applications. To enhance the effectiveness of HS algorithm and knock out the defects lie with feasible values of Harmony Memory Considering Rate (HMCR) and the Pitch Adjusting Rate (PAR). Fesanghary et al. (2007) introduced an effective HS algorithm that uses variable PAR and Band Width (BW) in improvisation step and also introduced different in harmony search, termed as global apex harmony search and main data is dumped from swarm to obtain the effective results from HS. The EHS adopted in this work have similar steps like conventional HS without step 3, and in this PAR has been dynamically changed show bellow.
$PAR=PA{{R}_{\min }}+\frac{PA{{R}_{\max }}PA{{R}_{\min }}}{NI}*gn$ (4)
where, PAR, Pitch Adjusting Rate for generation
, PAR_{min} is the min. adjusting rate, PAR_{max} is the max. Adjusting rate, is the generation number and NI is the number of solving vector generation. In addition, bandwidth for generation is dynamically updated as follows (Javadi et al., 2012).$bw=b{{w}_{\max }}{{e}^{\left( \frac{\ln \left( \frac{b{{w}_{\min }}}{b{{w}_{\max }}} \right)}{NI}*gn \right)}}$ (5)
where,
$b w$ is the generation bandwidth,
$b w _ { \min }$ is the min. bandwidth and $b w _ { \max }$ is the max. bandwidth
3.2. Application of harmony search algorithm to ED problem
After the initial development of HS algorithm in 2001, HS algorithm applications have been expanded in a huge range of obstacles. The scientific range is so broad, that one can conclude that Harmony Search Algorithm has generally accepted as a robust optimization technique (Coelhos & Mariani, 2009).
Step1. Initialize parameters.
Step2. Initialize the harmony memory.
Step3. Improvise a new harmony
Step4. Adaptive selection
Step5. Update harmony memory
Step6. Stopping criterion
3.3. Application of enhanced harmony search algorithm to ED problem
Although HSA has better capability in identifying the search space in sensible time it is not capable in finding local optimums in case of numerical issues. The drawbacks like fixed values of HMCR and fewer PAR values with large band widths may reduce the performance. Hence increase in iteration number may happen to find optimal solution. So 4 and 5 equations are utilized to change BW and PAR values.
The adopted algorithms HS, EHS were tested on 3 standard load dispatch problems consisting of 313 and 40units and implemented by MATLAB 8.5 R2016b. Here selection of parameters in adopted method is somewhat difficult. Optimal parameter setting for the all test systems is given in Table I & Table II.
Table 1. Optimal parameter setting of HS, EHS for 3 and 13 units
Parameters 
3 Units 
13 Units 

HS 
EHS 
HS 
EHS 

pop 
30 
30 
130 
130 
HMCR 
0.95 
0.95 
0.95 
0.95 
PAR 
0.45 
 
0.45 
 
PAR_{min} 
 
0.4 
 
0.4 
PAR_{max} 
 
0.99 
 
0.99 
BW 
0.01 
0.01 
0.01 
0.01 
BW_{min} 
 
0.00005 
 
0.00005 
BW_{max} 
 
0.05 
 
0.05 
iter 
100 
100 
200 
200 
Parameters 
13 Units 
40 Units 

HS 
EHS 
HS 
EHS 

pop 
130 
130 
400 
400 

HMCR 
0.95 
0.95 
0.95 
0.95 

PAR 
0.45 
 
0.45 
 

PAR_{min} 
 
0.4 
 
0.4 

PAR_{max} 
 
0.99 
 
0.99 

BW 
0.01 
0.01 
0.01 
0.01 

BW_{min} 
 
0.00005 
 
0.00005 

BW_{max} 
 
0.05 
 
0.05 

iter 
200 
200 
500 
500 
A system of 3thermal generating units with the quadratic cost function has been considered in this test. All generating units collectively meet the total demand of 850MW. Global optimal solution for this threeunit test system is attained as 8194.356124$/hr (Xia et al., 2013). The dispatch results using the proposed methods, HS, EHS are given Table 3. From Table 3, it is evident that the low cost gained by all the methods is same as the global solution. For this test system, hundred iteration (individual trails) have been made. Based on results obtained, the comparison of HS, EHS applied to thermal units are presented in Table III. The convergence criteria of HS, EHS methods for the three unit system are shown in Fig 2.
Table 3. Comparisons of simulation results for 3unit system
Unit 
Method 

HS 
EHS 

G1 
330.344330 
443.285943 
G2 
399.090808 
281.588517 
G3 
120.564862 
125.125540 
Minimum cost ($/h) 
8194.368755 
8194.356124 
Total power (MW) 
850 
850 
Figure 2. Convergence criterion for 3unit system
4.2. For thirteen unit system
Here, the proposed methods are applied to the 13generator unit test system that includes quadratic generator input cost functions. Two dissimilar power demands have been used to show the efficacy of adopted algorithms in obtaining optimum solutions. Two different case studies have been considered on this test system. At first a load demand of 1800MW is taken (Coelhos & Mariani, 2009) and later a load demand of 2520MW is considered.
Convergence criteria and the output power generations (MW) of HS and EHS methods applied to thirteen unit test system operating on a load demand of 1800MW is shown in Fig 3 and Table 4 respectively. From Table 4, it is clear that the low price gained from adopted approach is 17935.683284$/hr. Here we got the results by all the methods is same as the global solution.
Table 4. Comparisons of simulation results for 13unit system P_{D}=1800 MW
Unit 
Method 

HS 
EHS 

G1 
623.150720 
494.415492 
G2 
19.859718 
126.920922 
G3 
195.827057 
352.627500 
G4 
73.273771 
73.556964 
G5 
171.449723 
156.961100 
G6 
118.821822 
68.377544 
G7 
75.778269 
61.632842 
G8 
108.008892 
121.345118 
G9 
99.879446 
66.173336 
G10 
74.288292 
68.534340 
G11 
87.001353 
40.644621 
G12 
63.080884 
100.643458 
G13 
89.580050 
68.1 66764 
Minimum Cost($/hr) 
17935.418707 
17935.683284 
Total Power (MW) 
1800 
1800 
Table 5. Comparisons of simulation results for 13unit system P_{D}=2520 MW
Unit 
Method 

HS 
EHS 

G1 
680.000000 
680.000000 
G2 
277.380097 
209.329820 
G3 
357.803149 
314.965502 
G4 
101.803996 
205.115220 
G5 
125.653425 
203.759629 
G6 
161.343304 
94.586705 
G7 
134.837412 
118.625349 
G8 
151.287822 
144.707589 
G9 
104.156086 
105.136503 
G10 
117.409011 
76.195748 
G11 
93.878199 
133.645069 
G12 
115.492372 
126.241200 
G13 
98.955126 
107.691666 
Minimum cost($/hr) 
24062.989996 
24062.580327 
Total Power (MW) 
2520 
2520 
Figure 3. Convergence criterion for 13unit system with P_{D}=1800 MW
Figure 4. Convergence criterion for 13unit system with P_{D}=2520MW
Table 5. shows that dispatch solutions HS, EHS methods to a load of 2520MW, Output gained from all methods, it is evident that the power balance constraint is compromised after sixth decimal also. Result obtained by proposed approach is compared with the HS, EHS as presented in Table 5. The least price gotten by the adopted approach is 24062.580327$/hr. This can be achieved by balancing the power constraints. Fig 4 shows illustrate the convergence criterion of proposed algorithms for the 13unit system with a load demand of 2520 MW.
4.3. For forty unit system
Table 6. Simulation results of HS and EHS methods for 40unit system
Unit 
Method 

HS 
EHS 

G1 
113.999232 
113.999014 
G2 
114.000000 
114.000000 
G3 
120.000000 
120.000000 
G4 
190.000000 
190.000000 
G5 
97.000000 
97.000000 
G6 
140.000000 
140.000000 
G7 
300.000000 
300.000000 
G8 
300.000000 
300.000000 
G9 
300.000000 
300.000000 
G10 
130.000000 
130.000000 
G11 
94.000000 
94.000000 
G12 
94.000000 
94.000000 
G13 
125.000000 
125.000000 
G14 
271.853430 
270.842149 
G15 
265.890514 
266.432925 
G16 
267.256825 
267.726011 
G17 
500.000000 
500.000000 
G18 
500.000000 
500.000000 
G19 
550.000000 
550.000000 
G20 
550.000000 
550.000000 
G21 
550.000000 
550.000000 
G22 
550.000000 
550.000000 
G23 
550.000000 
550.000000 
G24 
550.000000 
550.000000 
G25 
550.000000 
549.999900 
G26 
550.000000 
550.000000 
G27 
10.000000 
10.000000 
G28 
10.000000 
10.000000 
G29 
10.000000 
10.000000 
G30 
97.000000 
97.000000 
G31 
190.000000 
190.000000 
G32 
190.000000 
190.000000 
G33 
190.000000 
190.000000 
G34 
200.000000 
200.000000 
G35 
200.000000 
200.000000 
G36 
200.000000 
200.000000 
G37 
110.000000 
110.000000 
G38 
110.000000 
110.000000 
G39 
110.000000 
110.000000 
G40 
550.000000 
550.000000 
Minimum Cost ($/hr.) 
119450.643035 
118660.253435 
Total Power (MW) 
10500 
10500 
Here adopted methods are implanted to forty thermal units with the quadratic input fuel cost function. Here the load demand expected to be met by generating units is 10500 MW.
Result obtained by implemented approach with 100 trails, on the 40 thermal units (Li & Wang, 2013) test is given in Table 6. It can be cleared that the adopted approach achieved in obtaining believable result of 40unit test system & has represented the importance to the other approaches for 40unit systems. The final low cost gained by the adopted approach for 40 unit system is 119441.588684$/hr. Fig 5 shows illustrate the convergence criterion of proposed algorithms for the 40unit system with a load demand of 10500 MW.
Figure 5. Convergence criterion for 40unit system with P_{D}=10500 MW
In this work heuristic approach such as Harmony Search and Enhanced Harmony Search algorithms effectively implemented to solve power system ED problem. The usefulness, perfectness of the adopted algorithm tested on electrical network having three, thirteen and forty units. From three systems simulation results we overcome the problem with reduction in cost of generation. Adopted method highly increased the searching capability and perfectly handled the system constraints. The successful global optimizing performance on the validation data set shows the efficiency of the adopted approach and show that it can be used as a dependable tool for solving ED problem.
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