A load economic dispatch based on ion motion optimization algorithm

Chapter 12A Load Economic Dispatch Based on Ion Motion Optimization Algorithm Trong-The Nguyen, Mei-Jin Wang, Jeng-Shyang Pan, Thi-kien Dao and Truong-Giang Ngo Abstract This paper prese

Chapter 12

A Load Economic Dispatch Based on Ion

Motion Optimization Algorithm

Trong-The Nguyen, Mei-Jin Wang, Jeng-Shyang Pan, Thi-kien Dao

and Truong-Giang Ngo

Abstract This paper presents a new approach for dispatch generating powers of

thermal plants based on ion motion optimization algorithm (IMA) Electrical power systems are determined by optimization in power balancing, transporting loss, and generating capacity The scheduling power generating units for stabilizing different dynamic responses of the control power system are mathematically modeled for the objective function Economic load dispatch (ELD) gains as the objective function is optimized by applying IMA In the experimental section, several cases of different units of thermal plants are used to test the performance of the proposed approach The preliminary results are compared with the other methods in the literature shows that the proposed plan offers higher effect performance

Keywords Ion motion optimization·Electric power generating plant outputs· Economic load dispatch

12.1 Introduction

Recently, electrical modes of renewable energy sources have increased rapidly [1] Fast rate load and fluctuation in the power grids needs stable balance Economic load dispatch (ELD) [2] refers to a scheduling method that rationally allocates the productive output of each generating unit to meet the constraints of power system

T.-T Nguyen · M.-J Wang · J.-S Pan (B)

Fujian Provincial Key Laboratory of Big Data Mining and Applications, Fujian University of Technology, Fuzhou 350118, Fujian, China

e-mail: jengshyangpan@gmail.com

T.-T Nguyen

e-mail: vnthe@hpu.edu.vn

M.-J Wang

e-mail: meijinwang0608@gmail.com

T.-T Nguyen · T Dao · T.-G Ngo

Department of Information Technology, Haiphong Private University, Haiphong, Vietnam

© Springer Nature Singapore Pte Ltd 2020

J.-S Pan et al (eds.), Advances in Intelligent Information Hiding and Multimedia

Signal Processing, Smart Innovation, Systems and Technologies 157,

https://doi.org/10.1007/978-981-13-9710-3_12

115

operation [3] A common single-objective optimization problem is to minimize power generation costs and to maximize power generation efficiency Economic scheduling problems are the nonlinear problems that constrained optimization due to excessive dependence on initial values and gradient information

Traditional methods such as Lagrangian method, linear programming [4,5], and internal penalty function [6] are not suitable for solving nonlinear-like economic scheduling problems In response to the shortcomings of traditional methods, recent techniques like metaheuristic algorithms have been applied to solve the power system optimization problems successfully [7,8,9] The algorithms such as particle swarm optimization (PSO) [10], genetic algorithm (GA) [11], simulated annealing (SA) [12], and neural network method (ANN) [13] refer to the metaheuristic These meth-ods have proven to be very effective in solving nonlinear and constrained problems [14] A recent metaheuristic algorithm, ion motion optimization algorithm (IMA) [15] that is inspired based on the essential characteristics of attracting and mutually exclusive anions and cations IMA’s structure is simple but effective, and easy to understand and to programming

This paper takes advanced IAM to consider its factors for solving the optimization

of the economic load scheduling problem of the power system

12.2 Related Work

12.2.1 Economic Load Dispatch

Economic load dispatch (ELD) is a kind of economic scheduling problems that minimize the total power generation cost under the operating constraints of the power system It is a critical mathematical optimization problem in power systems ELD

is a great significance for the economic and reliable operation of the power system [2] The main objective of the economic loading on generators is a minimum cost of producing and simultaneously met power demand (PD) under constraints of generator output limits, system losses, ramp rate limits, and prohibited operating zones The optimal dispatch is formulated as minimizing cost:

F =

m



i=1

According to the analysis the cost function of ith generating unit, f i (P) is a

quadratic polynomial function that is described a:

f i (P i ) = a i + b i P i + c i P i2

For different variables and notations are used, here, P D is power load demand P i

is active power deliver P is the minimum real power output P is maximum

12 A Load Economic Dispatch Based on Ion Motion … 117

power P0i is the previous real power output a i , b i and c iare coefficients of fuel cost

e i , f i are coefficients of the power grid system with valve-point effects m is a number

of the committed units P loss denotes the transporting load loss B ij , B 0i , B00are

B-matrix coefficients for transmission power loss U ri and D ri are the up ramp limits

and down ramp limits P imax is lower limits of the prohibited zone for generating

unit P iU pzk is upper limits of k th is the prohibited zone for ith generating unit Imaxis

a maximum number of iteration I is current iteration.

According to the equality constraints, total power generationm

i=1P iis equivalent

to load demand P Dand total loss as the following equation:

m



i=1

(P i ) = P D + P Loss (12.3) For the loss coefficient on transition load, the total loss may be derived as

P Loss=

m



i=1

n



j=1

P i B ij P j+

m



i=1

B 0i P i + B00 (12.4) The cost function of generation is also satisfied to below inequality constrained:

It is desired to control the generation power of each committed online generator smoothly and should be within the generator limits But the ramp rate limit restricts the limit for controlling the operation of generation in two operating periods The

ramp rate limit of ith generation unit is

Max

P imin , P0

i − D ri



≤ P i≤ MinP imax , P0

i + U ri



(12.6)

• (a) if generation increases,

P i − P0

• (b) if generation decreases,

The input–output characteristic of a generator is varied mainly when it has some valves in its steam turbine The ripples produced in the heat rate curve are the primary cause of valve point, and they are not expressed by a polynomial function

12.2.2 Ion Motion Optimization Algorithm

The ion motion optimization algorithm (IMA) [15] simulated the motion of the ions with anion (negative ion charge) set and cation (positive ion charge) set These two sets are employed as ion candidate solutions in the operation process They per-form different evolutionary strategies in the liquid phase and the solid phase and circulate between the liquid and the solid phase to achieve the purpose of optimiz-ing the ions Ions in the IMA algorithm can move toward best opposite charges It means that anions move toward the best cation; on the other hand, cations move toward the best anion The movement of ions in this algorithm can guarantee the improvement of all ions throughout iterations Their movement power depends on the attraction/repulsion forces between them The amount of this force specifies the momentum of each atom The following steps represent the process of the algorithm

Initialization

An initial random population is randomly generated according to a uniform

distri-bution within the lower and upper boundaries with D dimensions.

Liquid phase

In the liquid phase, the anion group (A) and the cation group (C) updated according

to the following patterns, respectively

A i ,j = A i ,j + AF i ,j×Cbest j − A j



(12.9)

C i ,j = C i ,j + CF i ,j×Abest j − C j



(12.10)

where Cbest and Abest are cation and anion optimization, respectively Subscript

i= 1, 2, 3, , NP/2, (NP/2 is the size of ions population), and j = 1, 2, 3, , D.

The optimal anion and cation are the anion and cation with the lowest fitness value

in the entire anion group and the Cation group, respectively, for a minimization problem

The resultant of anions attracted force AF i ,j and CF i ,jare mathematically modeled

as follows:

AF i ,j= 1

CF i ,j= 1

where AD i ,j and CD i ,j are the distances of ith anion from the best cation, and cation from the best anion in dimension, respectively AD i ,j =A i ,j − Cbest j, and CD i ,j=

C i ,j − Abest j.

Solid phase

The ion is gradually gathered with iteration near the optimal ion by the gravitational force The solid phase was set for breaking the phenomenon of excessive

concen-12 A Load Economic Dispatch Based on Ion Motion … 119 tration, and also providing diversity for the algorithm in case of over-concentration

of ions to make the algorithm fall into a local optimum The ion motion gradually slows down like the physical process as the iteration proceeds from the initial intense motion, and gradually, the liquid state ions will recrystallize into crystals The process

of recrystallization was simulated in IMA was known as a solid phase

A j=



A j + ϕ1× (Cbest − 1),

A j + ϕ1× (Cbest),

if rand > 0.5 otherwise (12.13)

C j=



C j + ϕ2× (Abest − 1),

C j + ϕ2× (Abest),

if rand > 0.5 otherwise (12.14)

Termination condition

Completion of the solid phase evolution strategy to determine whether to achieve the termination conditions of the algorithm The termination conditions include the presupposition accuracy, the number of iterations, and so on If it is reached, the optimal ion is directly output; otherwise, the anions and cations are returned to the liquid phase from the solid phase and continue to be iterated In such a process, anions and cations are circulated in the liquid phase and solid stage, and the optimal solution is gradually obtained with iteration

12.3 Scheduling Load Power Optimization Based on IMA

Search space optimization of the ELD includes both feasible and unfeasible scenarios that the main work is to identify the feasible points which produce close optimum results within the boundary framework It means the possible points have to satisfy all the constraints, while the unworkable aspects violate at least one of them As mentioned in the above section, the power system economic scheduling problems have multiple constraints, such as power balance constraints, operational constraints, slope limits, and prohibited operating space These constraints make the feasible domain space of the problem very complicated [3]

Therefore, the solution or set of optimized points must necessarily be feasible, i.e., the points must satisfy all constraints So, it is essential to design a suitable objective function, which results in success of an optimization problem The perfor-mance indices utilize in the area of optimization purposes with high acceptance rate The objective function characterized by the given different execution conditions and constraints [3]

To handle constraints, we use the penalty functions to deal with unfeasible points

We attempt figuring out an unconstrained problem in the search space points by modifying the objective function in Eq (12.1) The formula function is as follows:

Min f =



f (P i ), if P i ∈ F

f (P i ) + penalty(P i ), otherwise (12.15)

Ions mapping

ELD model

Success

Update local and

global best

Calculate The objectives

No

Yes

Feasible

Search feasible points for Optimization Yes

i< iterMax

End Yes

No

i=i+1

Modelling

Dispatch Space

The global best Optimization

Fig 12.1 Flowchart of the proposed IMA for dispatch power generation (ELD)

where F is optimum dispatch For dealing with constraints of prohibited zones, a

binary variable is used for adding to objective formula as follows:

V j=



1, if P jviolates the prohibited zones

The nearest distance points in the possible areas measure the effort to refine the solution

Min f =

n



i=1

F i (P i ) + q1

 n



i=1

P i − P L − P D

2

+ q2

n



j=1

12 A Load Economic Dispatch Based on Ion Motion … 121

Case study with a six-unit system

Iterations 1.483

1.484

1.485

1.486

1.487

1.488

1.489

104

Fig 12.2 Comparison of the proposed IMA for dispatching load scheduling generators with FA,

GA, and PSO approaches in the same condition

Parameters of penalty factors and constants associated with the power balance are

used to tune practically with values 1000 set to q1, and the value one set to q2in the simulation section

The necessary steps of IMA optimization for scheduling power generation dis-patch:

Step 1 Initialize the IMA population that associated model dispatch space.

Step 2 Update the Anion group (A) and the Cation group (C) updated according to

Eqs (12.10) and (12.11) as the patterns, respectively

figure current nearest solutions and then update the position as feasible archives

Step 4 If the termination condition met (e.g., max iterations), go to step 2, otherwise,

terminate the process and produces the result (Fig.12.1)

12.4 Experimental Results

To evaluate the performance of the proposed approach, we use the case study of six-unit and fifteen-unit systems to optimize the objective function in Eq (12.17) The outcome of the case testing for dispatch ELD is compared to other approaches, i.e., genetic algorithm(GA) [11], firefly algorithm (FA) [16], and particle swarm optimization (PSO) [10] Setting parameters for the approaches: population size N

is set to 40, and the dimension of the solution space D is set to 6 and 16 for the

six-unit system he fifteen-unit system, respectively The max-iteration is set to 200,

Table 12.1 Coefficients

setting for a six-unit system Units γ

$/MW 2 β $/MW α $ P min

MW

P max

MW

1 0.0075 7.60 250.0 110.0 500.0

2 0.0093 10.20 210.0 51.0 200.0

3 0.0091 8.50 210.0 82.0 300.0

4 0.0092 11.50 205.0 51.0 150.0

5 0.0082 10.50 210.0 51.0 150.0

6 0.0075 12.20 125.0 61.0 140.0

and number of runs is set to 15 The final obtained results averaged the outcomes from all runs The compared results for ELD are shown in Fig.12.2

A Case study of six units

The features of a system with six thermal units are listed in Table12.1 The power load demand is set to 1200 (MW)

The coefficients as Eq (12.2) for a six-unit system in the operating normally with capacity base 100 MVA are given as follows:

B ij= 10−3×

⎡

⎢

⎢

⎢

⎢

0.15 0.17 0.14

0.17 0.60 0.13

0.15 0.13 0.65

0.19 0.26 0.22

0.16 0.15 0.20

0.17 0.24 0.19

0.19 0.16 0.17

0.26 0.15 0.24

0.22 0.20 0.19

0.71 0.30 0.25

0.30 0.69 0.32

0.25 0.32 0.85

⎤

⎥

⎥

⎥

⎥,

B0= 10−3[−0.390 − 0.129 0.714 0.059 0.216 − 0.663],

B00= 0.056, and P D= 1200 MW

Table12.2shows the comparison results of the proposed approach with the FA,

GA, and PSO approach The solution has six generator outputs, including P1–P6 The average results of the runs for generating power outputs, making total cost, total power loss load, and total computing times, respectively

Figure 12.2 depicts the comparison of the proposed IMA for dispatch power generating outputs of a system six units with FA, GA, and PSO approaches in the same condition

B Case study of 15 units

The given coefficients for a system has 15 thermal units as its feature is listed in Table12.3

The power load demand is set to 1700 (MW) The features of a system with 15 thermal units are listed in Table12.3 There are 15 generator power outputs in each

solution listed as P 1 , P 2 , …, P 15 The dimension D of the search space equalizes to

15

12 A Load Economic Dispatch Based on Ion Motion … 123

Table 12.2 The best power outputs for six-generator systems

Total power output (MW) 1239.76 1235.76 1234.55 1234.53

Total generation cost ($/h) 14891.00 14861.00 14860.00 14844.00

Table 12.3 Coefficients

setting for a fifteen-unit

system

Units γ

$/MW2

β $/MW α $ P min MW P max MW

1 0.00230 10.51 671.12 150.0 445.0

2 0.00185 10.61 574.82 155.0 465.0

3 0.00125 9.51 374.98 29.0 135.0

4 0.00113 8.52 37.50 25.0 130.0

5 0.00205 10.51 461.02 149.0 475.0

6 0.00134 10.01 631.12 139.0 460.0

7 0.00136 10.76 548.98 130.0 455.0

8 0.00134 11.34 228.21 65.0 300.0

9 0.00281 12.24 173.12 25.0 165.0

10 0.00220 10.72 174.97 24.0 169.0

11 0.00259 11.39 188.12 23.0 85.0

12 0.00451 8.91 232.01 22.0 85.0

13 0.00137 12.13 224.12 22.0 85.0

14 0.00293 12.33 310.12 25.0 61.0

15 0.00355 11.43 326.12 19.0 56.0

B i0 = 10−3× [−0.1 − 0.22.8 − 0.10.1 − 0.3 − 0.2 − 0.20.63.9 − 1.70.0

− 3.26.7 − 6.4]; B00= 0.0055, P D = 1700 MW.

Table12.4depicts the comparison of the proposed approach with the other pro-cedures, e.g., FA, GA, and PSO methods in the same condition for the optimization

Table 12.4 The best power output for fifteen-generator systems

Outputs FA [ 14 ] GA [ 13 ] PSO [ 15 ] IMA

Total power output (MW) 1846.81 1837.81 1828.27 1827.60

Total generation cost ($/h) 1241.09 1236.09 1235.61 1234.61

Power loss (MW) 147.84 137.84 129.27 127.60

system with 15 generators The statistical results involved the generation cost, eval-uation value, and average CPU time are summarized in the table

Observed over Tables, the results of quality performance in terms of the cost, power loss and time consumption of the proposed method also produced better the other approaches The proposed IMA outperforms other methods

The observed results of quality performance in terms of convergence speed and time consumption show that the proposed method of parallel optimization outper-forms the other methods

12.5 Conclusion

In this paper, we presented a new approach based on ion motion optimization algo-rithm (IMA) for dispatching power generators outputs Economic load dispatch (ELD) is optimized with different responses of the control system in balancing, transporting loss, and generating capacity The linear equality and inequality con-straints were employed in modeling objective function The experimental section,