Solving optimal power dispatch using artificial algorithm for Nordic 44 network

Abstract

Optimal power flow (OPF) problems play a central role in planning and operating a transmission power system in a reliable, flexible and secure manner. In this paper, a significant intelligent algorithm based on the Particle Swarm Optimization method is proposed for solving the optimal reactive power dispatch (ORPD) problem by adjusting equality and inequality constraints. The objective function which is considered for optimal active and reactive power dispatch is the minimization of transmission power losses. In order to solve ORPD problems, Particle Swarm Optimization (PSO) based on the Newton Raphson method is coded with MATLAB software and the results of the simulation are compared to the Interior Point method as the classical method for minimization transmission losses in the Nordic 44 system. The Nordic 44 system consists of three countries, including Norway, Finland, and Sweden, and has 44 buses, 67 branches and 61 generators. The results illustrate the effectiveness of the proposed algorithm compared to the other methods for the minimization of total power losses for Nordic 44. The simulation results of the proposed method for the Nordic 44 system are compared to the results of the PSO algorithm for the ORPD problem in several networks, including IEEE 14 buses, 30 buses, and 57 buses.

Keywords

Optimal power flow classical method Particle Swarm Optimization total power losses minimization

1. Introduction

Optimal power flow is a nonlinear optimization problem, which focuses on the planning and operation of power systems [10]. Carpentrar was the first who introduced OPF problems in 1960s [4]. The main goal of OPF is the optimal operation of electrical power systems by minimizing the objective function through optimal settings of control variables [2]. The constraints of ORPF are classified as equality and inequality constraints, in which equality constraints are power flow equations, and inequality constraints of a power system are specified by upper and lower voltage limits, reactive power outputs of sources, shunt capacitor banks and transformer tapping [4].

Optimal power flow problems can be divided into two main groups; active power dispatch and reactive power dispatch [10]. Active power is defined as the energy consumed or lost, while reactive power is the energy circulated through all the network [7]. Reactive power dispatch plays the vital role of transferring real power throughout the circuit by enhancing the voltage stability and reliability of the power system [7]. Subsequently, the ORPD problem of improving the operation and planning of a power system has been the center of attention for many researchers. Control variables of ORPD are transformer tap settings, real power generated and generator bus voltages, which should be located on their boundaries, as well as the voltage magnitude of generators, etc. [10]. The main objectives of the reactive power dispatch problem are to reduce real power losses, improvement of the voltage profile and voltage stability of the power system, as well as to minimize system costs [10]. Throughout the last two decades, various techniques based on computer algorithms and mathematical formulation have been proposed for finding the optimal reactive power dispatch [4]. The techniques are classified based on conventional and intelligent methods. Some of the conventional techniques that are most commonly used in research papers are linear programming, nonlinear programming, the gradient method, the Interior Point method, and quadratic programming. The recently proposed intelligent methods are Genetic Algorithm, Particle Swarm Optimization, Ant Colony Algorithm, Imperialist Competitive Algorithm, Cuckoo Search Space Algorithm, and Whale Optimization Algorithm [3]. Because of the disadvantages of classical methods like slow running, non-convergence, the immense time of mathematical calculation, and only a single optimization solution per single simulation run, intelligent methods have been employed in many research papers in order to solve optimal reactive power dispatch problems [1].

In previous works on solving optimal active and reactive power dispatch problems, researchers have opted for different methods with consideration to a single objective function or multi-objective functions in real power networks, as follows; Goya and Mehta solved the OPF problem with a new intelligent algorithm called Cuckoo Search Method by optimal adjustment of control variables. Two separate objective functions, which have been considered for solving optimal active and reactive power dispatches, are minimization of the fuel cost of generation and of power losses. The proposed method was evaluated on IEEE 30 buses with MATLAB-2013 software. The results of the simulation illustrate the effectiveness of the proposed algorithm for the reduction of power losses and the fuel cost of generators compared other solutions [4].

Bhesdadiya and Parmar introduced a new intelligent algorithm that they named Whale Optimization Algorithm (WOA) for the optimization of OPF problems on the IEEE 30 buses system. WOA is inspired by the hunting strategy of humpback whales. Fuel cost reduction, active power minimization, and reactive power minimization are three objective functions, which are considered to solve optimal active and reactive power dispatches. The results of the WOA simulation are compared with other algorithms such as Flower Pollination Algorithm (FPA) and Particle Swarm Optimization (PSO). The results of the simulations illustrate the effectiveness of WOA compared to the other methods due to better optimization values, fast convergence and less computational time [1].

Nowadays, there are several effective methods introduced to solve ORPD problems in electrical power networks. Because the PSO algorithm offers more advantages compared the other intelligent algorithms, it was selected as the best one for solving the ORPD problem in widespread networks. In the present work, an intelligent optimization algorithm named Particle Swarm Optimization (PSO) is used to solve the ORPD problem of minimizing both real and reactive power losses in the electrical power system between Norway, Finland, Sweden and Denmark. The PSO algorithm is inspired by the social life of groups of animals like bird flocks or fish groups [6]. The advantages of using PSO are fast convergence, few control parameters, and its suitability for optimization multi-space for both integer and discrete optimization problems [6]. The proposed method is applied to the Nordic 44 electrical power system, which consists of 61 generators, 44 buses, 67 branches and 12 transformers. Two objective functions are introduced for solving the problem for the Nordic 44 system, including minimization of active and reactive power losses with PSO algorithm. The proposed method illustrates the high effectiveness and better optimization values to reduce power losses in large electrical power systems by increasing the number of buses and generators like Nordic 44, which is a widespread network between Norway, Finland, Sweden and Denmark.

This paper is organized as follows: Section 2 presents the description of optimal reactive power flow, the problem formulation, and two categories of optimal reactive power flow dispatch. Section 3 describes the Particle Swarm Optimization algorithm. Section 4 describes the implementation steps of the PSO algorithm in optimal reactive power flow. Section 5 illustrates the MATLAB-2015 simulation results with several tables and figures, and finally, Section 6 expresses the conclusions.

2. Optimal reactive power dispatch

For many researchers and planners today, OPF has become one of the most important challenges in planning and operating an electrical power system [10]. Generally, the problem of OPF can be divided into two main groups, namely, optimal active power dispatch and optimal reactive power dispatch [7]. Nowadays, optimal reactive power plays a central role in the reliable and flexible operation of a power system aiming for voltage stability, improvement of the voltage profile, fuel cost reduction, minimization of active and reactive power losses in the transmission network, and minimization of harmful environmental effects of the power systems [3]. Furthermore, optimization of reactive power dispatch can be divided into two main groups, namely single reactive optimization problems and multi-reactive optimization problems with the adjustment of control variables including active and reactive power generation, bus voltage magnitude and tap of transformers in their limitations between lower bound and upper bound [7,10]. The description of both single and multi-optimization problems can be described as follows; a single reactive optimization problem solves a single objective function like minimization transmission losses, improvement of the voltage stability, or minimization of the fuel cost of generators [7]. The mathematical formulation of single objective functions can be defined as follows [10]: $\begin{array}{l} (1) & Min/Max objective function f (x) \\ (2) & Subjected to Inequality constraints g (x) ⩽ 0 \\ (3) & Equality constraints h (x) = 0 \end{array}$ Multi-reactive optimization problems are considered multi-objective functions for solving optimization problems, including the minimization of power losses and reduction of the fuel cost of generators, improvement of the voltage profile or voltage stability [7]. Therefore, the mathematical formulation for multi-objective functions can be explained as follows [10]: $\begin{array}{l} (4) & Min/Max Objective function f (x) = [f_{1} (x), f_{2} (x), f_{3} (x), \dots] \\ (5) & Subjected to Inequality constraints g_{i} (x) ⩽ 0 where i = 1, 2, 3, \dots, i \\ (6) & Equality constraints h_{i} (x) = 0 where i = 1, 2, 3, \dots, i \end{array}$

2.1. Problem formulation

The objective of optimal reactive power dispatch is to identify control variables that minimize the objective functions. Examples of such control variables include voltages of generators, tap of transformers, active power of generators, and voltages of each of the buses that are considered between lower bound and upper bound [4]. The single objective function considered in this paper is defined as minimization of transmission power losses. In most research papers, the formulation of the minimization of transmission power losses is expressed as equation (7) [4]: $\begin{array}{l} (7) & M inf = \sum_{k = 1}^{N L} g_{k} [| V_{i} |^{2} + | V_{j} |^{2} - 2 | V_{i} | | V_{j} | cos (δ_{i} - δ_{j}) + \dots] \end{array}$ The constraints of optimization can be classified into two main groups as below [4]:

Equality constraints

Equality constraints are power flow equations, which are included in the active and reactive power equations, with $P_{D i}$ and $Q_{D i}$ being active and reactive power demand, respectively, and $P_{G i}$ and $Q_{G i}$ being injected active and reactive power, respectively [4]. For each line of the power system, active and reactive powers can be calculated as formulation (8) and (9) [4]: $\begin{array}{l} (8) & P_{i} = P_{G i} - P_{D i} = V_{i} \sum_{j = 1}^{n} V_{j} Y_{i j} cos (δ_{i} - δ_{j} - θ_{i j}) \\ (9) & Q_{i} = Q_{G i} - Q_{D i} = V_{i} \sum_{j = 1}^{n} V_{j} Y_{i j} sin (δ_{i} - δ_{j} - θ_{i j}) \end{array}$

Inequality constraints

Inequality constraints are defined as generator active power output, generator reactive power output, tap of transformers, and voltage of generators between their lower and upper bounds [8]. Equations (10) to (13) illustrate the control variables [4]: $\begin{array}{l} (10) & P_{g i (min)} ⩽ P_{g i} ⩽ P_{g i (max)} \\ (11) & V_{g i (min)} ⩽ V_{g i} ⩽ V_{g i (max)} \\ (12) & Q_{i (min)} ⩽ Q_{i} ⩽ Q_{i (max)} \\ (13) & T_{i (min)} ⩽ T_{i} ⩽ T_{i (max)} \end{array}$

3. Particle Swarm Optimization

Kennedy and Eberhard introduced the Particle Swarm Optimization algorithm based on the behavior of populations for solving optimal power flow problems in 1995 [6]. The PSO algorithm is inspired by the behavior of social groups of animals like birds, fishes, or insects within a group [5]. The PSO algorithm is modelled according to the movement of each particle in random search space. Each particle flows in random search space to find optimal solution for the operation of power systems. A collection of particles is called swarm [5].

In the basic description of the PSO algorithm, a group of birds located randomly, searching for a piece of food. None of the birds has a knowledge of the exact location of the food, but they have information about their distance to it. Each particle in the swarm flies to search for the optimum solution based on the velocity, their own experience and the experience of their neighbors [6]. One of the best strategies is following a bird that is nearer to the food. Figure 1 illustrates the random search space and the movement of each particle towards the best solution.

Each particle can find the optimal solution according to the three main items; best personal experience, global best experience, and velocity vector. Figure 2 illustrates the movement of each particle to the optimum solution [5].

The velocity and position for the next generation can be formulated as formulation (14) and (15) [6]: $\begin{array}{l} (14) & V_{j}^{k + 1} = W \times V_{j} + C_{1} \times {rand}_{1} \times (P_{best} - X_{j}^{k}) + C_{2} \times {rand}_{2} \times (G_{best} - X_{j}^{k}), \\ (15) & X_{j}^{k + 1} = X_{j}^{k} + V_{j}^{k + 1} . \end{array}$ Where,

$V_{j}^{k}$ : The velocity of individual i at iteration K.

${rand}_{1}$ and ${rand}_{2}$ : Random numbers between $(0, 1)$ .

$X_{j}^{k}$ : The location of individual at iteration k.

$C_{1}$ , $C_{2}$ : Velocity coefficients.

W: Inertial coefficient.

$V_{j}^{k + 1}$ : new velocity of particle i.

$X_{j}^{k + 1}$ : new position of particle i.

Fig. 1.

(A) Initial population of animals. (B) The movement of each particle towards the best solution according to the velocity vector.

Fig. 2.

The diagram of searching point based on PSO.

4. PSO in ORPF

The proposed algorithm has several advantages like fast convergence and reducing the time of calculation for solving optimal reactive power dispatch problems [9]. Our objective function is to minimize transmission power losses, including active and reactive power losses, by considering equality and non-equality constraints. The control variables are considered on the limitation of their boundary values of generator power (except slack generators), generator voltages and transformer taps. The steps for implementing the PSO algorithm in ORPD are described as follows [9]:

Producing the initial population by entering the input parameters of the systems between lower bound and upper bound, including ( $P_{G}$ , $V_{G}$ , $T_{trans}$ ).

Calculate the fitness function (minimization of transmission losses).

In the first iteration, identify $P_{best}$ and $G_{best}$ according to the fitness value of the objective function.

Initialize $C_{1}$ , $C_{2}$ , w and identify the initial velocity.

Modify the velocity and position of each particle according to the formulation (5) and (6) for producing the new population.

Compare $P_{best}$ of the objective function for each particle. If it is better than the previous calculation of $P_{best}$ , it is selected as new option after that the formula (14) and (15) for calculation of the new position of particles should be updated [9].

Finding the new $G_{best}$ , according to the fitness value,

Update the next iteration by $t = t + 1$ .

Checking the stopping criteria (Max iteration, time, convergence $\dots$ ), if it is Ok stop otherwise go to the step 2.

According to the objective functions, in the last iteration, the particle of the lowest amount of $G_{best}$ is selected as the optimum solution.

5. Simulation study and results

To verify the effectiveness of the proposed algorithm in the electrical power network for solving optimal reactive power dispatch problems, simulations are run on the Nordic 44 system and the results of the simulations will be compared to several standard IEEE systems, including IEEE 14 buses, 30 buses and 44 buses. Nordic 44 is a newly developed model, which connects several parts of European countries together, including Norway, Sweden and Finland. The system consists of 44 buses, 67 branches and 61 generators, 43 loads and 12 transformers. It consists of 33 generators in Norway, 26 in Sweden and 18 in Finland. Generally, Nordic 44 is developed to match a simplified model of the Nordic power system to the historical model of Nord Pool Spot. A single diagram of Nordic 44 with Modelica software is shown in the Fig. 3 [11].

Fig. 3.

Single diagram of Nordic 44 with Modelica software.

The system data of Nordic 44 such as generator data, bus data and load data are represented in Tables 1 and 2 respectively. Bus-9 is considered as slack bus.

Table 1

Bus data of Nordic 44 system

No. bus	Type	$V_{s p}$	θ	$P_{g i}$	$Q_{g i}$	$P_{l i}$	$Q_{l i}$	$Q_{min}$	$Q_{max}$
1	2	1	0	2000	1934	4262	1701	−1934	1934
2	3	1	0	0	0	1219	616	0	0
3	3	1	0	0	0	621	110	0	0
4	2	1	0	1700	324	621	650	−1866	1866
5	3	1	0	0	0	0	0	0	0
6	3	1	0	0	0	0	0	0	0
7	2	1	0	6599	670	0	0	−670	670
8	2	1	0	2048	469	2265	650	−1972	1972
9	1	1	0	2807	2688	2434.72	800	−2301	2301
10	2	1	0	5400	3140	5843.32	2400	−4915	4915
11	3	1	0	0	0	−330	262	0	0
12	3	1	0	0	0	0	0	0	0
13	2	1	0	972	850	1154.17	70	−850	850
14	3	1	0	0	0	0	0	0	0
15	3	1	0	0	0	0	0	0	0
16	3	1	0	0	0	0	0	0	0
17	2	1	0	6151	1614	2651	−70	−1700	1700
18	3	1	0	0	0	0	0	0	0
19	3	1	0	0	0	0	0	0	0
20	3	1	0	0	0	0	0	0	0
21	2	1	0	1858	1345	1149.77	100	−1800	1800
22	3	1	0	0	0	0	0	0	0
23	3	1	0	0	0	0	0	0	0
24	2	1	0	1132	555	4406.84	400	−1200	1200
25	3	1	0	0	0	0	0	0	0
26	2	1	0	1774	1109	1349.72	250	−1700	1700
27	3	1	0	0	0	0	0	0	0
28	3	1	0	0	0	0	0	0	0
29	3	1	0	0	0	0	0	0	0
30	3	1	0	0	0	1412	363	0	0
31	3	1	0	0	0	414	175	0	0
32	2	1	0	523	500	0	0	−500	500
33	3	1	0	0	0	0	0	0	0
34	2	1	0	4730	1255	2399.52	800	−4500	4500
35	2	1	0	2442	1164	3039	999	−2400	2400
36	2	1	0	3506	63	2489	150	−1800	1800
37	3	1	0	0	0	0	0	0	0
38	2	1	0	7039	787	7967.65	350	−6377	6377
39	3	1	0	0	0	−1219	600	0	0
40	3	1	0	0	0	343	−4	0	0
41	2	1	0	1620	558	2863.36	400	−1400	1400
42	2	1	0	754	917	3720	1299	−917	917
43	3	1	0	0	0	446	10	0	0
44	3	1	0	0	0	628	0	0	0

Table 2

Generator data of Nordic 44

No.	$P_{g}$	$Q_{g}$	$Q_{max}$	$Q_{min}$	$V_{g}$	$m_{Base}$	Status	$P_{max}$	$P_{min}$
1	2000	1934	1934	−1934	1	1000	1	2334	0
4	1700	324.392	1866	−1866	1	1000	1	2000	0
7	6599	670	670	−670	1	1000	1	7258	0
8	2048	469.25	1972	−1972	1	1000	1	2460	0
9	2807.49	2688.183	2301	−2301	1	1000	1	3000	0
10	5400	3139.955	4915	−4915	1	1000	1	6085	0
13	972	850	850	−850	1	1000	1	1100	0
17	6151	1613.912	1700	−1700	1	1000	1	6766	0
21	1858	1344.936	1800	−1800	1	1000	1	2200	0
24	1132	554.715	1200	−1200	1	1000	1	1280	0
26	1774	1109.34	1700	−1700	1	1000	1	2100	0
32	523	500	500	−500	1	1000	1	620	0
34	4730	1255.24	4500	−4500	1	1000	1	5600	0
35	2442	1164.135	2400	−2400	1	1000	1	3000	0
36	3506	63.41	1800	−1800	1	1000	1	3860	0
38	7038.5	787.878	6377	−6377	1	1000	1	8169	0
41	1620	558.004	1400	−1400	1	1000	1	1800	0
42	754	917	917	−917	1	1000	1	1183	0

To demonstrate the effectiveness of the PSO algorithm in an OPF problem, reducing transmission power losses is considered and implemented for the Nordic 44 system. Results of programming with MATLAB for Nordic 44 are shown below. The proposed algorithm is run with the objective function of minimization of transmission power losses for solving the optimal reactive power dispatch problem. In the simulation of Nordic 44, the Base MVA is defined as 1000 MVA, and control variables are voltage, bus of generators, and tap of transformers that are varied in their range (0.9–1.1). The convergence characteristic of the PSO algorithm for reducing power losses is shown in Fig. 3. According to the results of the simulation, power losses before implementing the PSO algorithm with the Interior Point method as OPF analysis are calculated as 358.606 MW, while after using the PSO algorithm, it reaches only 345.77 MW. Figures 4 and 5 illustrate the results of programming the PSO algorithm with MATLAB programming.

Fig. 4.

Graph of minimization of active power losses by iteration with PSO algorithm for Nordic 44.

Fig. 5.

The results of MATLAB programming for reducing power losses with the PSO algorithm based on the Newton Raphson method.

The results of PSO for Nordic 44 are compared with other classical methods like Newton Raphson and the Interior Point method with the objective function of the reduction of transmission power losses in the Table 3.

Table 4 illustrates the comparison of the objective function for reducing transmission losses with a PSO algorithm for several networks including Nordic 44, IEEE 14 buses and IEEE 30 buses and IEEE 57 buses. The data for IEEE standard systems are selected from the math power box in MATLAB.

Table 3

Comparison of different methods for minimization of transmission power losses

Methods	Active power loss (MW)	Reactive power loss (MVar)
Newton Raphson	861.30	6261.85
Interior Point method	358.606	5073.50
PSO algorithm	338.36	−1930.79

6. Conclusions

In this paper, Particle Swarm Optimization algorithm implemented for the Nordic 44 electrical power network, based on MATAB-2015 software, successfully solved the problem of optimal reactive power dispatch with the objective function of minimization of total power losses. Particle Swarm Optimization was introduced as one of the best heuristic algorithms, with several advantages like convergence, flexibility, reliability, and few control variables compared to other intelligent algorithms for solving OPF problems. The proposed method is examined on the Nordic 44 system and compared to different standard IEEE systems. The results of the simulation illustrate the reduction of power losses for Nordic 44 from 358 to 338 MW, which shows the effectiveness and robustness of the proposed algorithm for solving the problem of optimal reactive power dispatch.

Table 4

Implementation of PSO algorithm for minimization of transmission power losses on different electrical power networks

Networks	Active power loss (MW)	Reactive power loss (MVar)
Nordic 44	338.36	−1930.79
IEEE 14 bus	0.995	29.43
IEEE 30 bus	2.198	22.76
IEEE 57 bus	52.189	206.18

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