Employment of PSO algorithm to improve the neural network technique for radial distribution system state estimation

Conventional SE is the same as the objective function of distribution SE, which is expressed as follows: (1)min J ( x ) = ∑ i = 1 m w i ( z i − h i ( x ) ) 2 , where h_i is the state equation of measurement variable i; w_i the weighting factor of measurement variable i; x the state variable including the loads and C outputs; z_i the measurement value of measurement variable i.

Where: (2)x ¯ = [ v i , δ i , P g i , Q g i ] .

The difference between the measured and calculated state variables is minimized by using this function. One of the state variables is used as a load value at each section rather than a voltage utilized by conventional state estimators. Therefore, the reactive and active state power values are used as state variables. The input values V and δ are also used as state variables. In the study of Narvaez and Grijalva (2008), the state variables are obtained among the boundaries.

The constraints of the objective function are defined as follows: (3)x j , min ⩽ x j ⩽ x j , max , where x_j,max is the maximum value of the state variable j; x_j,min the minimum value of the state variable j. (4)0 ⩽ Q c i ⩽ Q c , max i , i = 1 , 2 , … , N c , Q}_{c}^{i}\leqslant {Q}_{c,\,\max }^{i},\,i=1,2,\ldots,{N}_{c},\end{eqnarray}?> where Q c i Q}_{c}^{i}$?> is the reactive power of ith capacitor bank; Q c , max i Q}_{c,\,\max }^{i}$?> the maximum reactive power of ith capacitor bank; N_c the number of capacitors bank installed on feeders. (5)v min , i P H ⩽ v i P H ⩽ v max , i P H , v}_{\min,i}^{PH}\leqslant {v}_{i}^{PH}\leqslant {v}_{\max,i}^{PH},\end{eqnarray}?> (6)δ min , i P H ⩽ δ i P H ⩽ δ max , i P H , \delta }_{\min,i}^{PH}\leqslant {\delta }_{i}^{PH}\leqslant {\delta }_{\max,i}^{PH},\end{eqnarray}?> where v i P H v}_{i}^{PH}$?> v min , i P H v}_{\min,i}^{PH}$?> v max , i P H v}_{\max,i}^{PH}$?> are the state variable of voltages minimum value and maximum voltages values; and δ i P H \delta }_{i}^{PH}$?> δ min , i P H \delta }_{\min,i}^{PH}$?> δ max , i P H \delta }_{\max,i}^{PH}$?> the state variable of bus voltage angles, minimum value, and maximum bus voltage angles values.

The ratio of the target section to the total load of the target network and the total power inputted into the target network are used to calculate the center value of the bound at each load. The average input is the center value of the bound of each variable. Fast power flow distribution is utilized to calculate the current and voltage (Shigenori et al., 2001).

A PMU can measure the current phasors of some or all of the lines connected to the installed bus and the voltage phasors of the installed bus. Efficient placement strategies (benefits derived from PMUs) should be applied to justify the costs of bringing the measurement system up to date. PMU placement methods are facing a huge challenge from the increasing dependence on renewable energy sources that are constantly added to modern distribution grids (Fukuyama et al., 1996).

A distribution grid is converted to an active system and its power path flow is changed by distribution generators. Literature review of papers (Manousakis et al., 2011; Ren and Jordan, 2018) published in the past addressed PMU placements in their classification of PMU placement problems on the basis of the algorithms and methods utilized to determine the difficulty of selecting the location and ideal number of installed PMUs. Graph theoretic procedure (GTP), integer programming, PSO, and generic algorithm methods have different constraints and different requirements for PMU installation objectives, which can be used to achieve different goals in different cases.

Rules that can be applied to PMU systems to make them observable are as follows:

Rule 1: the current phasor and bus voltage of all incident branches of installed PMU at a bus are all known.

Figure 2:

A simple approach for determining the locations and sizes of capacitor banks in RD feeders was presented in this study. In this approach, the total reactive loads and total reactive losses in all sections of the feeders were the same as the amount of required compensation for any feeder (Manousakis et al., 2012).

We installed capacitor banks on a bus through trial and error method to minimize power losses and improve voltage profiles. In the distribution network, the compensation of reactive power is a vital issue. Transformers, converters, and induction motors are combined to consume reactive power (More and Jadhav, 2013). A reasonable amount of reactive power is not transmitted through the network because the voltage drop and energy losses are increased. Thus, reactive power should be generated close to its consumers (Huang et al., 2014; Mohamed et al.). Figure 3 illustrates the effect of reactive power compensation.

Figure 3:

Compensating devise Q_c is connected to the consumer terminals, and the power delivered through a distribution line is as follows: (7)S L = P + j ( Q − Q c ) , where S_L is the power delivered; P the real power; Q and Q_c the reactive power and reactive power compensator.

Each NN in PSO–NN consists of its velocity and network position. The weight of the NN (W) is related to the network position. The network’s velocity (∆W) is equivalent to updating the NN’s weight. PSO’s main role in the NN is to attain the best set of weights (particle positions) where numerous particles are aiming for the best solution.

A PS is composed of numerous particles. Each particle monitors its own position, best position, velocity, best fitness, current fitness, and the particles nearby. To operate the NN successfully, the position vector should match the weight vector of the network. The fitness value should match the forward transmission through the network. The particle’s new solutions are guided by using its best neighbor and global best particle. The particle’s position serves as the answer at the end (Hamada et al., 2008). The PSO’s main task in the NN is to secure the best set of weights in a situation where numerous NNs compete for the best solutions. The velocity vector is given by the following equation: (8)v i k + 1 = w v i k + c 1 r 1 ( P best i k − X i k ) + c 2 r 2 ( g best i k − X i k ) . v}_{i}^{k+1}=w{v}_{i}^{k}+{c}_{1}{r}_{1}({P}_{{\rm{best}}i}^{k}-{X}_{i}^{k})+{c}_{2}{r}_{2}({g}_{{\rm{best}}i}^{k}-{X}_{i}^{k}).\end{eqnarray}?>

In each iteration the position of each particle is updated by using the velocity vector of the following equation: (9)X i k + 1 = X i k + V i k + 1 , X}_{i}^{k+1}={X}_{i}^{k}+{V}_{i}^{k+1},\end{eqnarray}?> where X i k X}_{i}^{k}$?> is the particle position at (k) iteration; X i k + 1 X}_{i}^{k+1}$?> the particle position at (k + 1) iteration; V i k V}_{i}^{k}$?> the particle velocity of at (k) iteration; V i k + 1 V}_{i}^{k+1}$?> the particle velocity at (k + 1) iteration; w the weight parameter; r₁ and r₂ the random number; c₁ and c₂ the learning factors [0, 1].

The development of the PSO–NN model with its angle and voltage magnitude is expressed as follows:

A population of NNs with various initial weights was assembled for the PSO–NN.

The PSO–ANN model aimed to estimate bus voltage and bus voltage angles.

A distribution system is characterized with short lines that have high R/X ratios and systems that are unbalanced. This system experiences more losses than a transmission network. An RDS normally spreads over a large area (Husham et al., 2018). This condition makes many customers’ domestic loads difficult to monitor and measure. The network’s observability and state estimator’s capability to provide realistic estimates are also hampered. This serious challenge can be tackled by using PMUs. This study compared the results of the proposed hybrid algorithm (PSO–NN) with the results of PMUs.

In this study, a new algorithm was tested on PSNs as an observational system to determine the increase and decrease of bus voltage and their angles. Some differences were observed between the values because the networks were monitored rather than performing aggregate analysis, as shown in Figures 7, 8, and 10.

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The proposed PSO–NN algorithm was designed on MATLAB. RPDSN IEEE (9, 33, and 69) bus standards were developed and simulated using a MATLAB/PSAT toolbox. PMUs were applied on the distribution networks by using the toolbox. Then, a GTP algorithm was adopted to find the PMU’s location. The capacitor bank’s size and location were determined through trial and error method. For each test, the number of capacitors and PMUs varied. As shown in Table 1, the measurements had low MSE.

The proposed PSO–NN algorithm and its installed PMU (Manousakis et al., 2012) were evaluated on three tests.

Test 3: IEEE 69-bus RDPSN

Four capacitor banks are installed on buses (24, 45, 49, and 63), each with a size of 0.55 p.u to improve the voltage profiles. In total, 27 PMUs are installed on buses 12, 15, 17, 19, 21, 23, 26, 3, 32, 34, 38, 40, 42, 45, 49, 5, 51, 54, 58, 60, 62, 64, 66, 69, 7, 69, and 9 to ensure network observability. As shown in Figures 10 and 11, similar results were obtained between the input and target when the PSO–NN algorithm was tested in the network. As presented in Table 1, errors appeared in the results. As shown in Figure 12, R = 0.99934 and R = 0.99583 were the regressions for the bus voltage and δ bus voltage, respectively.

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As previously mentioned, the simulation results proved that the hybrid PSO–NN algorithm can be tested on any distribution network because of its accurate and efficient SE of the input values and target.

A new algorithm was developed in this study for the SE of angle bus voltage and voltage magnitude of RDPSNs. Voltage was a good predictor of the proximity to voltage collapse, and phase angle was a good predictor of power flow. Capacitor banks were installed to enhance the voltage profiles. PMUs were installed to make the networks observable. Results indicated that redundancy occurred and the accuracy of the estimated states improved when synchronized PMUs were added into the measurement sets. Three distribution networks were used in the SE of IEEE (9, 33, and 69) buses. The simulation results showed that the proposed algorithm was accurate and fast and only required few variables. A small MSE difference was observed between the target output and the output obtained by utilizing PMUs and PSO–NN, as shown in the regression plots for each network. The target output was represented by a dotted line. MATLAB software and PSAT package were used to obtain the results. The SE problem was transformed to a constrained problem and solved using the PSO–NN algorithm. The effectiveness of the proposed method was confirmed by the simulation results.