Machine Learning Algorithm Based Static VAR Compensator to Enhance Voltage Stability of Multi-machine Power System

SVC [10] is the combination of Thyristor Controlled Reactor (TCR) and Thyristor Switched Capacitor (TSC) which is utilized for the dynamic compensation of transmission lines. It will likewise enhance the working adaptability and minimizes the losses in the power system. A basic SVC structure is as shown in Figure 1 which consists of three TSCs and one TCR. The quantity of TSCs required is principally relies upon working voltage, required most extreme power yield and current rating of the thyristor valves Inductive rating might be stretched out by including the number of TCRs dependent on the necessity. If the voltage is below the reference value TSC will act and create the required reactive power with the end goal that voltage takes back to reference level and if the voltage is more than reference value TCR will act.

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Figure 1. SVC FACTS controller

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Figure 2. V-I characteristics of SVC FACTS controller

The Volt-Amp characteristics of the SVC FACTS controller are as shown in Figure 2. The SVC can also be used to enhance the voltage profile at the midpoint and at the end of the lines, Improvement of dynamic and transient stability and Power oscillation damping.

2.1 Modelling of static VAR compensator

SVC can be mathematically modelled using susceptance and firing angle model as given in 2.1.1 and 2.1.2.

2.1.1 Susceptance method

The equivalent circuit is used to obtain the SVC FACTS controller power equations and the equations required by NR method [11]. In this method susceptance is varied until the voltage return back to a reference level.

The current drawn by the SVC is given at bus k is given as 

${{I}_{SV{{C}^{.}}}}=j*{{B}_{SVC}}*{{V}_{k}}$                         (1)

The reactive power injected at bus k.    

  ${{Q}^{.}}_{SVC}={{Q}_{k}}=-V_{k}^{2}*{{B}_{SVC}}^{.}$                       (2)

The linearized equation is given by the following Eq. 3

${{\left[ \begin{matrix}   \Delta {{P}_{{{k}^{.}}}}  \\  \Delta {{Q}_{{{k}^{.}}}}  \\\end{matrix} \right]}^{(i)}}={{\left[ \begin{matrix}   {{0}^{.}} & {{0}^{.}}  \\   {{0}^{.}} & {{Q}_{{{k}^{.}}}}  \\\end{matrix} \right]}^{(i)}}{{\left[ \begin{matrix}   \Delta {{\theta }_{{{k}^{.}}}}  \\   {}^{\Delta {{B}_{SVC}}^{.}}/{}_{{{B}_{SVC}}^{.}}  \\\end{matrix} \right]}^{(i)}}$       (3)

2.1.2 Firing angle method

In firing angle method $B_{S V C}$  is given by Eq. 4

$\begin{array}{l}{\mathrm{B}_{\mathrm{svc}}=\mathrm{B}_{\mathrm{C}}-\mathrm{B}_{\mathrm{TCR}}=\frac{1}{X_{C} * X_{L}}\left\{X_{L}-\frac{X_{C}}{\prod} *[2 *(\Pi-\alpha)+\sin 2 \alpha]\right\}} \\ {X_{L}=\omega^{*} L} \\ {X_{C}=\frac{1}{\omega^{*} C}}\end{array}$      (4)                                               

In firing angle method reactive power generated at bus k is given as Eq. 5

$Q_{k}=-\frac{V_{k}^{2}}{X_{C} * X_{L}}\left\{X_{L}-\frac{X_{C}}{\Pi} *[2 *(\Pi-\alpha)+\sin 2 \alpha]\right\}$     (5)      

From Eq. 5, the linearised SVC equation can be written as Eq. 6

$\left[\begin{array}{c}{\Delta P_{k}} \\ {\Delta Q_{k}}\end{array}\right]=\left[\begin{array}{cc}{0} & {0} \\ {0} & {\frac{2 * V^{2}}{\Pi^{*} X_{L}}[\cos (2 \alpha)-1]}\end{array}\right]\left[\begin{array}{c}{\Delta \theta_{k}} \\ {\Delta \alpha}\end{array}\right]$           (6)

ANNs are developed to impersonate the characteristics of the human brain. ANN is a blend of different Artificial Neurons. A single neuron is a processing unit and like a summing unit. At each neuron the weighted sum is calculated and the output is determined by processing through activation function. The ANN has been extensively used for massive parallelism, fault tolerance distributed representation, generalization ability, adaptively, learning ability, inherent contextual information processing and Low energy consumption. Artificial Neural Networks also used for forecasting of some parameters or controlling of some other parameters. Most of the applications use MNNs rather than single layer Neural networks which will be trained using BPNN.

4.1 Back propagation algorithm

The architecture of MNN is as given in Figure 3 [14, 15]. It has an input, hidden and output layers with n,p and m number of neurons.

The input vector X is applied to neurons in the input layer and the activation values at the hidden layer can be calculated. The activation value of the j^th neuron is given by Eq. 10.

  ${{Z}_{-inj}}=Voj+\sum\limits_{i=1}^{n}{({{X}_{i}}*{{V}_{ij}}})$                      (10)

The output of the j^th hidden neuron can be calculated using sigmoidal activation function as given in Eq. 11.

  $Z_{j}=f\left(Z_{\text {-inj }}\right)=\frac{1}{1+e^{Z_{-i n j}}}$                         (11)

The hidden layer neurons output will be the inputs for each neuron in the output layer. The activation function at the k^th neuron can be calculated using Eq. 12.

${{Y}_{-ink}}^{.}={{W}_{o{{k}^{.}}}}+{{\sum\limits_{j=1}^{p}{({{Z}_{j}}^{.}*{{W}_{jk}}}}^{.}})$                        (12) 

The response of the neuron in the k^th output layer can be determined using sigmoidal activation function as given in Eq. 13.

  Y_k=f(Y_-ink) = $\frac{1}{1+{{e}^{{{Y}_{-ink}}}}}$                           (13)

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Figure 3. Multi layer Neural Network (MNN)

The actual output of each neuron is compared with the expected output and squared error is determined. The weights are adjusted to decrease the squared error. Once outputs of the neurons in the output layer calculated, the error is calculated and the weights are adjusted to minimize the error. The updation of the weights are given as follows.

The weights between the output and hidden layer, bias neuron can be updated as follows.

W_jk^.(new)=W_jk^.(old)+ ΔW_jk^.,

                                     W_ok^.(new)=W_ok^.(old)+ ΔW_ok                                (14)

where, ΔW_jk= $\eta *\frac{\partial E}{\partial {{W}_{jk}}}$ = $\eta \text{*}{{\delta }_{\text{k}}}\text{*}{{\text{Z}}_{\text{j}}}$

   ΔW_ok= $\eta *\frac{\partial E}{\partial {{W}_{ok}}}$ = $\eta $ * $\delta_{\mathrm{k}}$                                     (15)

The weights between the hidden and input layer, bias neuron will be updated as follows.

V_ij (new)=V_ij^.(old)+ΔV_ij

V_oj^.(new) =V_oj (old) + ΔV_oj

where,

ΔV_ij= $\eta $ * δ_j*X_i

                                               ΔV_ok=alpha*δ_i                                                (16)

4.1.1 Methodology

BPNN is a procedural method for training MNNs. It is a multilayer feed forward network using extended gradient-descent based method for training which is popularly known as back propagation rule. It is a computationally efficient method for optimizing the weights in a feed-forward network. In this derivation of the squared error with respect to the weights is calculated and the weights are changed to minimize the squared error. So, the continuous activation functions are used which are differentiable.  Sigmoidal activation function is mainly used which is the continuous approximation of the step function.  The network is trained by supervised learning method. The aim of this algorithm is to train the multi layer neural network to achieve a balance between the output and expected output and ability to respond correctly to the input patterns that is used for training and the ability to provide better responses to the input that are similar. The BPNN Algorithm is as given below.

Step 1: Initialize all weights to random values

Step 2: Apply the deviation in voltage as input to the BPNN.

Step 3: Calculate the outputs of the hidden and output layers in forward pass.

Step 4: Determine the squared error which is to be minimised for better training.

Step 5: Update the weights between output and hidden, hidden and input layers in the backward process.

Step 6: Repeat the pocedure till the squared error is blow 0.001.

Several drawbacks in the Back propagation algorithm are

1.  It converges slowly If learning rate is chosen small and becomes unstable if it is cho sen high.
2. It is agonized with local minima problem.
3. ANNs some times over trained so that learning algorithm gives worse generalization performance.
4. Gradient based algorithms ingest more time for most of the applications.

4.2 Extreme learning machine algorithm

The bottlenecks in the BPNN Algorithm, ELM is proposed [16]. ELM network is a single hidden layer feed forward NN as shown in Figure. In this network, input weights and biases are initialized randomly and the output weights are calculated analytically.

Mathematically, let (x_i, y_i)_i=1^N  are the patterns formulated to train ELM network. Where x_i is input variable contains p number of attributes and y_i is output variable contains q number of attributes. The mathematical model of ELM with h number of neurons and with activation function G(.) is expressed as:

$Y({{x}_{k}})=\sum\limits_{i=1}^{h}{{{C}_{i}}}G({{x}_{i}}.{{a}_{k}}+{{b}_{i}})\,\,\,\,,k=1,2,....N$          (17)

where, c matrix is output weight matrix, a is input weight matrix and b is the bias matrix.

The ELM network is depicted as shown in Figure 4.

The Eq. (17) can be written in matrix notation as:

  $\mathrm{G} \mathrm{C}=\mathrm{Y}$                                 (18)

where, G is matrix of outputs of hidden layer and Y is output

matrix of ELM n $\left[ \begin{matrix}   g(a1*x1+b1) & \circ  & \circ  & g(an*x1+bn)  \\   \circ  & \circ  & \circ  & \circ   \\   \circ  & \circ  & \circ  & \circ   \\   g(a1*xn+b1) & {} & {} & g(an*xn+bn  \\\end{matrix} \right]$ (19)

$\beta =\left[ \begin{matrix}   C_{1}^{T}  \\   \circ   \\   \circ   \\   C_{n}^{T}  \\\end{matrix} \right]$    $Y=\left[ \begin{matrix}   y_{1}^{T}  \\   \circ   \\  \circ   \\   y_{n}^{T}  \\\end{matrix} \right]$

This can be compared similar to the cost function minimization in gradient based back propagation algorithm.

${{F}_{BP}}=\sum\limits_{i=1}^{N}{\left[ \sum\limits_{j=1}^{h}{{{\beta }_{j}}}G({{x}_{i}}.{{a}_{j}}+{{b}_{i}})-{{y}_{i}} \right]}$                  (20)

For randomly assigned input weights and biases the output weight matrix elements are calculated as:

$C={{G}^{+}}Y$                               (21)

where, G⁺ is the Moore – Penrose inverse of G matrix, which is evaluated from Singular value decomposition.

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Figure 4. Layout of single hidden layer ELM network

4.2.1 Methodology

In this work, the voltage variation fed as input to the ELM network and the corresponding firing angle and susceptance is forecasted for SVC FACTS controller to bring back the voltage to the actual reference value. The implementation of ELM is given in following steps

Step 1: The voltage variation at the busses is given as input for ELM network

Step 2: Input weights and biases are generated randomly and the output weights are calculated analytically.

Step 3: With the obtained output weights, the output firing angle or susceptance will be predicted.

Step 4: Using the updated firing angle or susceptance the voltage profile of the system is improved.

The ELM is having diverse assets when compared to the classical learning techniques in terms of high speed learning, greater generalization performance, uses non differential activation function, not suffered with over fitting, local minima and imprecision in learning.