Bat algorithm-based back propagation approach for short-term load forecasting considering weather factors

Abstract

This paper proposes a Bat algorithm-based back propagation approach for solving the short-term load forecasting considering the weather factors such as temperature and humidity. The accuracy of load forecasting is very important in the restructured power system as it plays a major role in providing a better cost-effective risk management and operation plans. Load/demand forecasting is difficult due to the nonlinear and randomness properties load itself, its dependency on factors like weather conditions, variations of social and economic environments, and price of the electrical power in deregulated environment. Load forecasting accuracy significantly impacts the cost of power utilities in operational planning of the energy supply. To show the effectiveness of the proposed approach, the PJM (Pennsylvania–New Jersey–Maryland) system load demand data are considered. The simulation results obtained have shown that the day-ahead hourly forecasts of load demand using the proposed method is very accurate with very less error and well forecasted.

Appendix: back propagation neural network(BPNN)

Artificial neural networks (ANNs) have the capability of handling nonlinear dependencies between the load demands and the factors influencing it. ANN maps the input and output relationships by approximating the linear/nonlinear mathematical functions. The structure of an ANN can be classified into 3 groups as per the arrangement of neurons and the connection patterns of the layers, and they are feed forward (back propagation networks), feedback (recurrent neural networks and adaptive resonance memories) and self-organizing (Kohonen networks). In this paper, back propagation neural networks (BPNNs) are used, which utilizes the available input and output data and then adjusts weights with the help of some observable functions (i.e., loss functions). The FFNNs consist of three or more layers of nodes, i.e., one input layer, one output layer and one or more hidden layers [41].

BPNN is composed by one input layer and one or more hidden layers and one output layer. The learning process of NN includes two courses, one is the input information transmitting in forward direction and another is the error transmitting in backward direction. In the forward action, the input information goes to the hidden layers from input layer and goes to the output layer. If the output of output layer is different with the wishful output result then the output error will be calculated, the error will be transmitted backward direction and then the weights between the neurons of every layers will be modified in order to make the error as minimum as possible. Then, the network is said to be trained for the given data or application. Figure 4 depicts the structure of 3-layer BPNN. From this figure, it can be observed that ’i’ represents the input layer, ’j’ represents the hidden layer, ’k’ represents the output layer, ’I’ is the input and ’O’ is the output.

Fig. 4

Structure of back propagation neural network (BPNN)

The input to jth neuron of hidden layer is expressed as [42],

$$\begin{aligned} \mathrm{net}_j =\mathop \sum \limits _i W_{ji} O_i \qquad i=1,2,3,\ldots ,N_I \end{aligned}$$

(9)

where $N_I$ is the number of input nodes. The output of $j{\mathrm{th}}$ neuron is expressed as,

$$\begin{aligned} O_j =g\left( {\mathrm{net}_j } \right) \end{aligned}$$

(10)

The input to $k{\mathrm{th}}$ neuron of output layer is expressed as,

$$\begin{aligned} \mathrm{net}_k =\mathop \sum \limits _j W_{kj} O_j \qquad j=1,2,3,\ldots ,N_H \end{aligned}$$

(11)

where $N_H$ is the number of hidden nodes. The output of $k{\mathrm{th}}$ neuron is expressed as,

$$\begin{aligned} O_k =g\left( {\mathrm{net}_k } \right) \end{aligned}$$

(12)

where g is Sigmoid function, and it is expressed as,

$$\begin{aligned} g\left( x \right) =\frac{1}{1+e^{-x}} \end{aligned}$$

(13)

The principle of BPNN is the error back propagation during the learning process. The learning process is accomplished through minimization of an object function, which is the sum of squares of the errors between the actual output of network and the target output. Gradient descent algorithm is used to derive the computing formula. In the learning course, the target output of $k{\mathrm{th}}$ neuron of output layer is $T_{pk}$, the corresponding actual output of network is $O_{pk}$ and the average of sum of squares of error of system is expressed as,

$$\begin{aligned} E=\frac{1}{2p}\mathop \sum \limits _p \mathop \sum \limits _k \left( {T_{pk} -O_{pk} } \right) ^{2}=\frac{1}{2}\mathop \sum \limits _k \left( {T_k -O_k } \right) \end{aligned}$$

(14)

where p is the number of training samples employed in training the network and E is the objective function. According to the gradient descent algorithm, we derive the increment or adjustment value of every weight and they are expressed as,

$$\begin{aligned} \Delta W_{kj}= & {} \eta \left( {T_k -O_k } \right) \delta _k \left( {1-O_k } \right) O_j \end{aligned}$$

(15)

$$\begin{aligned} \Delta W_{ji}= & {} \eta \delta _j O_i \end{aligned}$$

(16)

where $\eta$ is the learning rate, and $\delta _k$, $\delta _j$ are expressed as,

$$\begin{aligned} \delta _k= & {} \left( {T_k -O_k } \right) O_k \left( {1-O_k } \right) \end{aligned}$$

(17)

$$\begin{aligned} \delta _j= & {} O_j \left( {1-O_j } \right) \mathop \sum \limits _k \delta _k W_{kj} \end{aligned}$$

(18)

If the rate of learning ($\eta )$ is more, the adjustment or increment value of every weight will also become more, and this can accelerate the training process of the network, but this result can generate the oscillations. In order to avoid the oscillations due to the increased rate of learning ($\eta )$, a momentum term is added, and it is expressed as,

$$\begin{aligned} \Delta W_{ji} \left( {n+1} \right) =\eta \delta _j O_i +\alpha \Delta W_{ji} \left( n \right) \end{aligned}$$

(19)

where $\alpha$ is the proportionality constant or momentum factor. Through the back propagation network training, once the accuracy requirement is satisfied, then the interconnection weight between all nodes are ascertained or stored. This completes the training of BPNN. Then, the trained network can be used to identify the unknown sample. Usually, this part is termed as testing.