Shortterm load forecasting model based on quantum Elman neural networks was constructed in this paper. The quantum computation and Elman feedback mechanism were integrated into quantum Elman neural networks. Quantum computation can effectively improve the approximation capability and the information processing ability of the neural networks. Quantum Elman neural networks have not only the feedforward connection but also the feedback connection. The feedback connection between the hidden nodes and the context nodes belongs to the state feedback in the internal system, which has formed specific dynamic memory performance. Phase space reconstruction theory is the theoretical basis of constructing the forecasting model. The training samples are formed by means of
Shortterm load forecasting (STLF) is the basis for the normal and safe operation of power system. Since the introduction of competition mechanism in power market, the power company has put forward higher requirements on the accuracy and rapidity for load forecasting [
The process of human brain information processing may be related to the quantum phenomenon, and quantum mechanical effects may exist in the brain, which had been shown by some research results [
The STLF model based on quantum Elman neural networks (QENN) is constructed in this paper. QENN is composed of the quantum neurons. The quantum neuron model consists of the weighting part, the aggregating part, the activating part, and the excitation part. Different from the traditional multilayer feedforward neural network, QENN has not only the feedforward connection but also the Elman feedback connection. The Elman feedback connection between the hidden nodes and the context nodes belongs to the state feedback in the internal system, which has formed specific dynamic memory performance [
Some studies had shown that the load time series is nonlinear and chaotic [
Through the actual example simulation, it is proved that the proposed model can effectively improve the prediction accuracy and has adaptability to different load time series.
QENN is composed of quantum neurons, and quantum neurons are the basic elements of building QENN.
In the classical computation, the binary numbers “0” and “1” are used to represent the information, and they are usually called bits. In the quantum computation,
In quantum computation, a qubit state
In order to further understand the superposition of the qubits, an electron is used to illustrate. An electron can be in the ground state
Schematic diagram of superposition state of ground State
If the electron is in the superposition state
Schematic diagram of the ground state and excitation state after the superposition state is collapsing.
The quantum neuron model adopted in the paper consists of the weighting part, the aggregating part, the activating part, and the excitation part. The weighting part is the simulation of the bond strength between synapses of nerve cells. The aggregating part is the simulation of the spatial and temporal integration of stimuli received by multiple dendrites. The activating part is the simulation of the interaction between the current activation values and membrane potential change of nerve cells. The excitation part is the simulation of the nonlinear characteristics of nerve cells, such as excitation, inhibition, fatigue, refractory period, and threshold. According to [
The quantum neuron model is shown in Figure
Quantum neuron model.
The weight and activation values are qubits.
The activation values are the important parameters for the activating part. The simulation of the interaction between the current activation values and membrane potential change of nerve cells is the core of the activating part. The activation values affect the accuracy of prediction. The parameters can be optimized by genetic algorithm.
In Figure
The relationship between input and output can be expressed as
Considering Figure
Quantum Elman neural networks.
For QENN, each node in the hidden layer has a corresponding node connected to it in the context layer.
The input nodes and the hidden nodes, the hidden nodes and the output nodes, and the context nodes and the hidden nodes are connected by adjustable weights.
Compared with the feedforward network structure, QENN has the context layer. The role of the nodes in the context layer is to store the internal state of neural network. They are used to store the state of the hidden nodes at the current time, and the state will be passed to the hidden nodes at the next time. The connection between the hidden nodes and the context nodes belongs to the state feedback in the internal system, which has formed specific dynamic memory performance.
QENN can not only store the information in the current input data but also store some historical information. In the training process, it can dynamically backtrack to the input at the past time related to the expected output at the current time.
According to (
If
This shows that many groups of
Genetic algorithm is used to adjust the parameters of the QENN, which include
The chromosome can be constituted by these parameters, which can be expressed as
Genetic algorithm uses the roulette method as selecting operator. The crossover operator adopts the nonuniform arithmetic crossover method, and the mutation operator uses the noncoherence mutation method. The fitness function mainly considers the accuracy of the output of the forecasting model; that is,
The process is shown in Figure
The process of parameters optimization of QENN by genetic algorithm.
The operation object of arithmetic crossover operator is chromosome
If two chromosome individuals are
For the nonuniform arithmetic crossover operator,
Supposing the chromosome individual
The chaotic time series is usually analyzed by PSRT. The embedding theorem of Takens and Saner et al. is the basis of the prediction theory to chaotic time series.
To chaotic time series
According to the GP method, the embedding dimension
PSRT provides a theoretical basis for the construction of the QENN model. For the model, the number of input nodes is decided by the embedding dimension
The
In the reconstructed phase space with
According to (
Four STLF models are constructed, which are modelI based on the QENN, modelII based on the quantum feedforward neural network, modelIII based on the conventional Elman neural network, and modelIV based on the conventional feedforward neural network.
The input nodes number of four STLF models for actual simulation load system is 7, which is acquired by the GP method. The delay time
The number of the neighbor phase points affects the prediction accuracy and calculation of the forecasting model. If the number of the neighbor phase points is too much, it will not only increase the amount of computation but also increase the number of false neighbor phase points, which has a great impact on the prediction performance. If the number of the neighbor phase points is too small, the effective information is not fully utilized, which also affects the prediction performance. In general, the relationship between the number of the neighbor phase points
For modelI and modelIII, the number of the input nodes is 7, the number of the hidden nodes is 10, the number of the context nodes is 10, and the number of the output points is 1.
For modelII and modelIV, the number of the input nodes is 7, the number of the hidden nodes is 10, and the number of the output points is 1.
The maximum permissible error of the four STLF models is
Four models are used to predict the actual load. The daily load forecasting errors and comparison of four models for workday and weekend are shown in Figures
The daily load forecasting errors of four STLF models for workday.
The daily load forecasting errors of four STLF models for weekend.
The mean absolute percentage error and the maximum relative error of four STLF models for workday and weekend are described in Table
The error comparison of four STLF models for two day types.
Day type  ModelI  ModelII  ModelIII  ModelIV  








 
Workday  1.025  2.977  1.826  4.450  1.813  4.521  2.903  5.327 
Weekend  1.210  3.225  2.153  4.705  2.275  4.860  3.152  5.515 
The maximum relative error is the maximum value of relative error of all predicted points. The relative error can be calculated as
The error performance of the modelIV is the worst in the all models, because the conventional feedforward neural network cannot effectively characterize the chaotic dynamic behavior of the load system. The error performance of modelII is better than that of modelIV, because quantum computation can enhance information processing capability and generalization ability of quantum feedforward neural networks. The error performance of modelIII is better than that of modelIV, because the conventional Elman neural network has formed specific dynamic memory performance. The forecasting performance of modelI is the best, which is the fusion of quantum computation and feedback mechanism.
From Table
In order to verify how sensitive are the simulation results on
The forecasting error of modelI with different
 

5  6  7  8  9  









 
Monday  1.629  4.512  1.466  4.011  1.025  2.977  1.358  3.965  1.540  4.250 
The forecasting error of modelI with different
 

1  2  3  4  







 
Monday  1.025  2.977  1.376  3.852  1.405  4.009  1.466  4.542 
From Tables
In order to further verify the performance of modelI, the real example are used to test in a week. The mean absolute percentage error and the maximum relative error in the week are described in Table
The daily load forecasting errors in a week.
Day type  ModelI  


 
Monday  1.025  2.977 
Tuesday  1.109  3.012 
Wednesday  1.153  2.985 
Thursday  1.128  2.944 
Friday  1.090  2.870 
Saturday  1.210  3.225 
Sunday  1.227  3.412 
This paper constructs a STLF model based on QENN. The quantum computation and Elman feedback mechanism were integrated into QENN. Quantum computation can effectively improve the approximation capability and the information processing ability of the neural networks. QENN has not only the feedforward connection but also the feedback connection. Phase space reconstruction theory is the theoretical basis of constructing the forecasting model. The training samples are formed by means of
The authors declare that they have no competing interests.
This work was supported by the project Supported by National Nature Science Foundation of China (51477078).