Ultra-short-term power forecast of photovoltaic power station based on VMD–LSTM model optimised by SSA

VMD adopts a completely non-recursive modal decomposition method, which decomposes data sequence f into multiple modal sub-functions u_k and solves the problem of loss of high-frequency signal and aliasing in traditional modal decomposition. The formula for calculating u_k bandwidth is as follows: (1) {min{uk},{wk}∑k‖∂t[(δ(t)+jπt)×uk(t)]e−jωkt‖2s.t. ∑kuk(t)=f(t), \left\{{\matrix{{\mathop {\min}\limits_{\{{u_k}\},\{{w_k}\}} \sum\limits_k {{\left\| {{\partial _t}\left[{\left({\delta (t) + {j \over {\pi t}}} \right) \times {u_k}(t)} \right]{e^{- j{\omega _k}t}}} \right\|}^2}} \hfill \cr {s.t.\;\sum\limits_k {u_k}(t) = f(t),} \hfill \cr}} \right. where {u_k} represents the modal component and {ω_k} is the centre frequency of the modal component.

The formula in Eq. (1) is solved by using the augmented Lagrange function to obtain the following expression: (2) L({uk},{ωk},λ)=α∑k=1K‖∂t[(δ(t)+jπt)×uk(t)]e−jωkt‖2+‖f(t)−∑k=1Kuk(t)‖2+〈λ(t),f(t)−∑k=1Kuk(t)〉 \matrix{{L\left({\{{u_k}\},\{{\omega _k}\},\lambda} \right)} \hfill & {= \alpha \sum\limits_{k = 1}^K {{\left\| {{\partial _t}\left[{\left({\delta (t) + {j \over {\pi t}}} \right) \times {u_k}(t)} \right]{e^{- j{\omega _k}t}}} \right\|}^2} + {{\left\| {f(t) - \sum\limits_{k = 1}^K {u_k}(t)} \right\|}^2}} \hfill \cr {} \hfill & {+ \left\langle {\lambda (t),f(t) - \sum\limits_{k = 1}^K {u_k}(t)} \right\rangle} \hfill \cr} The alternating direction method of multipliers (ADMM) algorithm is used to further solve Eq. (2) to obtain Eqs (3) and (4): (3) u^kn+1(ω)=f^(ω)−∑i≠ku^i(ω)+λ^(ω)/21+2α(ω−ωk)2; \hat u_k^{n + 1}(\omega) = {{\hat f(\omega) - \sum\limits_{i \ne k} {{\hat u}_i}(\omega) + \hat \lambda (\omega)/2} \over {1 + 2\alpha {{(\omega - {\omega _k})}^2}}}; (4) ωkn+1=∫0∞ω|u^k(ω)|dω∫0∞|u^k(ω)|2dω; \omega _k^{n + 1} = {{\int_0^\infty \omega \left| {{{\hat u}_k}(\omega)} \right|d\omega} \over {\int_0^\infty {{\left| {{{\hat u}_k}(\omega)} \right|}^2}d\omega}}; where u^kn+1 \hat u_k^{n + 1} and ωkn+1 \omega _k^{n + 1} are the Wiener filter and frequency centre of each component, respectively.

The SSA is a bionic intelligent algorithm, and its behaviour characteristics are modelled after sparrows’ foraging and anti-predation behaviours. The sparrow population is composed of two sub-populations: joiner and discoverer. The search and foraging activities of the whole sparrow population is based on the change in the discoverer's position. In order to increase the predation rate, participants follow the discoverer to get food while constantly monitoring the discoverer. Sparrows in different positions will choose different escape strategies after realising that the population is in danger. The comprehensive performance of common algorithms, such as moth–flame optimisation algorithm, grey wolf optimisation algorithm and PSO algorithm, is inferior to hat of the SSA in terms of accuracy and stability; so, SSA with better searching ability is used to optimise VMD.

The VMD decomposition method decomposes data series by setting variables, such as decomposition number, fidelity coefficient and convergence conditions. The decomposition quantity K and penalty factor α control the accuracy of the VMD decomposition method. The setting of the decomposition quantity K should be reasonable. When the K setting is large, decomposition is excessive, while data are lost when the K setting is small. It is determined by the bandwidth penalty variable α, and the bandwidth scale affects the extraction effect of the sequence. When the VMD method is used to decompose non-stationary and non-linear sequences, the subjective settings of the decomposition number K and the penalty factor α affect the accuracy of decomposition. Therefore, the SSA algorithm is proposed in this paper to optimise VMD parameters.

The aim of optimising VMD using the SSA algorithm is to decompose sequence f (t) into K eigenmode functions u_i, and the decomposition process is shown in Eq. (5). (5) f(t)=u1(t)+u2(t)+⋯+uk(t)=∑i=1kui(t). f(t) = {u_1}(t) + {u_2}(t) + \cdots + {u_k}(t) = \sum\limits_{i = 1}^k {u_i}(t).

When the actual decomposition components of the signal are all orthogonal, the energy of the original data sequence f (t) is equal to the sum of the energy of the K decomposition sequences, as shown in Eq. (6). (6) Ef1=∫−∞∞f2(t)dt {E_{f1}} = \int\limits_{- \infty}^\infty {f^2}(t)dt (7) EBIMF=∫−∞∞u1(t)dt+…+∫−∞∞uk(t)dt {E_{BIMF}} = \int\limits_{- \infty}^\infty {u_1}(t)dt + \ldots + \int\limits_{- \infty}^\infty {u_k}(t)dt (8) Ef1=EBIMF {E_{f1}} = {E_{BIMF}}

When the BIMF is not completely orthogonal, then E_f1 and E_BIMF have energy error E_err. (9) Eerr=|Ef1−EBIMF|. {E_{err}} = \left| {{E_{f1}} - {E_{BIMF}}} \right|.

The orthogonality of the BIMF component is determined by the value of E_err. The smaller the E_err is, the better is the orthogonality effect and the better is the representation effect of the sub-components on the sequence f (t).

To sum up, the solution process of [K,α] by the VMD decomposition method is as follows:

Setting the initial population and related parameters of the SSA and selecting E_err as fitness function.

For selecting a photovoltaic power sequence to optimise the VMD decomposition method by SSA, the population number of SSA is set to 20; the number of iterations is set to 30; the search range of the decomposition quantity K is [2, 16] and the optimisation range of the penalty factor α is [0,3000]. According to the change in the optimal fitness value, the optimal [K,α] combination is [6,1578], and the minimum energy error is 0.551. The VMD decomposition results of SSA optimisation of a photovoltaic power sequence are shown in Figure 2.

Fig. 2

When a recurrent neural network (RNN) processes long-time series data, the gradient is easy to explode, resulting in poor processing effect. The LSTM network solves the gradient problem when an RNN network processes long sequence data by setting the cell state and three-door structure in the hidden layer, so that the memory information can be saved for a long time. The internal structure of an LSTM network is shown in Figure 3.

Fig. 3

The inputs of the LSTM network are the sequence x_t, hidden layer state h_t−1 and the memory cell state c_t−1 at the previous moment, and the outputs are the hidden layer state h_t, candidate state c_t, forgetting gate t time result f_t, input gate t time result and output gate t time result o_t. In the LSTM network, the forgetting gate is used to determine whether information is retained, the input gate determines the data input, and the output gate determines the data output. The LSTM calculation formula is as follows: (10) {it=σ(Wxixt+Whiht−1+Wcict−1+bi)ft=σ(Wxfxt+Whfht−1+Wcfct−1+bf)ct=ftht−1+ittanh(Wxcxt+Whcht−1+bc)ot=σ(Wxoxt+Whoht−1+Wcoct−1+bc)ht=ottanh(ct), \left\{{\matrix{{{{\boldsymbol{i}}_t} = \sigma \left({{{\boldsymbol{W}}_{{\rm{xi}}}}{{\boldsymbol{x}}_t} + {{\boldsymbol{W}}_{{\rm{hi}}}}{{\boldsymbol{h}}_{t - 1}} + {{\boldsymbol{W}}_{{\rm{ci}}}}{{\boldsymbol{c}}_{t - 1}} + {{\boldsymbol{b}}_i}} \right)} \hfill \cr {{{\boldsymbol{f}}_t} = \sigma \left({{{\boldsymbol{W}}_{{\rm{xf}}}}{{\boldsymbol{x}}_t} + {{\boldsymbol{W}}_{{\rm{hf}}}}{{\boldsymbol{h}}_{t - 1}} + {{\boldsymbol{W}}_{{\rm{cf}}}}{{\boldsymbol{c}}_{t - 1}} + {{\boldsymbol{b}}_f}} \right)} \hfill \cr {{{\boldsymbol{c}}_t} = {{\boldsymbol{f}}_t}{{\boldsymbol{h}}_{t - 1}} + {{\boldsymbol{i}}_t}\tanh \left({{{\boldsymbol{W}}_{{\rm{xc}}}}{{\boldsymbol{x}}_t} + {{\boldsymbol{W}}_{{\rm{hc}}}}{{\boldsymbol{h}}_{t - 1}} + {{\boldsymbol{b}}_c}} \right)} \hfill \cr {{{\boldsymbol{o}}_t} = \sigma \left({{{\boldsymbol{W}}_{{\rm{xo}}}}{{\boldsymbol{x}}_t} + {{\boldsymbol{W}}_{{\rm{ho}}}}{{\boldsymbol{h}}_{t - 1}} + {{\boldsymbol{W}}_{{\rm{co}}}}{c_{t - 1}} + {{\boldsymbol{b}}_c}} \right)} \hfill \cr {{{\boldsymbol{h}}_t} = {{\boldsymbol{o}}_t}\tanh \left({{{\boldsymbol{c}}_t}} \right),} \hfill \cr}} \right. where W_x, W_h and W_c are different weight matrices; b_i, b_c, b_f and b_o represent the offset vectors; σ and tanh are the activation functions.

Both LSTM network and BP neural network use gradient optimisation algorithm to transform weights. In order to solve the problem of large parameter dimension, Adam algorithm is used to optimise the random objective function.

The parameters of the LSTM network model of the first and second layers are as follows: dropout value is 0.2; the function is optimised to Adam; the activation function takes tanh; the number of nodes is 50; training times are 1,000; batch size is 72; and the average absolute error function is selected as the loss function.

The forecast results of each forecast model for sunny weather are shown in Figure 5. In sunny weather, the fluctuation of photovoltaic power is small. It can be clearly seen in the figure that there is a big deviation between the predicted value and the real value of the BP neural network. When the power value is larger, the error is more obvious. Statistical errors of the five models are plotted in Table 1. Comparing the prediction errors of BP, ANN and LSTM models, the e_MAPE values of the LSTM model are 0.096 and 0.082 lower than those of BP and ANN, respectively, which indicates that the deep learning model has obvious precision advantages over the traditional neural network and can extract more detailed characteristics of the photovoltaic data series. On November 7, e_MAPE, e_RMSE and e_TIC of the SSA–VMD–LSTM prediction model were 0.179, 56.833 and 0.076, respectively. Among the five models, the three error evaluation indices were the smallest, and when the photovoltaic power curve fluctuated at 11 am, the SSA–VMD–LSTM power prediction curve was still consistent with the actual power curve.

Fig. 5

Figure 6 shows the photovoltaic output forecast of each forecast model in rain and snow weather, and Table 2 shows the model errors. Compared with the photovoltaic power in cloudy weather, the photovoltaic power in rainy and snowy weather decreases and fluctuates obviously. At this time, the prediction accuracy of each model has decreased, and there are obvious errors between the predicted power and the actual power of the BP and ANN models, which are no longer suitable for the prediction of photovoltaic power. On June 5, compared with BP, ANN and LSTM, the e_MAPE values of the VMD–LSTM and SSA–VMD–LSTM models decreased obviously. The main reason was that the complexity of data sequence was reduced by VMD, and redundant interference was avoided.

Fig. 6

When the weather changes from sunny to cloudy, the photovoltaic power prediction results of each model are shown in Figure 7, and the errors are shown in Table 3. The complexity of photovoltaic power data series in abrupt weather is higher than that in non-abrupt weather, and the deviation between the predicted power of BP and ANN models and the actual power is greater, which obviously reduces the prediction accuracy. As can be seen from Table 3, on January 8, the e_MAPE value of the SSA–VMD–LSTM prediction model decreased by 0.350, 0.318, 0.242 and 0.067, respectively compared with BP, ANN, LSTM and VMD–LSTM models, i.e. the prediction accuracy of the SSA–VMD–LSTM model is still the highest in sudden weather.

Fig. 7