GCN-LSTM spatiotemporal-network-based method for post-disturbance frequency prediction of power systems

0 Introduction

Owing to the expansion of the interconnection scale,the operating characteristics of power grids have become increasingly complex.For example, the active power disturbances caused by the switching of generators and load changes propagate in the power grid as waves with a speed considerably less than that of light and exhibit distinct spatiotemporal distribution characteristics [1-2].This may cause cascading failures, produce severe frequency-stability problems, and ultimately lead to the occurrence of blackouts[3-4].If the response characteristics of the disturbance frequency can be quickly predicted after the occurrence of disturbances, frequency-modulation resources can be used,and energy storage can be switched on.The energy storage devices help to restore the frequency characteristics of the system and has a positive effect on the safe and stable operation of the system.

The prediction of frequency-response characteristics based on power-system disturbances has been widely investigated.Frequency-response-prediction methods can be categorized as methods based on physical models and those that are data-driven.The frequency-response-prediction methods based on physical models include full-timedomain differential algebraic equation (DAE) models [5],frequency divider (FD) models [6], center of inertia (COI)frequency-estimation methods [7], closed-loop average system frequency (ASF)-response models [8], open-loop ASF-response models [9], system frequency response (SFR)models [10], and electromechanical wave models [11-12].As a classic approach for power-system dynamics, the fulltime-domain DAE model requires the establishment of accurate nonlinear differential equations of power-system components.Numerical methods are then used to solve these equations to predict the frequency response [5].Milano and Ortega [6] combined the rotor speed of a synchronous machine and the admittance matrix of the network.The FD model was then proposed, and it was used to approximate the quasi-steady-state bus frequency.However, these models inevitably require the numerical solution of the differential equations representing the components, and the calculation time is long.Based on the measured bus frequency and the FD model, Milano proposed a frequency-estimation method under the COI coordinate system based on weighted aggregation [7].In other studies [8-10], single-machine equivalent modeling of a power system was performed.The transfer function of the frequency response was derived to predict the frequency-response curve by considering the multi-speed and equivalent-speed governor models.With the increase in power-system interconnections, the scale of the system is gradually expanding.The frequencyprediction method under the COI coordinate system cannot describe the spatiotemporal frequency-distribution characteristics, and this is not conducive to the participation of distributed power sources in frequency-stability control[7].The electromechanical-wave model considers the spatiotemporal distribution characteristics of the frequencyresponse process.The process of disturbance propagation in large-scale power grids has been described by using the continuum-modeling method [11-12].However, the continuum-modeling process does not consider the dynamic response characteristics of the governor.

With the development of artificial intelligence and deep-learning technology, data-driven algorithms are being widely used in image reconstruction [13], speech recognition [14], and natural language processing [15].Unlike model-driven frequency-prediction methods, datadriven methods do not rely on accurate system modeling.The regression models for prediction are constructed by mining the internal connections between the data using these methods.Data-driven frequency-response-prediction models include artificial neural networks [16], V-support vector regression [17], multi-layer support vector machines(SVMs) [18], multivariate random forest regression [19],stacked denoising autoencoders [20], deep belief networks(DBNs) [21], and restricted Boltzmann machines.These models are characterized by a high calculation speed and robustness.However, they are based on the COI frequency,and the spatiotemporal distribution characteristics of the frequency response are not considered; thus, optimizing the control of spatially distributed frequency-modulation resources is difficult.

In summary, the DAE and FD models based on physical models require numerical calculations, making them timeintensive.Moreover, the ASF and SFR models and the data-driven models based on the premise of single-machine equivalent simplification in the COI coordinate system do not consider the spatiotemporal distribution characteristics of the frequency.Finally, the physical-model-based electromechanical-wave propagation model does not consider the dynamic characteristics of the governor.

To address these limitations, the rapidity of the datadriven methods, spatiotemporal distribution characteristics of the frequency response, and response characteristics of the governor are combined in the present study.A novel dynamic frequency-prediction methodology based on the graph convolutional network (GCN) and long short-term memory (LSTM) spatiotemporal network is developed for use in power systems, and it is named STGCN-LSTM.The contributions of this study are summarized as follows.

1) The form of the hyperbolic partial differential equations of frequency propagation is derived by formulating the spatiotemporal frequency distribution model, and the factors affecting the spatiotemporal frequency-distribution characteristics are clarified.The multi-machine swing equations are considered to determine the effect of the spinning-reserve capacity on the frequency response.The spatiotemporal features of the network input are then constructed.

2) The graph structure of the phasor measurement units (PMUs) in the power grid is constructed, the network parameters are embedded in the adjacency matrix, and an improved GCN (IGCN) model is proposed to extract the spatial characteristics of the frequency response.The LSTM network is then used to extract the temporal characteristics of the frequency response.

3) The GCN-LSTM spatiotemporal regression prediction model is trained by applying the rolling update of the measurement data window and asynchronous sequence mapping, to predict an accurate frequency curve after the occurrence of a disturbance, while meeting the needs of online applications.

The remainder of this paper is organized as follows.Section 1 describes the motivation behind the framework of the proposed spatiotemporal-network model.The spatiotemporal network consisting of an IGCN for spatialfeature extraction and a LSTM network for temporal-feature extraction is proposed in Section 2.The simulation results are discussed in Section 3.Finally, Section 4 concludes the paper.

1 Framework of frequency prediction by considering the spatiotemporal distribution characteristics

1.1 Spatiotemporal distribution model of the frequency response

The spatiotemporal distribution characteristics of the frequency response are manifested in the response times of different measurement devices to disturbances in a multi-machine power grid [22], and these characteristics are widely used to locate disturbance sources [23-24].Thorp [11] proposed a continuum-modeling method for disturbance propagation.A non-uniform frame-structure model was proposed in [12] by considering the anisotropy of the power grid.The voltage phase angle propagation equation is as follows:

where ji is the distributed inertia, ω0 is the frequency of the system, Vi is the bus voltage, bi is the line susceptance,Δφ(x, t) is the voltage phase-angle variation, and Δ2 is the Laplace operator.The active power and frequency obtained from continuous modeling can be expressed as follows:

where p(x, t) is the active-power injection, ω(x, t) is the speed, and f (x, t) is the frequency.The substitution of Eq.(2)into Eq.(1) yields the following.

Equations (1)-(3) can be combined to derive the following.

Thus, the spatiotemporal-distribution model of the frequency response can be expressed as follows.

Equation (5) shows that the frequency response after a power-grid disturbance has the same formation as the propagation of the electromechanical wave; it satisfies the space-time hyperbolic partial differential equation,and the propagation speed satisfies the electromechanical disturbance wave velocity described in previous studies[1-2].Equation (5) indicates that the system inertia, bus voltage, and line susceptance affect the frequency-response characteristics.Because the solution of the wave equation satisfies d’Alembert’s solution, the frequency-response solutions of different spatial positions are a continuous time-series [2].Therefore, the time-series characteristics of the frequency information must be considered during the frequency prediction.In addition, because the spatial location of the measurement information is included in the solution, the relationship between the measurement information at different spatial locations needs to be considered in the prediction model; this is the unique advantage of the GCN model applied in this study and described in the following sections.

1.2 Factors affecting the spatiotemporal distribution of the frequency response

After the power system is subjected to a disturbance,the unbalanced power is distributed according to the electrical distance of each bus, which causes the frequency at each generator bus to differ.Subsequently, the voltage phase-angle difference between the buses changes,affecting the variation in the unbalanced power flow in the system until the frequency at each bus attains the value of the synchronous frequency.Before the system attains synchronization, the relative deviation between the speeds of different generators causes differences in the frequency of different nodes, as described by Anderson [10].The distribution of the unbalanced power and generator-motion equation are as follows:

where PLΔ(0+)is the disturbance power, PkΔ is the active power allocated to the generator bus, Ksku is the synchronous torque coefficient between bus k and disturbance position u, and Uk and Eu are the terminal voltages of the k-th and u-th buses,respectively.Moreover, Hk, Dk, and δk are the inertia time constant, damping coefficient, and power angle of the k-th generator, respectively.The inertial time constant is related to the generator inertia.Plk is the load power equivalent to the terminal bus, and Pek and Pmk are the active power and mechanical power of the k-th generator, respectively.

The frequency-response characteristics can be divided into three stages, as shown in (6).In the first stage, the unbalanced active power is distributed according to the synchronous torque coefficient, thereby generating the initial rate of change of frequency.The synchronous torque coefficient is related to the electrical distance and system voltage level.In the second stage, each generator responds to the unbalanced power as per the rotor motion equation.If the governor crosses the dead zone, the governor and rotor act simultaneously until the frequency reaches the lowest point.The third stage comprises the process from the lowest frequency point to frequency recovery.The inertial response is still the dominant factor; therefore,for the frequency-response prediction, parameters Uk,Eu, Hk, Dk, δk, Plk, Pek, and Pmk affect the frequency curve.Among these parameters, mechanical power cannot be obtained by using the measurement devices.However, the variation in the mechanical power is related to the governor parameters, which are not considered in the hyperbolic spatiotemporal-distribution model given in Eq.(5).In addition, the transfer function of the governor in the actual system cannot be accurately obtained owing to the governor type and controller parameters.Research has shown that the regulating ability of the governor is constrained by the spinning reserve capacity of the unit [25].Therefore, the mechanical power can be converted into system reserve capacities, which can be used by operators as an input of the frequency- prediction model.

1.3 Framework of the proposed online-rolling prediction method of disturbance frequency

As mentioned previously, the set of frequency-response data after a disturbance has spatiotemporal characteristics,that is, at the determined bus positions, the frequencyresponse characteristics have time-series properties.Moreover, for the same timestamp, the frequency-response characteristics of different spatial positions have spatial attributes.In addition, for online applications, the method for the prediction of frequency-response characteristics must have the ability to roll along the time axis, thereby providing a technical guarantee for the coordination of spatially distributed frequency-modulation resources after disturbances occur.Therefore, this paper presents a prediction method based on the GCN-LSTM spatiotemporal network for use after a disturbance.The framework of the proposed method is presented in Fig.1.

The proposed method comprises the following five steps.

1) Feature construction and normalization

The spatiotemporal features must be constructed and normalized as the input of the prediction model.

2) Spatial feature extraction

The electrical network model is embedded into the GCN as a graph structure, and the network parameters are converted into an adjacency matrix in the IGCN, thereby extracting the spatial features.

3) Temporal feature extraction

The output of the IGCN is used as the input of the LSTM network, and the frequency-response feature is extracted in the temporal dimension to solve the gradientdisappearance problem of the recurrent neural network(RNN).

4) Spatiotemporal-network model training

The rolling update of the measurement data window and asynchronous sequence mapping are used to train the regression prediction model, enhancing the robustness of the GCN-LSTM model.

5) Online prediction

Disturbances are detected by employing singular value decomposition, as described in [24].The features in the measurement data window are input into the trained model,and the frequency in the prediction time window is output.The newest measurement information is obtained at the next moment, and the corresponding frequency is predicted online.

2 Rolling prediction model of frequency based on the GCN-LSTM spatiotemporal network

2.1 Input features of the spatiotemporal network

As described in the analysis presented in Section 1, the input for the prediction of the frequency response are the spatiotemporal data, which are composed of various PMU measurement information and system operating parameters.The PMU measurement information includes the data on the voltage amplitude, angle, active power, and frequency.The system operating parameters include the generator inertia time constant, generator spinning reserve capacity, and damping coefficient, which do not change along the time axis.The network structure and parameters are embedded in the graph neural network; therefore, they do not act as the input features.

For an arbitrary p-th bus with PMU measurement, the input feature is Xp(t) = [Up(t), φp(t), Pp(t), fp(t), Hp, Psp, Dp],where Up(t), φp(t), Pp(t), and fp(t) represent the measurement data in the time series, which are derived from the PMU measurement data.Because the configuration buses of the PMU include generator buses, load buses, and non-generator non-load buses, for the three different types of nodes, Xp(t)corresponds to different parameters but the same formation.For example, parameters Up(t), φp(t), Pp(t), and fp(t) of the generator bus respectively represent the internal electric potential amplitude, internal electric potential angle, output active power, and generator speed measured by the PMU during a period of time; Dp is the damping coefficient.Pp(t)is the active power load.For other types of buses, only Up(t), φp(t), and fp(t) are defined.For non-generator nodes,the corresponding operating parameters, Hp, Psp, and Dp,are zero-filled.For a system in which the spatiotemporal features are measured by multiple PMUs, the input is X(p, t)= [X1(t), X2(t),…, Xp(t)].

2.2 Improved graph convolutional network

The GCN is a neural-network structure that has become popular in recent years.Compared with the traditional network model, namely the convolutional neural network,which can only be used for grid-structured data, the GCN can process data with a non-Euclidean structure and deeply explore the characteristic information in the spatial dimension.

Owing to the different topological connections and PMU configurations of power systems, the graph structure is irregular and can be regarded as a non-Euclidean structure.The GCN is used to process the electrical-network structure and parameters; thus, the spatial correlation of the frequency response can be captured.

Owing to the PMU installation cost and communication failure, data-quality problems are inevitably encountered.According to California ISO statistics, 1-17% of PMU data is bad [26], causing the measurement information in the system topology to be non-redundant.Thus, Kron reduction can be used to reduce the network to the graph structure of the available PMU measurement information, namely G = (V, E) [27].Node V is configured by the PMU, and edge E is determined based on the topological connection relationship, as illustrated in Fig.2.

Each blue node in Fig.2 represents an available PMU bus in the topology, which forms V = [V1, V2,…, Vp].E is a set of edges, and it represents the topological connection parameters between different PMUs.According to the definition of graph theory, the adjacency matrix, A∈ℝp×p,of the graph structure is a symmetric square matrix of size p, and it describes the connection relationship between the nodes and edges in G and can be expressed as follows:

If two PMUs in the graph structure are connected, the corresponding element in A is 1; otherwise, it is 0.The degree matrix of the nodes in the graph structure is defined as

Thus, the degree matrix of a node is a diagonal matrix whose diagonal elements represent the number of edges connected to the node.To extract the spatial features of the graph, the GCN must perform a convolution operation,which is simplified by mapping to the frequency domain.Given the attributes of the node itself, the convolution process can be represented by a Laplacian matrix (L) as follows:

where represents the adjacency matrix after considering the node's own attributes, I is the identity matrix, D~ is the updated degree matrix, and L is the Laplacian matrix.Therefore, the layered propagation formula of the graph convolution is

where ReLU is the linear rectification function, H(l) is the input of the hidden layer, H(l+1) is the output of the l-th hidden layer, and W(l) is the weight matrix of the l-th layer.

After the GCN performs the convolution operations in space, the measurement characteristics of the PMUs in different spatial positions can be extracted.The traditional GCN only considers the connection characteristics of the graph structure, that is, the impedance parameters of the network are ignored.Therefore, an IGCN structure is proposed by embedding the network parameters into the graph structure, as illustrated in Fig.3.

The adjacency matrix obtained after embedding the network parameters is as follows:

represents the adjacency matrix after the network parameters are embedded, and represents the matrix of network parameters.The corresponding element in matrixis the impedance between two nodes in the graph structure.Worgin and Wtopology are the parameters that the IGCN needs to learn.Therefore, the layered propagation formula of the improved graph convolution can be expressed as follows:

2.3 Long short-term memory network

The LSTM is a gated RNN that can extract features in the temporal dimension and effectively solve the gradient explosion and gradient disappearance problems encountered by traditional RNNs.An LSTM unit is shown in Fig.4.

In Fig.4, xt represents the input of the LSTM unit, that is, the output of the GCN, Xo(p, t).Moreover, Ct-1 represents the state at the previous time, Ct represents the state at the current time, ht-1 represents the output at the previous time,and ht represents the current-time output.Finally, tanh is the hyperbolic tangent activation function, and σ is the sigmoid activation function.The LSTM unit includes input, forget,and output gates, as shown in Fig.5.

Fig.5(a) presents the forget gate of the LSTM unit;this gate controls the state content of the upper layer of the forgetting cell.According to the value of ht-1 of the previous sequence and the value of xt of this sequence as the input,the sigmoid activation function is used to determine the information removed in the former state, as given in the following equation:

where Wf and bf are the weight matrix and the bias term,respectively.

Fig.5(b) presents the input gate that processes the input of the current sequence.It determines the information that needs to be updated and can be divided into two parts.In the first part, the sigmoid layer determines the new information that should be added to the current state; in the second part, the tanh function is used to generate a new vector, by applying the following equation.

The forget and input gates are used to update the state at the last moment to the current state, as given in the following equation:

where ⊗ represents the element-wise multiplication of the matrices.

The output gate determines the output based on the saved content of the state.The sigmoid activation function is used to determine the content that needs to be output, and the tanh activation function is used to calculate the content of the state at the previous moment.The two parts are then multiplied, as shown in Eq.(16).

Multiple LSTM units forming an LSTM network can be used to learn the characteristics of the temporal dimension of the input spatiotemporal data.

2.4 Spatiotemporal network model via rolling training approach

To ensure the real-time operation of the proposed model based on GCN-LSTM, a spatiotemporal network is proposed by employing rolling training, as presented in Fig.6.

The structure and training method of the proposed spatiotemporal network model are presented in Fig.6.The spatiotemporal data measured by multiple PMUs are divided into the measurement time window, DWme, and prediction time window, DWpr.For the measurement time window,the input of spatiotemporal features is used as the input of the spatiotemporal network, that is, DWme = X(p, Tme).0-Tme is the length of the measurement time window, Tpr is the length of the prediction time window, and the output information is the frequency of the bus in the prediction time window, Tme+1-Tpr.As DWme slides on the time axis, the input spatiotemporal features also slide, thereby ensuring online frequency prediction.

The minimized prediction error is used during the training process to train the regression model.Because the proposed model uses rolling training, the prediction frequency error in DWpr does not affect the prediction accuracy at the next moment.

3 Case study

The IEEE 10-machine 39-bus New-England system was adopted to verify the effectiveness of the proposed method.To avoid generality, the configuration of the PMUs was the same as that described in [28], as illustrated in Fig.7.

The system consisted of 10 generators, among which generator on 39-th bus was considered as an equivalent machine representing the entire large-scale connected system.Moreover, the system had 19 load buses and 22 spatial positions for PMU measurement.

3.1 Scenario generation

The frequency-response characteristic is mainly related to an active power disturbance.The model of the IEEE 10-machine 39-bus system was simulated by employing the Power System Simulator for Engineering (PSS/E).The scenarios were generated by considering the generation loss (GL) under different operating conditions (load levels from 51, 52 to 100%) [21].The GL was introduced at 5 s,simulation step size was 0.0167 s, and total simulation time was 30 s.The GL capacity was 40%, 60%, 80%,and 100% of the different generators, excluding the 39th generator.A total of 1800 scenarios was simulated.As per the proposed rolling prediction scheme, 30 cycles of measurement data were used for the measurement time window, and the prediction time window was set to 120 cycles.The prediction information within 1.5 s can meet the preventive control strategy described by Jin et al.[29].The sliding step of the data window was one cycle, that is, after the measurement data of a new cycle were obtained, the frequency response prediction was performed in the next data window, thereby realizing the online operation of the frequency prediction.

Based on the feature construction method described in Section 1, the corresponding features were constructed under the 1800 scenarios, and the features were normalized by using Eq.(17).

The training and validation sets were randomly selected at a ratio of 8:2.The training sets were used to train the proposed GCN-LSTM spatiotemporal network,and the validation sets were used to verify the prediction effectiveness.Rolling prediction was adopted for both the training and validation sets.

3.2 Evaluation index

To comprehensively evaluate the performance of the online frequency-prediction method after disturbances, the mean absolute error (MAE), mean absolute percentage error(MAPE), and root-mean-square error (RMSE) were selected as the evaluation indices.The MAE represents the average error between the predicted and true values, MAPE denotes the degree of deviation between the predicted and true values, and RMSE indicates the degree of dispersion of the predicted values relative to the true values and the degree of concentration of the prediction error.The definitions of these three indicators are given in the following equations:

where n is the length of the time series, yi is the accurate value of the frequency, and f(xi) is the predicted value of the frequency.

3.3 Performance of different methods

Machine-learning methods, including time convolutional neural networks (TCNs), LSTM networks, DBNs, stacked autoencoders (SAEs), SVMs, and GCNs, are widely used in time-series forecasting.These methods were all developed based on the PyTorch architecture.In the training process of the model, the Adam optimizer was used to train the network parameters.The models were run on a Windows 10 system and a server equipped with an Intel i9-9900k 3.6 GHz CPU, an NVIDIA RTX 3080 GPU, and 16 GB memory.

The network structure settings for different algorithms are as follows.The TCN had three hidden layers with [128,64, 64] filters and a kernel size of 3.The LSTM consisted of two LSTM layers with 128 neurons.The DBN had three hidden layers with 120-128-256-600-4840 neurons.The SAE had three hidden layers with 120-128-256-600-4840 neurons.For the SVM, penalty factor C, maximum error ε,and RBF kernel coefficient γ were set to 1.463, 0.1, and 2,respectively.The GCN had three GCN layers with [128, 64,64] filters and a kernel size of 3.The proposed GCN-LSTM had a GCN with [128, 64, 64] filters, a kernel size of 3, and a LSTM with 128 neurons.

For the scenario in which a 0.4 p.u.GL occurs at bus 32,the frequency prediction of different algorithms at bus 26 is depicted in Fig.8.

The blue line in Fig.8 is the curve of the original frequency simulated by using the PSS/E.The other dashed lines of different colors represent the frequency curves predicted by the different machine-learning methods.The red dashed line represents the prediction effect of the proposed GCN-LSTM spatiotemporal model.As shown in Fig.8,the proposed method learns the time-series characteristics of the frequency data at a certain spatial position, and the predicted values are the closest to the reference values.The values predicted by the TCN and LSTM are approximately identical to the reference values, although with small errors.This is because a purely sequential network structure cannot learn spatial information.

To verify the applicability of the proposed method for spatiotemporal feature extraction, for the same calculation example as earlier, the three indicators described in Section 2.2 were used to calculate the prediction errors of 22 PMUs,as shown in Fig.9.

The red, green, and blue curves in Fig.9 respectively represent the MAE, MAPE, and RMSE for different PMU positions.The figure indicates that all the prediction results of the proposed method at different spatial positions exhibit good performance, owing to the spatial feature extraction of the power-system structure of the IGCN model.The maximum MAE value is 0.947×10-3 Hz, maximum MAPE value is 1.696×10-5%, and maximum RMSE value is 1.846×10-3 Hz.Moreover, the average MAE value is 0.863×10-3 Hz, average MAPE value is 1.351×10-5%,and average RMSE value is 1.688×10-3 Hz.Therefore,the proposed method achieves good performance in both the spatial and temporal dimensions and can accurately predict the frequency curve considering the spatiotemporal distribution characteristics.

Table 1 lists the prediction accuracies of different methods on the validation sets.

As shown in Table 1, compared with the other machinelearning algorithms, the proposed method yields the lowest MAE, MAPE, and RMSE values, further illustrating the advantages of GCN-LSTM in the extraction of spatiotemporal features.

3.4 Performance under different noise conditions

Gaussian white noise was added to the PMU measurement data to verify the noise immunity of the proposed method.The method proposed by Yadav et al.[30]was adopted to determine the signal-to-noise ratio (SNR),and the SNR range was 40-70 dB.The MAE values of the different prediction algorithms under different noise levels are listed in Table 2.

Table 2 reveals that all the machine-learning algorithms are significantly robust to noise, which is an advantage of data-driven algorithms.However, compared with the other machine-learning algorithms, the proposed GCNLSTM spatiotemporal network model achieves the best performance under different noise levels; the average MAE value under different noise levels is 1.142×10-3 Hz.Thus,the proposed method can better capture the spatiotemporal correlation between the PMU measurement data.

3.5 Comparison of calculation times

All the algorithms were run on the same server, and the average calculation times of the different machine-learning methods are listed in Table 3.

Machine-learning algorithms have an advantage of substantially shortening the calculation time for prediction,making them suitable for online applications.The average calculation time of the proposed algorithm was 3.483 ms,which was the longest among those of the other algorithms.This is ascribed to the complexity of the proposed spatiotemporal network model.Nevertheless, the calculation time of 3.483 ms can still satisfy the requirement for online frequency prediction.

3.6 Verification on IEEE 118-bus system

The IEEE 118-bus system has 54 generators, among which the generator on bus 69 is the reference bus.Moreover, it has 65 spatial positions for PMU measurement,as shown in Fig.10.

The IEEE 118-bus system was subjected to the same scenarios as those presented in Section 3.1, and 1200 groups of scenes were generated for the GCN-LSTM spatiotemporal network.The prediction accuracy of the proposed method is presented in Table 4.

Table 4 indicates that for the large-scale IEEE 118-bus system, the proposed GCN-LSTM spatiotemporal network still exhibits the best performance compared with the other traditional methods.The MAE is 0.9221×10-3 Hz, MAPE is 1.52×10-5%, and RMSE is 1.6748×10-3 Hz, owing to the proposed method combining the spatiotemporal relationship.

The MAE of the proposed method in the IEEE 118-bus system under different noise levels is presented in Table 5.

Table 5 shows that at a noise level of 40 dB, the MAE of the proposed model is the largest, which is 1.3952×10-3 Hz.Furthermore, the accuracy of the prediction increases with the increase in the number of decibels; however, the change is not large.This indicates that the proposed method is robust to noise.

4 Conclusion

This study presents a frequency-response prediction method based on a GCN-LSTM spatiotemporal network to evaluate the spatiotemporal frequency distribution characteristics after a power grid is disturbed.The spatiotemporal data measured by the PMUs were used as the input for the proposed model.The IGCN and LSTM network were used to extract features in the spatial and temporal domains, respectively.Furthermore, an online frequency-prediction model based on a rolling data window is proposed.The simulation results indicated that the proposed method can accurately characterize the spatiotemporal frequency distribution and predict the frequency online after a disturbance.This method lays the foundation for an active control strategy of cooperative frequency modulation resources.

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos.51627811,51725702) and the Science and Technology Project of State Grid Corporation of Beijing (Grant No.SGBJDK00DWJS2100164).

Declaration of Competing Interest

We declare that we have no conflict of interest.