2.1. Basics of state estimation

Fred Schweppe first introduced state estimation to the power system in 1970 (Schweppe and Wildes, Reference Schweppe and Wildes1970). This data processing algorithm converts available information like meter readings into an estimate of the static-state vector, which represents the magnitudes and angles of voltage at all network buses.

The mathematical relationship of the related static model is often represented as

(1) $$ \boldsymbol{z}=h\left(\boldsymbol{x}\right)+\boldsymbol{v}, $$

The fundamental mathematical model of state estimation is built upon the relationship between measured variables and state variables, where z stands for the set of network measurements, x signifies the vector of the state variables, h indicates the connection between the measured values z and the state variables x, and v is the measurement error vector, accounting for all the discrepancies or errors in the observed values. Measured variables can encompass real and reactive power flows on overhead lines/underground cables, real and reactive power injections at buses, and voltage magnitudes at specific buses. State variables, which typically include bus voltage magnitudes and phase angles, define the state of the system, however, state variables are not directly measured. Instead, they are estimates based on the measured variables and the system model. The error vector v, which can be represented as a 1d vector $ \left\{{v}_1,{v}_2,{v}_3,\dots, {v}_m\right\} $ , is assumed to consist of zero mean Gaussian noise, an assumption which may not hold in practice. The variance of the error, R, provides an indication of certainty about the measurement. Occurrences of high variances on the diagonal of the covariance matrix are indicative that the measurement is not accurate.

In the measurement-state relationship (1), x represents the unknown true state, a deterministic quantity. Since the errors v are random variables, this makes the measurements z random variables as well. As such, z is assumed to follow a Gaussian distribution with mean h(x) and covariance R. The goal of state estimation is minimizing the error, which is equal to the maximum likelihood estimate of minimizing the squared error weighted by the measurement accuracy. The solution for the weighted least squares performance index, J, is given by

(2) $$ \min J(x)={\left[z-h(x)\right]}^T{R}^{-1}\left[z-h(x)\right] $$

which is equivalent to

(3) $$ \min\;J(x)=\frac{1}{2}\sum \limits_{i=1}^m\frac{{\left\{{z}_i-{h}_i(x)\right\}}^2}{{\sigma_i}^2}, $$

where m is the total number of measurements and $ {\sigma_i}^2 $ is the measurement error variance at ith measurement.

In the state estimation model, the main assumption is that there is not a full bus and line representation of the network of interest. Figure 1 illustrates a typical 14-bus test network, featuring 5 generators and 11 loads. To demonstrate the effectiveness of a state estimator, load data can be set within a network model such as this, and a full observation can be obtained from power flow calculations using the topology and line characteristics. Censoring this ground truth data and performing state estimation to understand the network operating parameters, yields accuracy measures for the estimator in an experimental environment.

Figure 1. IEEE 14-bus test network with 5 generators connected.

Traditional state estimation methods assume that the system state remains constant during the estimation period. However, in modern distribution power systems, these assumptions often come into question, for example, integration of renewable energy sources, such as solar and wind, brings about variability and unpredictability from changing weather conditions and lack of diversity on distribution level loads. Additionally, system disturbances, which can range from unexpected equipment failures to sudden demand surges, add to the challenges. Given these complexities, there is a pressing need for a means to allow the network to be reconfigured accordingly.

2.2. Dynamic state estimation

Dynamic state estimation (DSE) emerged in response to the inadequacies of traditional state estimation methods. Unlike its predecessors, DSE acknowledges the dynamic nature of power systems, furnishing more precise estimates of the system state (Zhao et al., Reference Zhao, Gómez-Expósito, Netto, Mili, Abur, Terzija and Meliopoulos2019). It provides essential near-real-time data for proficient system operation and control. Utilizing time-series data, including voltage, current, and power flow measurements, DSE forecasts the power system’s future state based on both historical and current data. These dynamic measurements, such as phase angle difference measurements, highlight the system’s future generation-load balance and provide real-time insights into its behavior (Liu et al., Reference Liu, Singh, Zhao, Meliopoulos, Pal, Ariff, Van Cutsem, Glavic, Huang, Kamwa, Mili, Mir, Taha, Terzija and Yu2021). This prediction hinges on a dynamic system model and the presently estimated state. DSE incorporates system dynamics into the estimation by employing mathematical models of components like generators and transformers. These models account for the physical laws that dictate the operation of these components and their operational constraints.

One notable DSE method is the forecasting-aided state estimation (FASE) (Filho et al., Reference Filho, Souza and Freund2009). While it typically yields satisfactory results for smooth-evolving input vectors, it outperforms simpler estimation techniques (Zhao et al., Reference Zhao, Gómez-Expósito, Netto, Mili, Abur, Terzija and Meliopoulos2019). However, its assumption of Gaussian-distributed load data assume might not always be accurate at the distribution level. Despite this, FASE remains a valuable tool for security analysis and preventive control functions at higher voltage levels.

Most state estimation tools play a crucial role in real-time power system monitoring and control. However, its application in distribution networks is not widespread, primarily because customer loads change dynamically and are non-linear. This variability makes it challenging to obtain accurate pseudo-measurement estimates using Gaussian distributions, as commonly done at the transmission level. Typically, load values derived from standard load profiles serve as the foundation for these estimates (Manitsas et al., Reference Manitsas, Singh, Pal and Strbac2012). Yet, within the distribution system, these values are often completely unknown due to their high variability. This has led to a thorough examination of the characteristics of pseudo-measurements in the distribution network. Consequently, a number of statistical characteristics of pseudo-measurements in distribution networks have been modeled in various ways. A Gaussian mixture model, employed to estimate the accuracy of state estimation, is used in Singh et al. (Reference Singh, Pal and Jabr2010a, Reference Singh, Pal and Jabr2010b), and a time-varying variance and mean model in Angioni et al., Reference Angioni, Schlösser, Ponci and Monti2016). Furthermore, several machine learning methodologies have been deployed: an artificial neural network (ANN) model in tandem with a load profile approach is put forward in Manitsas et al. (Reference Manitsas, Singh, Pal and Strbac2012), and a probabilistic neural network (PNN) is outlined in Gerbec et al. (Reference Gerbec, Gasperic, Smon and Gubina2005).

To address this challenge, this paper proposes an innovative approach: employing transfer learning to predict these pseudo-measurements. This method expedites the training process and requires less data, making it both efficient and cost-effective. The development of this approach, which is elaborated in Section 3, shows great promise for the safe and cost-efficient operation of power systems at the distribution level.

4.1. Day ahead forecast benchmark model

Before describing the transfer learning methodology, it is essential to define the “local” learning method, which serves as a base case representing no transfer. The benchmark for the day ahead forecast in load forecasting is derived from the model by Hong et al. (Reference Hong, Wang and Willis2011). This model, grounded in multiple linear regression, serves as a foundational approach for load forecasting. The mathematical representation of this model is provided below:

(4) $$ L( load)={\displaystyle \begin{array}{l}{\beta}_0+{\beta}_1\times Trend+{\beta}_2\times Day\times Hour+{\beta}_3\times Month+{\beta}_4\times Month\times TMP+{\beta}_5\\ {}\times Month\times {TMP}^2+{\beta}_6\times Month\times {TMP}^3+{\beta}_7\times Month\times TMP+{\beta}_7\times Month\\ {}\times TMP+{\beta}_8\times Month\times {TMP}^2+{\beta}_9\times Hour\times {TMP}^3.\end{array}} $$

In the model, $ {\beta}_0,{\beta}_1,\bullet \bullet \bullet, {\beta}_9 $ are the regression coefficients, “Trend” represents the hour-by-hour trend, designated by natural numbers in ascending order. The variables “Hour”, “Day”, and “Month” correspond to the 24 hours in a day, the 7 days in a week, and the 12 months in a year. “TMP” is the local temperature (in °C). L represents data after the day following the day that all other parameters pertain to. The rationale for using this model is that it solely relies on temperature and calendar variables in load forecasting models, excluding past loads, which could be accessible due to regulatory restrictions or privacy concerns in many areas. The exclusion of past loads also serves to maintain the interpretability of the model (Wang et al., Reference Wang, Liu and Hong2016). Therefore, it is well-suited for transfer learning since it relates to weather and time (factors that do not hinder the transfer process with additional local data requirements), as these are common features in both the source and target domains. Moreover, it contains only the instantaneous load measurement (Singh et al., Reference Singh, Pal and Jabr2010a).

4.2. Updating Bayesian transfer learning method

Here, a running estimate of the forecast error at a target substation is derived to adjust the forecast model output at a source substation. Defining the observed load measurement for a substation as $ L $ and the predicted load for the same substation as $ \hat{L} $ . The real data for the source substation is then $ {L}_S $ . The real data for the source substation is then $ {L}_S $ and for the target substation is $ {L}_T $ . For a single substation, a day-ahead load forecast residual is defined as the difference between the real and predicted load. Hence, the residual $ e $ of a load $ L $ , for the transfer learning source will be

(5) $$ {e}_S={L}_S-{\hat{L}}_S. $$

Similarly, for the target substation, the residual is

(6) $$ {e}_T={L}_T-{\hat{L}}_T. $$

The relationship between $ {\hat{L}}_S $ and $ {\hat{L}}_T $ is that both utilize the same fitting model based on the source data, as defined in equation (4). However, each substation is influenced by different local weather conditions. This distinction can lead to significant errors when applying the same fitted model across different substations.

The day ahead residual of the load forecast is assumed to be drawn from a multivariate normal distribution with mean $ m $ and covariance $ E $ (Stephen et al., Reference Stephen, Telford and Galloway2020):

(7) $$ {e}_T\sim N\left(m,E\right). $$

To infer substation level values of $ {e}_T $ , a novel transfer learning model using a running Bayesian estimate of error is proposed. It assumes the prior sample is $ {e}_T $ follows a multivariate normal distribution with mean and precision (which is the inverse of the covariance) m and E and with the posteriors over given $ {e}_T $ error distribution follows a Normal–Wishart distribution with mean $ \mu $ and prior precision $ \Lambda $ .

(8) $$ p\left(\mu |\Lambda, {e}_T\right)=N\left(\mu; {\mu}_N,{\unicode{x03B8}}_N\Lambda \right), $$

(9) $$ p\left(\Lambda |{e}_T\right)=W\left(\Lambda; {\alpha}_N,{B}_N\right). $$

For a half-hour resolution, day ahead forecast, which outputs a 48-dimensional forecast vector, the priors are set with the assumption that the value of the error is initially unknown. The initial parameters are defined as follows:

1. 1. Mean prior: $ {\mu}_0=0 $ (This represents a 48-dimensional vector of zeros.)

2. 2. Prior precision of the mean: $ {\unicode{x03B8}}_0=1 $ (This scalar value is applied across all 48 dimensions.)

3. 3. Prior precision: $ {\Lambda}_0={I}_{48}\;\left(\mathrm{This}\ \mathrm{represents}\;\mathrm{a}\;48\mathrm{x}48\;\mathrm{identity}\ \mathrm{matrix}.\right) $

4. 4. Wishart distribution parameters: $ {\alpha}_0 $ and $ {B}_0 $ , $ {\alpha}_0 $ is 49 and $ {B}_0 $ is a 48 x 48 identity matrix.

Then, the posterior mean of the observation of error will become

(10) $$ {\mu}_N=\frac{\Lambda_0{\mu}_0+N\overline{e_T}}{\Lambda_N}, $$

where N is the number of observed single-day load forecast error vectors.

(11) $$ {\unicode{x03B8}}_N={\unicode{x03B8}}_0+N. $$

From equation (8), it is evident that with each instance of newly observed data, its mean denoted as $ \overline{e_T} $ , the expected error data distribution $ {\mu}_N $ , is updated. This update is based on the discrepancy between the most recent prior data and the observed data. Consequently, the value of the expected error data is refined each time new true data is received, thereby enhancing its accuracy.

The updating precision matrix will become

(12) $$ {\Lambda}_N=\left[\frac{1}{\unicode{x03B8}_N\left({\alpha}_N-1\right)}\right]{B}_N, $$

where

(13) $$ {\alpha}_N={\alpha}_0+\frac{N}{2} $$

and

(14) $$ {B}_N={B}_0+\frac{N}{2}\left[\overline{\unicode{x03B8}}+\frac{\Lambda_0}{\Lambda_N}\left(\overline{e_T}-{\mu}_0\right){\left(\overline{e_T}-{\mu}_0\right)}^T\right], $$

(15) $$ \overline{\theta}=\frac{1}{N}\sum \limits_{n=1}^N\left({e}_N-\overline{e_T}\right){\left({e}_N-\overline{e_T}\right)}^T. $$

The posterior mean of the transfer learning result based on the observed data is in equation (10) and its updating precision matrix is equation (12).

4.3. Conditional Bayesian transfer learning method

If a limited number of observations are gathered each day, subsequent predictions can be enhanced under Gaussian conditions, where both the data and errors are assumed to follow a normal distribution, characterized by a Gaussian distribution. Utilizing these observations, an improved method is proposed that employs a conditionally Gaussian distributed residual. For the joint residual distribution for a whole day, assume it follows a 48-dimensional multivariate Gaussian distribution. This distribution is represented by the matrix X, where the mean vector of the distribution is $ \mu $ and its covariance matrix is $ \Sigma $ . The multivariate Gaussian distribution can be expressed as

(16) $$ f\left(x;\mu, \Sigma \right)=\frac{1}{{\left(2\pi \right)}^{\frac{n}{2}}{\left|\Sigma \right|}^{\frac{n}{2}}}\mathit{\exp}\left[-\frac{1}{2}{\left(x-\mu \right)}^T{\Sigma}^{-1}\left(x-\mu \right)\right]. $$

This distribution may change based on specific conditions or observed data. Let $ \omega $ denote the actual observed error data, represented as $ {e}_h $ to maintain clarity in the time format with other parameters. Here h represents the time interval during which the data is observed, and $ {\mu}_h $ is its corresponding predicted value vector. Consider $ {\mu}_t $ as the mean vector and $ {\Sigma}_t $ as the variance for a different time interval t of the predicted residual on the same day. Also, consider $ {\Sigma}_{t,h} $ as the covariance between the intervals t and h. Adopting this approach offers a nuanced model that can refine residual predictions by estimating the discrepancies between observed and model-predicted values implied by the intra-day dependency structure. Formulating the conditional multivariate Gaussian with data and parameters can be expressed as (Stephen et al., Reference Stephen, Telford and Galloway2020)

(17) $$ f\left({e}_t|{e}_h=\omega \right)=N\left(\overline{\mu},\overline{\Sigma}\right), $$

(18) $$ \overline{\mu}={\mu}_t+{\Sigma}_{t,h}{\Sigma_{h,h}}^{-1}\left(\omega -{\mu}_h\right), $$

(19) $$ \overline{\Sigma}={\Sigma}_t+{\varSigma}_{t,h}{\varSigma_{h,h}}^{-1}{\varSigma}_{h,t}, $$

where $ \overline{\mu} $ represents the conditional predict mean and $ \overline{\Sigma} $ represents the conditional predict variance at time interval t. This observation serves as a critical reference point in adjusting and refining the predictive model.

4.4. Process in Bayesian transfer learning

The flowchart in Figure 2 illustrates the process of the novel Bayesian transfer learning method. It begins with initial data collection and preprocessing in data-rich areas. Subsequently, a day ahead forecast benchmark model is applied in the data-rich area data, proceeds to the generation of errors generated from inputting data from data sparse into the model fit in data-rich area, serving as priors. The online Bayesian update error model is then employed to determine the day ahead posterior error. If the data-rich is supported by real time monitoring via a sim based modern, a conditional multivariate Gaussian model is utilized to further refine the model. The final prediction is obtained by the selection and application of the transfer model, integrating this error prediction into the original prediction.

Figure 2. Transfer learning of day ahead forecast for data-sparse substations from data-rich substation models using online Bayesian estimate of forecast error distribution.

5.1. Forecast performance metrics

It is necessary to quantify the transferability of instance pairs with the same label from the source to the target domains. After testing the model, four different metrics are used to evaluate the effectiveness of the proposed transfer learning methodology against the baseline prediction methods. The four metrics (equations A, B, C, D) are used to gauge an improvement in performance (Weiss et al., Reference Weiss, Khoshgoftaar and Wang2016). At model testing, if all four error metrics show better performance when associated with the proposed transfer learning methodology, as compared to the baseline, this is taken as evidence of a positive transfer learning effect.

Mean absolute error (MAE) quantifies the average size of the error, regardless of its direction, to make sure there are no biases in the prediction model. Additionally, to assess whether the model tends to have larger errors, the root mean square error (RMSE) is used, as it gives larger errors more weight, indicating the model’s sensitivity to large discrepancies. Furthermore, since predictions occur at different scales, the mean absolute percentage error (MAPE) is also utilized. It expresses the error as a percentage of the actual value, offering an intuitive measure of prediction accuracy across various scales. Finally, R-squared (R ²) statistically represents the extent to which the predicted and actual values are correlated, offering insights into the model’s goodness of fit across its entire range of outputs beyond mere error size. If the four error metrics showed better performance compared to the baseline, which involves less error measured and a higher R ² value measured than the baseline model, which assumes the model is the same in the two areas, this is further evidence of a positive transfer learning effect. Additionally, a comparison is made between the transfer learning model and the locally trained model to determine whether transfer learning can achieve results that are nearly identical to the baseline model’s best performance.

(20) $$ \mathrm{MAE}=\frac{\sum_{k=1}^N\left|{y}_k-\hat{y_k}\right|}{N}, $$

(21) $$ \mathrm{RMSE}=\sqrt{\frac{\sum \limits_{k=1}^N{\left({y}_k-\hat{y_k}\right)}^2}{N}}, $$

(22) $$ {R}^2=1-\frac{\sum \limits_k{\left({y}_k-\hat{y_k}\right)}^2}{\sum \limits_k{\left({y}_k-\overline{y}\right)}^2}, $$

(23) $$ \mathrm{MAPE}=\frac{1}{n}\sum \limits_{t=1}^n\left|\frac{A_t-{F}_t}{A_t}\right|. $$

5.2. Substation data and transfer learning result

The source substation data used in this study originates from an 11 kV substation in a typical rural area in GB – at 30 min resolution, over 12 month period, this can be considered data rich. As illustrated in Figure 3, the main load, represented by the red line, averages around 15 kW every half-hour. Conversely, the target dataset, represented by the blue line, is recorded at an urban substation in GB, with a load averaging around 200 kW.

Figure 3. Example source and target substation daily load profiles with varying scales.

The transfer learning methodology outlined in Section 4 is now applied to determine the accuracy of predicting/forecasting load profiles in the target domain and assessed against the metric from Section 5.1. Equation (4) is used to forecast the source data 1 day ahead. In subsequent steps, equations (10) and (15) are employed to predict errors associated with transfer learning. Additionally, to validate the practicality of transfer learning, the benchmark model is also employed as a comparative measure in local learning to highlight the effectiveness of the transfer learning methodology, and localized forecasting is undertaken to demonstrate what the best-case scenario could be.

In Figure 4a, the benefits of the transfer learning approach are clearly demonstrated. Across all 10 substations, there is a significant reduction in prediction MAE error compared to the baseline, no negative transfer learning result was exhibited, decreasing the original forecast error from 200 to approximately 20 kW – almost in line with the ideal case of localized forecasting.

Figure 4. a) 10 substations MAE comparison over four prediction methods; b) MAE comparison across three prediction methods at the substation. c) RMSE comparison across three prediction methods at the substation.

Based on Figure 4b and c, with data observed transmitted via a SIM-based modem at the start of the day, the conditional transfer learning method proves to be the most accurate for all substations. It exhibits a lower MAE and RMSE compared to other methods, even surpassing the benchmark in the same area. The reason for this is that with continuous observation of the target substation, the model can update daily to reflect short-term substation performance trends relative to short-term calendar and weather condition drivers. Meanwhile, the benchmark model remains static, retaining the training data characteristics as a linear regression model. This demonstrates the advantage of transfer learning, underscoring the efficacy of integrating condition monitoring into transfer learning models. Additionally, the plots highlight variations in prediction errors across substations, suggesting that some substations may have load patterns that are inherently more challenging to forecast.

Figure 5 provides a visualization of the true versus predicted values for each prediction method. Each point in the scatter plot represents a half-hourly load data point, with its true value given by the x-coordinate and its predicted value by the y-coordinate. In the upper tail of the transfer learning comparative scatter plot, there is a noticeable deviation between the y and x values. At peak values, the predicted values are significantly lower than the actual values. However, with the assistance of a few observations, conditional transfer learning addresses this issue effectively, highlighting the advantages of this method.

Figure 5. Comparative prediction analysis for the four forecasting approaches.

Figure 6 presents the temporal evolution of transfer learning performance, underscoring its ability to capitalize on previously acquired error knowledge. The trend indicates that standard transfer learning methods generally need a period of 3 to 4 days to integrate this knowledge efficiently. On the other hand, the conditional transfer learning method demonstrates notable efficacy from the beginning. Contrary to expectations, the local training approach fails to show the predicted incremental daily enhancement.

Figure 6. MAE across days for the four distinct prediction methodologies for one substation.

6.1. Representative GB test case

An urban neighborhood network is constructed based on a subset of feeders from a GB distribution license area - it serves a residential and light commercial customer base. The network is typical of GB Distribution networks: it is fed at 33 kV, with a circuit voltage of 11 kV and the LV feeders branch off at 415 V before connecting to premises. The primary substation is designated as the Slack bus, which is used to balance the active and reactive power of the overall system. It is set as a reference point to measure angle and voltage throughout the network. This substation features a transformer line that steps down the voltage from 33 kV to 11 kV. Twenty loads are directly connected to the low-voltage bus of the transformer. Figure 7 presents the network map.

Figure 7. GB urban neighborhood area 22-bus network.

For the performance evaluation of the state estimators, the power flow result is designated as the true value or perfect information. The transfer learning value, derived from the active and reactive power of 20 low-voltage buses, is used as the pseudo-measurement for state estimation. Concurrently, the power flow values of the voltage magnitude and angles at bus 0 and 1 serve as the true measurements. These correspond to both the high and low-voltage transformer sides, which can be measured in the real world. Specifically, the slack bus 0 is set to have a voltage magnitude of 1 p.u. and an angle of 0, “p.u.” means “per unit”, representing normalized values relative to a base value. This normalization ensures consistency in representing quantities across different voltage scales. The transformers are set to be ideal, ignoring minor losses like core and winding losses. This means the bus on the low-voltage side of the transformer neither produces nor consumes power (Grainger and Stevenson, Reference Grainger and Stevenson1994). For the parameters in the state estimator, the true measurements are determined with a 0.5% standard deviation error, while a 5% standard deviation error is set for the pseudo-measurement. The state estimation voltage magnitude and angle comparisons for one low-voltage substation (bus #10), estimated over a period of 10 days, are displayed in Figures 8a, b and 9, respectively.

Figure 8. a) Comparison of state estimation voltage magnitude results with those obtained through power flow, that is, the ideal case. b) Comparison of state estimation magnitude results with those obtained through power flow across the three transfer learning methods.

Figure 9. Comparison of State Estimation Angle Results with those obtained through power flow.

Figures 8a, b and 9 display the results of 10-day-long state estimations compared to the near ideal results obtained from the power flow calculations. Figure 9a presents the voltage magnitude results from four methods. It is evident that the Hong-Baseline performs the poorest and is unable to provide meaningful estimation for the localized substation cases – demonstrating the ineffectiveness of a global forecast model solution. Figure 9b demonstrates that the other three methods closely follow the power flow results, indicating high accuracy. In Figure 10, the Hong-Baseline again performs the worst in terms of voltage angle, while the other three methods excel. These findings affirm the utility and effectiveness of transfer learning in forecasting network performance resulting from an anticipated load.

Figure 10. Voltage magnitude error distributions on GB urban local network.

Figure 10 displays the error distribution results between each method and the power flow, depicted as a violin plot. This plot represents the kernel density estimate of the data distribution, showcasing a symmetrical spread and indicating data density at different values. The inner lines within the violin plot denote the data’s quartiles, representing the 75%, 50%, and 25% quartiles. The results reveal that the Hong-Baseline has the highest error, ranging from 0.1 to 0.2. Given that the data is in per unit (p.u.), this error is substantial, equating to a real voltage error multiplied by 11,000. The remaining three methods exhibit considerably smaller errors.

Figure 11 further underscores the inferior performance of Hong-Baseline in voltage angle. However, Hong-Baseline-local primarily distributes between 0.005 to 0.015, highlighting the potential of linear regression. Both the transfer and conditional transfer methods yield similar results.

Figure 11. Voltage Angle Error distributions on GB urban local network.

6.1.1 UKGDS network result

To achieve a more general representation of actual GB distribution networks, especially in urban areas or larger networks within the UK, and to test the scalability of the algorithm for deriving more robust and generalizable conclusions, it is also crucial to test the extremes, as UKGDS is intended to be a stylized representation of extremes. The UKGDS 77-bus network was also employed for testing. The UKGDS is a compilation of power system network models representative of UK distribution networks(UKGDS, 2015). Developed by the Centre for Sustainable Electricity and Distributed Generation (SEDG), the comprehensive UKGDS comprises overhead lines and cables at 132, 33, and 11 kV voltage levels with 281 bus bars and 322 branches, all supplied by four grid supply points. This research configures a segment of the UKGDS network, specifically utilizing 77 buses and 75 lines at the 11 kV level. The network map is shown in Figure 12.

Figure 12. UKGDS 77-bus low voltage test network.

Compared to the 22-bus model used in Section 6.1, this network features a transformer that steps down from 33 kV to 11 kV. Additionally, the 75-bus model is divided into several levels to connect to the transformer’s low-voltage side. The transformer’s high-voltage side is designated as the slack bus, with a voltage of 1 p.u. The intraday predicted load is input as pseudo-measurements into the 75-bus model. In this case study, the voltage magnitude and angle on both the high and low sides of the transformer are set as the true measurements. Similar to the 22-bus model, the true measurements are determined with a standard deviation error of 0.5%, while a 5% standard deviation error is applied to the pseudo-measurements.

Figures 13 and 14 depict the outcomes of 10-day-long state estimations in comparison to the power flow within the UKGDS network. Figure 8a illustrates the voltage magnitude results derived from four distinct methods. Even within a larger network, state estimation remains effective, closely mirroring the fluctuations of the power flow. This attests to the success of the purposed transfer learning methodology. Additionally, when compared with local training, it is evident that transfer learning can achieve comparable results.

Figure 13. Comparison of state estimation voltage magnitude results with power flow in UKGDS network.

Figure 14. Comparison of state estimation voltage angle results with power flow in UKGDS network.

Figure 15 presents the error distribution using a violin plot, with the layout and definition being the same as those in Figure 10. The results are also similar; the Hong-Baseline method exhibits the highest error, while the errors in other methods are significantly lower. The Baseline model exhibits the longest tails and a larger error magnitude, implying a significant possibility of incorrect forecasts and outliers. The medians of the remaining three methods are close to zero, indicating the accuracy of the predictions. The distribution from local training resembles a Gaussian distribution, validating the effectiveness of the benchmark. Transfer learning demonstrates more substantial errors in the upper tail, suggesting the model consistently underestimates the load profile’s peak behavior. However, with the integration of additional observations, the conditional transfer learning method notably reduces the tail lengths, indicating its superior identification of peak values. Additionally, the error associated with the 75-bus model is greater compared to the 22-bus model. This indicates that the 75-bus model is less tolerant than the 22-bus model due to its increased complexity and reduced network redundancy.

Figure 15. Voltage magnitude error distributions in UKGDS network.

Figure 16 further demonstrates the results in terms of voltage angle, highlighting the inferior performance of the Hong-Baseline method. Similar to the magnitude results, the transfer learning method may yield more extreme errors, whereas the conditional transfer learning method can reduce them by observation. Consistent with the outcomes from the 22-bus model, local training demonstrates superior performance. This suggests that, in comparison to active power, reactive power exhibits greater variability across different regions, potentially owing to the diverse mix of industrial and commercial premises.

Figure 16. Voltage angle error distributions in UKGDS network.

Table 1 presents the transfer learning result for the load forecasting in four methods, illustrating that with the help of a few data monitoring assist, the conditional Bayesian transfer learning has the highest accuracy in all three metrics, while the Bayesian transfer learning also has similar performance to the result compared to the local training, highlighting the effectiveness of the transfer learning across substations.

Table 1. Performance comparison of substation day ahead active power forecast models (bold value represent best performance model)

Table 2, 3, 4 and 5 show the results of transfer learning in dynamic state estimation for voltage magnitude and voltage angle within the local network and UKGDS 77-bus network, respectively. When compared to the local network, the errors in the 77-bus network are significantly higher, which can be attributed to reduced network redundancy and increased complexity. The results indicate that local training yields the best outcomes in both categories. However, the results of transfer learning for voltage magnitude are also very good, with only a 0.05% increase in percentage error. This level of accuracy can lead network operators to make the right decisions. An incorrect voltage magnitude prediction can be a source of misinformation, potentially reducing the operators’ situational awareness. Similar to the findings in Table 3, the performance in voltage angle estimation is notably poorer than expected. Further study is required to mitigate inefficiencies in power transfer in cables and to prevent system oscillations, which could potentially lead to network instabilities (Meegahapola and Littler, Reference Meegahapola and Littler2015).

Table 2. Performance comparison of transfer learning methods in state estimation of voltage magnitude in local network (bold value represent best performance model)

Table 3. Performance comparison of transfer learning methods in state estimation of voltage angle in local network (bold value represent best performance model)

Table 4. Performance comparison of transfer learning methods in state estimation voltage magnitude in UKGDS network (bold value represent best performance model)

Table 5. Performance comparison of transfer learning methods in state estimation voltage angle in UKGDS network (bold value represent best performance model)