A power distribution network partition bus voltage prediction method based on longitudinal federated learning

Abstract

The application discloses a power distribution network partition bus voltage prediction method based on longitudinal federal learning, and belongs to the field of power distribution network partition voltage prediction. The power distribution network partition bus voltage prediction method based on longitudinal federal learning comprises the following steps: selecting partition bus voltage and partition power information as the output and input of a prediction model respectively, and forming samples; identifying the samples by using an ID-based encrypted sample alignment technology; identifying and replacing abnormal data in the aligned samples, and filling in missing data to complete sample data preprocessing; training an XGBoost model based on longitudinal federal learning by using the preprocessed sample data; and performing inference by using the trained XGBoost model to predict the partition bus voltage.

Description

Technical Field

[0001] This invention belongs to the field of distribution network zone voltage prediction, specifically involving a distribution network zone bus voltage prediction method based on vertical federated learning. Background Technology

[0002] Because the operation of active distribution networks is influenced by the natural environment and involves numerous uncertainties, such as those from photovoltaic and wind power generation, the uncertainty of active distribution networks is far greater than that of transmission networks, with random voltage fluctuations being particularly significant. Therefore, predicting the voltage of zoned busbars in advance is beneficial for improving power quality, promoting renewable energy output, and guiding operational control. As the smallest sample for operational control in the distribution network, zoned busbars, when sensed in advance, facilitate decentralized control. Different zones have different stakeholders with varying ownership. While integrating data from other zones to improve voltage prediction accuracy, privacy protection techniques are needed to safeguard the privacy of different stakeholders. Voltage control is mainly divided into centralized control, decentralized control, and local control. Decentralized control combines the advantages of centralized and local control, balancing control efficiency and difficulty. Furthermore, zoned voltage control aligns with the concept of local reactive power balancing; therefore, establishing a distribution network zoned busbar voltage prediction method based on vertical federated learning has strong theoretical and practical basis.

[0003] Currently, extensive research has been conducted both domestically and internationally on voltage prediction for distribution network sub-zones, but a universal prediction system and method have not yet been established. The main challenges in voltage prediction are threefold: 1) Blind spots exist in low-voltage distribution networks, making it difficult for distribution system operators to obtain global information; 2) Numerous distributed power sources cause power flow variations, increasing the difficulty of prediction; 3) Privacy protection between sub-zones belonging to different entities increases the difficulty of data sharing. While domestic and international scholars have offered solutions to the first two challenges, none have considered privacy protection. To accurately predict sub-zone bus voltage, the influence of nearby sub-zones needs to be comprehensively considered. Although sub-zones are physically connected to the active distribution network, their operators are different, meaning their ownership is different. If a centralized (non-privacy) prediction method is still used, privacy leaks due to the direct transmission of raw data will inevitably occur. Summary of the Invention

[0004] To address the shortcomings of existing technologies, the present invention aims to provide a method for predicting the voltage of distribution network partition buses based on vertical federated learning.

[0005] The objective of this invention can be achieved through the following technical solutions:

[0006] A method for predicting the voltage of a distribution network zone bus based on longitudinal federated learning includes the following steps:

[0007] The zone bus voltage and zone power information are selected as the output and input of the prediction model, respectively, and samples are formed.

[0008] The samples are identified using an ID-based encrypted sample alignment technique;

[0009] Abnormal data in the aligned samples are identified and replaced, and missing data is filled in to complete the preprocessing of the sample data.

[0010] The XGBoost model based on longitudinal federated learning was trained using the preprocessed sample data.

[0011] The trained XGBoost model is used for inference to predict the voltage of the partition bus.

[0012] Furthermore, the abnormal data is processed using the 3σ criterion, where σ is the standard deviation. If the absolute error between the measured value and the mean in the sample data is greater than 3σ, the measured value is discarded.

[0013] Furthermore, the missing data processing employs cubic spline interpolation; the missing data filling step includes: arranging the active power, reactive power, and voltage data of the grid connection points of the partition bus in chronological order, locating the missing data points, and using the data before and after the missing position, filling the data at the corresponding missing time points through a cubic spline interpolation fitting function to obtain the time series data after missing data filling.

[0014] Furthermore, the feature is that the difference formula is:

[0015] D(t miss )=f θ (t miss (1)

[0016] In the formula, t miss For the time points where load data is missing, the number f θ (x) is the cubic spline interpolation fitting function.

[0017] Furthermore, the feature is that the model training steps include:

[0018] S41, the active party sends the first-order and second-order encryption gradients corresponding to the samples in the current split node to the passive party;

[0019] S42, the passive side finds all possible splitting thresholds based on the sorted feature values, sends the multiplicative gradient to the active side, and records the splitting rules locally.

[0020] S43, the active party decrypts the multiplication gradient using its local private key and calculates the split gain; at the same time, the active party calculates the possible split gains based on the first and second order of the plaintext, compares all the split gains, and finds the largest split gain.

[0021] S44, move to the selected child node, loop from S41 to S44 until the maximum depth of the tree model is reached or the classification gain is ≤0;

[0022] S45 uses RMSE and MAE to comprehensively evaluate the model's performance, and determines the optimal hyperparameters of the model by comparing RMSE and MAE based on cross-validation and grid search.

[0023] Furthermore, the formula for calculating the splitting gain is:

[0024]

[0025] In the formula, the subscript I L I R Representing the left node set and the right node set respectively, I L ∈{I L I R};o i and s i γ and λ represent the first and second gradients, respectively; both γ and λ are regularization coefficients.

[0026] Furthermore, the active party employs the Paillier encryption algorithm to achieve the additive property between ciphertexts. The Paillier encryption algorithm is shown in the following equation:

[0027] [[u]]·[[v]]=[[u+v]] (5)

[0028] In the formula, u and v represent plaintext, respectively; [[u]] and [[v]] represent ciphertext, respectively.

[0029] Furthermore, during the reasoning process, for any split node in the tree, the initiator only needs to send a request to the corresponding party based on the recorded participant ID; the participant determines whether to classify it as a left child node or a right child node based on local features and gain threshold; the above steps are repeated until a leaf node is reached.

[0030] A distribution network zone bus voltage prediction system based on vertical federated learning includes:

[0031] Sample generation module: Selects the zone bus voltage and zone power information as the output and input of the prediction model, respectively, and forms samples;

[0032] Sample alignment module: Uses ID-based encrypted sample alignment technology to identify the sample;

[0033] Data preprocessing module: Identifies and replaces outlier data in the aligned samples and fills in missing data to complete the preprocessing of the sample data;

[0034] Model training module: Uses preprocessed sample data to train the XGBoost model based on longitudinal federated learning;

[0035] And, the inference prediction module: uses the trained XGBoost model to perform inference to predict the voltage of the partition bus.

[0036] A computer storage medium storing a readable program that executes the above method when the program is run.

[0037] The beneficial effects of this invention are:

[0038] 1. This invention constructs a partitioned bus voltage prediction framework. To ensure privacy in partitions, the centralized multi-task problem is transformed into a federated multi-task problem. Furthermore, to ensure the model is lossless and to reduce training costs, the federated multi-task problem is transformed into multiple federated single-task problems.

[0039] 2. This invention proposes an Extreme Gradient Boosting Tree (XGBoost) algorithm based on vertical federation for joint modeling of multiple partitions. Compared with the traditional centralized XGBoost algorithm, the performance of the proposed method is demonstrated in terms of privacy, losslessness and scalability.

[0040] 3. Through comparative analysis of gain and voltage sensitivity, this invention verifies that the proposed XGBoost method based on vertical federation has good interpretability.

[0041] 4. The distribution network zone bus voltage prediction method based on vertical federated learning in this invention provides data support for voltage control and power quality improvement. Attached Figure Description

[0042] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0043] Figure 1 This is a flowchart of the method of the present invention;

[0044] Figure 2 This is the IEEE 9-node distribution network topology diagram used in this embodiment of the invention;

[0045] Figure 3 This is a ranking diagram of gain and sensitivity in an embodiment of the present invention;

[0046] Figure 4 This is a structural diagram of the 100th decision tree from the perspective of partition 1 in this embodiment of the invention;

[0047] Figure 5 This is a scatter plot of the actual voltage value and the predicted voltage value in an embodiment of the present invention; Detailed Implementation

[0048] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0049] like Figure 1 As shown, a method for predicting the voltage of a distribution network zone bus based on longitudinal federated learning includes the following steps:

[0050] S1, select the zone bus voltage and zone power information as the output and input of the prediction model respectively, and form samples;

[0051] Based on the prediction target, the zone bus voltage is selected as the model output. The model input selection should consider the actual structure of the current active distribution network: First, the low-voltage distribution network still has many monitoring blind spots, leading to data gaps. Second, the power information of each zone belongs to different entities, and the distribution network operator cannot grasp all the power information. If all nodes in a zone are selected as model input, the input sample for the entire model will be extremely large and difficult to integrate. Clearly, the topology of a zone is more stable than the topology of all nodes. Based on these factors, zone power information is selected as the input to the prediction model, forming a universal data input template. Based on this, a federated mechanism for cooperation with neighboring zones is proposed. The selection of the passive party can be determined based on the k-th order neighbor of the active party.

[0052] S2 uses an ID-based encrypted sample alignment technique to align the model input and output samples;

[0053] Sample alignment is necessary before vertical federated learning. The sample content of this invention consists of the power information of each partition and its corresponding partition bus voltage. The sample ID is a timestamp, such as 2022.08.21.31.46.30. Encrypted sample alignment techniques are quite mature, such as Privacy-Preserving Set Intersection (PSI). ID-based encrypted sample alignment is used to identify model inputs and outputs, ensuring that non-overlapping samples from each party are not exposed. In the following steps, it is assumed that encrypted sample alignment has already been completed.

[0054] S3, Data Preprocessing

[0055] Human and natural factors can both lead to erroneous and missing data, which, when used as model input, will naturally affect the model's accuracy. To ensure the validity, usability, and completeness of the data, outlier data needs to be identified and replaced, and missing data needs to be imputed before modeling. Outlier data is handled using the Laida criterion (3σ criterion), and missing data is handled using cubic spline interpolation. It is worth noting that the above interpolation only applies to cases where a single element is missing from the sample sequence; the entire sample sequence being missing at a given time is not within the scope of this discussion. Furthermore, the prediction model used is the XGBoost algorithm, which uses a binary tree as a classifier, therefore, normalization of the input data is not required.

[0056] The 3σ criterion is based on the premise that the data follows a normal distribution, where σ is the standard deviation. This criterion removes outliers or noise from the training sample set. If the absolute error of a measurement in a partition of sample data from the mean is greater than 3σ, then that measurement is bad data and should be removed.

[0057] The steps for filling in missing data include: arranging the active power, reactive power, and voltage data of the grid-connected points of the partitioned bus in chronological order; locating the missing data points; and using data before and after the missing points, fitting the function f through cubic spline interpolation. θ (x), the data at the corresponding missing time points are filled to obtain the time series data after missing data filling; the interpolation formula can be expressed as:

[0058] D(t miss )=f θ (t miss (1)

[0059] In the formula, t miss f represents the time point where load data is missing. θ (x) is the cubic spline interpolation fitting function.

[0060] S4 uses the data preprocessed in S3 to train the XGBoost model based on longitudinal federated learning.

[0061] XGBoost is an enhancement-based ensemble learning algorithm consisting of a set of base learners stacked together. It typically uses Classification and Regression Trees (CART) to handle regression problems. By fitting the residuals of the previous tree, a new tree is iteratively generated, giving the tree model higher accuracy and stronger generalization ability.

[0062] There are some differences between the XGBoost algorithm based on vertical federated learning and the centralized XGBoost algorithm. The main execution steps of vertical federated XGBoost are performed on the active side. The passive side plays the role of having the computation rules but not the computation of the label data. The main process of model training is as follows:

[0063] Assuming XGBoost is composed of K stacked trees, its objective function consists of a loss function and a regularization term, as shown in the following equation:

[0064]

[0065] in, f is the predicted value. k (x i ) represents the output of the k-th regression tree, T is the number of leaf nodes, w is the leaf node score, and γ and λ are regularization coefficients.

[0066] A second-order Taylor expansion is used to minimize the overall objective function. To learn the construction of the k-th regression tree, the objective function is expressed as follows:

[0067]

[0068] Among them, I j Let o be the sample set at the j-th leaf node. i and s i These are the first and second gradients of the (k-1) tree, respectively;

[0069] For a deterministic tree structure, the optimal leaf node score and the corresponding optimal loss function can be derived from the above equation; however, enumerating all possible tree models is impractical. For the current node, a greedy algorithm is used to enumerate all feature separation results to obtain the segment with the maximum gain, as shown in the following equation:

[0070]

[0071] In the formula, the subscript I L I R Representing the left node set and the right node set respectively, I L ∈{I L I R};o i and s i γ and λ represent the first and second gradients, respectively; both γ and λ are regularization coefficients.

[0072] The active party uses the Paillier encryption algorithm to achieve the additive property between ciphertexts. The Paillier encryption algorithm is shown in the following formula:

[0073] [[u]]·[[v]]=[[u+v]] (5)

[0074] In the formula, u and v represent plaintext, respectively; [[u]] and [[v]] represent ciphertext, respectively.

[0075] The steps of model training include:

[0076] S41, the active party sends the first-order and second-order encryption gradients corresponding to the samples in the current split node to the passive party;

[0077] S42, the passive side finds all possible splitting thresholds based on the sorted feature values, sends the multiplicative gradient to the active side, and records the splitting rules (features and thresholds) locally.

[0078] S43, the active party decrypts the multiplication gradient according to the local private key and calculates the split gain according to equation (4); at the same time, the active party calculates the possible split gains according to the first and second order of the plaintext, compares all the split gains, and finds the largest split gain.

[0079] S44, move to the selected child node, loop from S41 to S44 until the maximum depth of the tree model is reached or the classification gain is ≤0;

[0080] S45, the root mean square error (RMSE) and mean absolute error (MAE) are used to comprehensively evaluate the performance of the model; and the optimal hyperparameters of the model are determined by comparing RMSE and MAE based on cross-validation and grid search; the smaller the RMSE, the better the model performance; the trend of MAE is consistent with the trend of RMSE; RMSE and MAE are shown in Equation (6) and Equation (7), respectively.

[0081]

[0082]

[0083] S5 uses a trained XGBoost model based on vertical federation for inference to predict the partition bus voltage.

[0084] For a trained model, the active party records the participant ID and index corresponding to the maximum gain at each split node. It should be noted that a party can be either active or passive. During inference, for any split node in the tree, the active party only needs to send a request to the corresponding party based on the recorded participant ID. The participant determines whether to classify the node as a left or right child based on local features and the gain threshold. This process is repeated until a leaf node is reached.

[0085] All trees in the XGBoost model follow the reasoning process described above.

[0086] The above steps pertain to the prediction process for the bus voltage of a single zone. For the prediction of the bus voltage of other zones, the same steps are followed, only the role of the participant is changed from passive to active.

[0087] Example

[0088] To validate the effectiveness of the proposed framework, a case study was conducted on a modified IEEE 9-node system. All experiments were performed on a laptop equipped with an AMD R7-5800H CPU, 3.3GHz, and 16GB of RAM. The centralized XGBoost model was implemented in Python 3.7.0, and the federated XGBoost model was implemented in FATE 1.8.0.

[0089] The modified configuration of the IEEE 9-node system is as follows: Figure 2 As shown. Four zones are connected to the system via buses 4, 6, and 8, respectively, with zones 1 and 2 connected to bus 4. An IEEE 9-bus system is used, and modeling data is generated using the Newton-Raphson method. Power information between the zones and the distribution network is obtained by randomly varying the power output of distributed generators and loads within each zone. i and Q i These represent the active power and reactive power of partition i, respectively.

[0090] The specific steps are as follows:

[0091] S1, power information (P) of different partitions i and Q i The voltage of the partitioned bus is used as the input to the XGBoost model, and the voltage of the partitioned bus is used as the output of the XGBoost model. A voltage prediction model is built using 10,000 samples, which are divided into training and test sets in a 7:3 ratio; the size of the training data and the test data are 7,000 and 3,000 respectively.

[0092] S2 employs an ID-based encrypted sample alignment technique to identify identical samples and ensures that non-overlapping samples from each party are not exposed. The sample ID is a timestamp, such as 2022.08.21.31.46.30. Subsequent steps assume that encrypted sample alignment has already been completed.

[0093] S3. For actual power grid operation data, outliers and missing values ​​exist. Outliers and missing values ​​can be processed according to the 3σ criterion and cubic spline interpolation, respectively. In this embodiment, the data is sample data generated using the Newton-Raphson method and does not contain outliers or missing values; therefore, this step is omitted. Furthermore, the prediction model used is the XGBoost algorithm, which uses a binary tree as a classifier; therefore, normalization of the input data is not required.

[0094] S4. Training was performed on the selected sample data. To verify the lossless and privacy-preserving nature of the designed partitioned bus voltage prediction method based on longitudinal federated learning, five scenarios were analyzed and compared. The following analysis focuses on partition 1. Partition 1 uses both centralized and federated XGBoost algorithms for voltage prediction. The specific feature selections for different scenarios are summarized in Table 1.

[0095] The optimal parameters for the centralized model (scenarios 1-3) were determined using grid search and 5-fold cross-validation. Taking scenario 3 as an example, the hyperparameter settings for the XGBoost algorithm are shown in Table 2. The specific parameters for scenarios 4 and 5 are the same as those for scenarios 2 and 3, respectively. Furthermore, Paillier encryption was used as the encryption method for federated learning.

[0096] Table 1 Feature selection for different scenarios

[0097]

[0098] Table 2 shows the optimal hyperparameters of the XGBoost algorithm in scenario 3.

[0099]

[0100]

[0101] The order of average gain and voltage sensitivity is as follows: Figure 3 As shown. Gain represents the degree of dependence on each feature during model training. High-gain features are more important for model training. It can be found that the reactive power of the partition is dominant and is the most important indicator. The closer the electrical distance to bus 4, the higher the average gain. In addition, the voltage sensitivity of bus 4 is calculated and ranked in descending order as a reference. The average gain ranking basically overlaps with the voltage sensitivity ranking, indicating that the well-trained model conforms to the circuit principle. The inconsistencies are due to the randomness of sample sampling and feature sampling to avoid overfitting.

[0102] From the perspective of partition 1, the 100th decision tree in Federated XGBoost is as follows: Figure 4 As shown, the power information of partitions 1 and 2 is used for three sample segmentations, while the power information of partitions 3 and 4 is used for two sample segmentations. This phenomenon is largely consistent with the trends in sensitivity and gain. From the perspective of partition 1, it only possesses its own specific characteristics and corresponding segmentation thresholds; the segmentation information of other partitions is unknown. The active party (the proxy server of partition 1) is only responsible for encrypting and sending the first and second gradients of the samples on the separation nodes to the passive party (the proxy clients of other partitions).

[0103] S5, based on the vertical federated XGBoost inference process, obtained the test set sample prediction values ​​for scenarios 1 to 5 respectively.

[0104] Figure 5 The prediction accuracy of the test set was demonstrated in different scenarios. Figure 5 (a) in the table represents the prediction results of the centralized method. Figure 5 (b) in the table shows the prediction results of the federated method. Each test set has 3000 samples. Figure 5 As shown in (a), the prediction result for Scene 3 is closer to the diagonal, indicating that Scene 3 performs better than Scenes 1 and 2. Furthermore, the scatter distributions of Scenes 3 and 5 tend to be consistent, such as... Figure 5 As shown in (b) above, this demonstrates that the performance of centralized XGBoost and federated XGBoost is comparable.

[0105] This invention establishes a framework for predicting bus voltage in distribution network zones and proposes a method for predicting bus voltage in distribution network zones based on longitudinal federated learning. This method achieves prediction performance comparable to centralized models while protecting the data privacy of each zone. This method provides sufficient data support for improving zoned voltage power and power quality.

[0106] In the description of this specification, references to terms such as "an embodiment," "example," "specific example," etc., indicate that a specific feature, structure, material, or characteristic described in connection with that embodiment or example is included in at least one embodiment or example of the invention. In this specification, illustrative expressions of the above terms do not necessarily refer to the same embodiment or example. Furthermore, the specific features, structures, materials, or characteristics described may be combined in any suitable manner in one or more embodiments or examples.

[0107] The foregoing has shown and described the basic principles, main features, and advantages of the present invention. Those skilled in the art should understand that the present invention is not limited to the above embodiments. The embodiments and descriptions in the specification are merely illustrative of the principles of the invention. Various changes and modifications can be made to the invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the claimed invention.

Claims

1. A power distribution network partition bus voltage prediction method based on longitudinal federated learning, characterized in that, Includes the following steps: The zone bus voltage and zone power information are selected as the output and input of the prediction model, respectively, and samples are formed. The samples are identified using an ID-based encrypted sample alignment technique; Abnormal data in the aligned samples are identified and replaced, and missing data is filled in to complete the preprocessing of the sample data. The XGBoost model based on longitudinal federated learning was trained using the preprocessed sample data. The trained XGBoost model is used for inference to predict the voltage of the partition bus. The steps of model training include: S41, the active party sends the first-order and second-order encryption gradients corresponding to the samples in the current split node to the passive party; S42, the passive side finds all possible splitting thresholds based on the sorted feature values, sends the multiplicative gradient to the active side, and records the splitting rules locally. S43, the active party decrypts the multiplication gradient using its local private key and calculates the split gain; at the same time, the active party calculates the possible split gains based on the first and second order of the plaintext, compares all the split gains, and finds the largest split gain. S44, move to the selected child node, loop from S41 to S44 until the maximum depth of the tree model is reached or the classification gain is ≤0; S45. The performance of the model is comprehensively evaluated using RMSE and MAE, and the optimal hyperparameters of the model are determined by comparing RMSE and MAE based on cross-validation and grid search. The formula for calculating the splitting gain is: （4） where the subscript I L , I R represent the left and right node sets, respectively, ; o i and s i denote the first and second order gradients, respectively; and γ and λ are positive regularization coefficients.

2. The power distribution network partition bus voltage prediction method based on longitudinal federated learning according to claim 1, characterized in that, The abnormal data is processed using the 3σ criterion, where σ is the standard deviation. If the absolute error between a measurement value and the mean in the sample data is greater than 3σ, the measurement value is removed.

3. The power distribution network partition bus voltage prediction method based on longitudinal federated learning according to claim 1, characterized in that, The missing data processing adopts cubic spline interpolation; the missing data filling steps include: arranging the active power, reactive power and voltage data of the grid connection points of the partition bus in chronological order, locating the missing data points, and using the data before and after the missing position, filling the data at the corresponding missing time points through a cubic spline interpolation fitting function to obtain the time series data after missing data filling.

4. The power distribution network partition bus voltage prediction method based on longitudinal federated learning according to claim 3, characterized in that, The formula for the difference is: （1） In the formula, t miss is the missing time point of load data, and (x) is a cubic spline interpolation fitting function.

5. The power distribution network partitioned bus voltage prediction method based on longitudinal federated learning according to claim 1, characterized in that, The active party uses the Paillier encryption algorithm to achieve the addition property between ciphertexts. The Paillier encryption algorithm is shown in the following formula: （5） wherein, u and v respectively represent plaintexts; and respectively represent ciphertexts.

6. The power distribution network partitioned bus voltage prediction method based on longitudinal federated learning according to claim 1, characterized in that, During the reasoning process, for any split node in the tree, the initiator only needs to send a request to the corresponding party based on the recorded participant ID; the participant decides whether to classify it as a left child node or a right child node based on local features and gain threshold; repeat the above steps until a leaf node is reached.

7. A power distribution network zoned bus voltage prediction system based on federated learning in longitudinal direction, performing the method of any one of claims 1-6, characterized in that, include: Sample generation module: Selects the zone bus voltage and zone power information as the output and input of the prediction model, respectively, and forms samples; Sample alignment module: Uses ID-based encrypted sample alignment technology to identify the sample; Data preprocessing module: Identifies and replaces outlier data in the aligned samples and fills in missing data to complete the preprocessing of the sample data; Model training module: Uses preprocessed sample data to train the XGBoost model based on longitudinal federated learning; And, an inference prediction module: using the trained XGBoost model to make inferences to predict the partition bus voltage.

8. A computer storage medium storing a readable program, characterized in that, When the program is run, the method of any of claims 1-6 is performed.

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