Within the federated learning framework, each participating node conducts local training and uploads model parameters to the server. The server aggregates parameters and updates the model, distributing the global model to each node. These procedures are iterative and continue until convergence is achieved. The most commonly used model aggregation algorithm for federated learning is FedAvg. The total number of nodes is K , the kth node contributes data samples as S k , then the number of samples of this node is S k , and the total number of data samples is S = ∑ k = 1 K S k .

As in Eq. 6, the global objective function that requires optimization is min ⁡ F ω , where ω ∈ R d : F ω = 1 S ∑ i = 1 S f i ω (6) where f i ω = l x i , y i ; ω represents the prediction loss for the sample x i , y i under the model parameter ω , then Eq. 6 can be rewritten as Eqs 7, 8: F k v k = 1 S k ∑ x i , y i ∈ S k f i v k (7) F ω = ∑ k = 1 K S k S F k v k (8) where F k v k is the loss function of the kth node, while F ω is the overall federated loss function.

Each iteration process of FedAvg comprises a local update and a global update. During every iteration, the following process is undertaken:

Initially, the latest global model from the server is received as the local model for the current round during the local update of each node. Then, e iterations of the local model are carried out, updating the model parameter ω k for the kth node in each iteration as in Eq. 9: v k t = v k t − η ∇ F k v k t (9) where, η is the learning rate of the local update at each node.

Secondly, once the local update is executed, the model parameters are uploaded to the server. Subsequently, a server-side global aggregation update is implemented using the following aggregation formula as in Eq. 10: ω t + 1 = ∑ k = 1 K S k S v k t (10)

The asynchronous adaptive federated learning algorithm proposed in this paper makes improvements based on the FedAvg algorithm. In the process of global model aggregation, an asynchronous mechanism and adaptive adjustment of node model weights based on the determination of time obsolescence are introduced. In the process of updating local node models, personalized learning and adaptive step length adjustment are introduced to ensure asynchronous real-time and regional personalization, and to improve the prediction accuracy while enhancing the generalization ability of the global model. The specific implementation process of the asynchronous adaptive federal learning algorithm is shown as follow.

First, the overall objective function is determined based on Eq. 10. In the asynchronous adaptive federal learning algorithm, the overall objective function is Eq. 11: min ∑ k = 1 K S k S F k v k (11)

Let λ k = S k S denote the weight of each node. Then add model personalization constraints as in Eq. 12: s . t .   v k − ω 2 ≤ T k (12) where T k , k = 1,2 , … , K is the model personalization tolerance threshold, which is used to reflect the difference between the global model and the local model to achieve the personalized learning of the node’s local model. The above optimization objective function with constraints can be rewritten by using the Lagrange equation as in Eq. 13: L v k , ω , μ k = ∑ k = 1 K λ k F k v k + u k 2 v k − ω 2 − T k (13)

The partial derivative of this objective function can be obtained: ∇ v k L = λ k ∇ F k v k + u k v k − ω (14) ∇ ω L = ∑ k = 1 K u k v k − ω (15) ∇ u k L = 1 2 v k − ω 2 − T k (16) where k = 1,2 , … , K .

Secondly, in the aggregation of global models, the asynchronous characteristics of multi-node collaboration should be considered. Asynchronous training allows each node to update model parameters independently with the server without worrying about the calculation pace of other nodes. In the case of asynchronous training, the model parameter update on the server side is carried out asynchronously, that is, there is no need to wait for the progress of other nodes that are being calculated. Considering the different time required for each node to complete the local task, this paper assumes that global aggregation begins whenever a single edge node completes uploading model parameters. Therefore, the variable φ k t ∈ 0,1 is introduced to indicate the completion of node updates as in Eq. 17: φ k t = 1 , k t h   n o d e   c o m p l e t e s   u p d a t e   a t   t 0 , o t h e r s (17)

The above variables satisfy the constraints as in Eq. 18: ∑ k = 1 K φ k t = 1 (18)

According to Eq. 15, the global model aggregation is given by: ω t 1 = ω t 0 + ∑ k = 1 K φ k t 1 β k t k , t 1 v k t 1 − ω t 0 (19) where t 1 is the generation moment of the global model of the current generation, which is also the moment when the local model of the kth node completes this update, t 0 is the generation moment of the global model of the previous generation, t k is the generation moment of the global model adopted by the local model of the kth node when it performs the current update, β k is the weight adaptive adjustment function based on the determination of the temporal staleness, which is determined according to the before and after of the local update of the kth node. If this model is outdated, the current node model is given a smaller weight, the specific form can be defined as: β k t k , t 1 = β m ⁡ cos π t 1 − t k 2 t 1 + ε (20) where β m is the maximum learning weight, ε is a constant value close to zero. And when t k is similar to t 1 , β k takes a value close to β m . ∂ β k ∂ t k = β m ⁡ sin π t 1 − t k 2 t 1 + ε π 2 t 1 + ε > 0 (21) ∂ β k ∂ t 1 = − β m ⁡ sin π t 1 − t k 2 t 1 + ε π t k + ε 2 t 1 + ε 2 < 0 (22) ∂ 2 β k ∂ t k 2 = − β m ⁡ cos π t 1 − t k 2 t 1 + ε π 2 t 1 + ε 2 < 0 (23) ∂ 2 β k ∂ t 1 2 = − β m ⁡ cos π t 1 − t k 2 t 1 + ε π t k + ε 2 t 1 + ε 2 2 + 2 β m ⁡ sin π t 1 − t k 2 t 1 + ε π t k + ε 2 t 1 + ε 3 (24)

The values of β k vary with t k and t 1 as shown in Figure 2. According to Eqs 21–24, β k increases with the increase of t k , and the magnitude of the increase decreases gradually, indicating that when the local model is updated with the newer global model, the weight of the updated node model should be increased accordingly. Also β k decreases with the increase of t 1 . When t k is smaller, ∂ 2 β k ∂ t 1 2 > 0 , which means β k is convex. When t k is larger, ∂ 2 β k ∂ t 1 2 < 0 , which means β k is concave. It indicates that the global model parameters obtained at the early stage of prediction have large weights only at the early stage, and their weights decrease rapidly in the later stage of prediction.

Convergence analysis is carried out on the aggregation of the global model. When T → ∞ , the algorithm is considered convergent if the statistic R T = ∑ t = 1 T F ω t − F ω * meets the predetermined convergence criteria that R T / T → 0 , which means ω → ⁡ arg ⁡ min ∑ t = 1 T F ω t ≜ ω * , ω converges to ω * to minimize the objective function.

The convergence analysis is shown as in Eq. 25. R T = ∑ t = 1 T F ω t − F ω * ≤ ∑ t = 1 T < ω t − v , ω t − ω * > ≤ ∑ t = 1 T 1 2 β t ω t − ω * 2 − ω t ′ − ω * 2 + β t 2 ω t − v 2 (25a)

According to (22), it can be inferred that β t decreases monotonically. Using the assumption that ω t − ω * 2 ≤ D 2 and local node personalization constraints ω t − v 2 ≤ G 2 , the above equation can be simplified and transformed as follows: R T ≤ 1 2 β 1 D 2 + D 2 ∑ t = 2 T 1 2 β t − 1 2 β t − 1 + G 2 2 β T = D 2 2 β T + G 2 2 β T ≤ D 2 2 β min + G 2 β max 2 (25b)

Consequently, R T has an upper bound and mean value of the statistic R T / T → 0 when T → ∞ , ensuring the convergence of the global model aggregation.

Furthermore, for the update process of the edge model, according to Eqs 14, 16, the update process can be expressed as follow: v k t 1 = v k t k − η 1 ∇ F k v k t k × λ k ∇ F k v k t k + u k v k t k − ω t k (26) u k t 1 = u k t k − η 2 v k t k − ω t k 2 × v k t k − ω t k 2 − T k (27) where η 1 and η 2 are the adaptive adjustment function of the step size, which can be expressed as η z = ⁡ max η min , ⁡ min η max , p q * z (28) where η min is the minimum value of the step, η max is the maximum value of the step, p > 1 and q > 0 , z is the independent variable of the adaptive function, and η z is a non-decreasing function taking values in the interval η min , η max . As in Eq. 28, it is indicated that when the gradient of the loss function is small and the local model is close to the global model, the step size is adaptively reduced to prevent non-convergence in the learning of the local model. Compared with the fixed step size, the adaptive step size can adjust the amplitude of the parameter changing direction along the gradient according to the training stage, and improve the convergence rate while ensuring the convergence accuracy.

Convergence analysis of the update process of the edge model is performed. Since the parameters of the above process are solved according to a variant of gradient descent, a proof is required: E v k t 1 − ω * 2 ≤ l v k t k − ω * (29)

As in Eq. 29, the distance between the parameters of the current iteration and the optimal parameters ω * is less than the distance between the parameters of the previous iteration and the optimal parameters ω * , which means that the upper bound of the distance between the parameters of the current iteration and the optimal parameters is decreasing.

There are some assumptions on the functions F 1 , … , F K for the algorithm as the premise of convergence analysis:

1) As in Eq. 30, F 1 , … , F K are all L-smooth: for all v and ω ,

Meanwhile, we measure the system heterogeneity by Γ as in Eq. 34, which denotes the difference between the optimal value of the global objective function and the weighted local objective function. Γ converges to 0 when the data are distributed iid independently and identically. Γ = F * − ∑ k λ k F k * (34)

Let v ′ = v k t 1 and v = v k t k , then E v ′ − ω * 2 = E v − ω * − η 1 ∇ F k v × λ k ∇ F k v + u k v − ω 2 = E ( v − ω * 2 + η 1 2 λ k ∇ F k v + u k v − ω 2 − 2 η 1 × < v − ω * , λ k ∇ F k v + u k v − ω ∇ F k v > (35) in which − 2 η 1 < v − ω * , λ k ∇ F k v + u k v − ω ∇ F k v > = − 2 η 1 < v − ω + ω − ω * , λ k ∇ F k v + u k v − ω ∇ F k v > ≤ − 2 η 1 u k T k + μ 2 v − ω * 2 + 6 Γ L η 1 2 − λ k T k (36) λ k ∇ F k v + u k v − ω 2 = λ k 2 ∇ F k v 2 + u k 2 v − ω 2 + 2 λ k μ k ∇ F k v v − ω ≤ λ k 2 G 2 + u k 2 T k + 2 λ k u k μ 2 T k (37)

In summary, the original equation can be converted as follow: E v ′ − ω * 2 ≤ 1 − η 1 μ E v − ω * 2 + η 1 2 B − ( λ k + 2 η 1 u k ) T k (38) B = 6 Γ L + λ k 2 G 2 + u k 2 T k + λ k u k μ T k (39)

As in Eqs 35–39, this convergence analysis is formally similar to that of FedAvg. However the equation in this paper subtracts one more positive term, which show that the iterative process is convergent.

Considering the demand for distributed power time-series data prediction and the disadvantages of centralized processing, the approach of applying blockchain technology in this case is proposed in this paper in order to decentralize prediction, leveraging its global data ledger sharing and non-tamperability, which can further ensure the trust, security, transparency and traceability of data in the power industry.

In this paper, the federated learning global model parameter updating process described in the previous subsection is implemented by placing it on a blockchain. The blockchain guarantees the operation of a shared and trusted distributed ledger on a peer-to-peer network, which in the prediction scenario of this paper is the shared set of global model parameters blocks. The global model used for prediction is placed on the blockchain for updating so that it is not under the control of any single node, but each regional node has equal rights to validate and access the global model in the blockchain in such a way that it not only avoids a single point of failure, but also protects against data attacks in terms of security.

The workflow of the communication layer of the decentralized federated learning framework used in this paper broadly includes the processes of block generation, consensus verification and global maintenance. The decentralized implementation of the prediction framework includes the following:

1) Setting permissions for each regional participant, only participants with permissions for electricity data prediction can jointly train the federation model, and new nodes can join and receive the current federation model only if they meet the permission conditions when accessing.

The local update is completed to form a data packet: T X 0 = t k , t 1 , v k t 1 , … (40)

3) Each edge node collects the latest packet T X 0 broadcast in the network, completes the update of the global model by block mining, calculates the global model aggregation ω t 1 , and packages the data before and after the aggregation.

The global aggregation is completed to form a data packet: T X 1 = φ k t 1 , ω t 1 , … (41)

4) The node that completes block mining packages the above packets to form a block, and broadcasts it to the network of nodes waiting for verification. The block consists of a block body and a block header, where the block body contains the data before and after aggregation, and the block header contains the predecessor hash and the root hash, and the specific process of block generation includes:

Local updating and global aggregation of data packets together form the block body: B o d y N = T X = T X 0 , T X 1 , … (42)

Hash algorithm achieves the irreversible mapping from plaintext to ciphertext, this paper uses SHA256 to calculate the hash value. The successor hash is the hash value of the previous block and the root hash is generated based on the hash value of the packet T X : P r e v H a s h = H a s h H e a d e r N − 1 (43) R o o t H a s h = M e r k l e T r e e T X (44)

The successor hash and the root hash together form the block header of the block: H e a d e r N = P r e v H a s h , R o o t H a s h (45)

The block header and the block body together form the block: B l o c k N = H e a d e r N , B o d y N (46)

5) After receiving blocks, all nodes in the network collaboratively verify updates to the global model. The fastest node to complete verification within each region sends its signature to the other nodes for verification. Once all nodes have reached a consensus, the block is added to the global model block set. The most recent global model, along with signatures and timestamps, form new blocks and attach to the blockchain. Finally, each node receives a copy of the most recent global model block to synchronise the model. Additionally, all nodes collectively store and maintain the block set of the global model.

At the same time, as shown in Eqs 47, 48, the node that successfully mines a new block can obtain the block reward B l o c k R e w a r d k , which in turn encourages the nodes to jointly maintain the update of the global model. Meanwhile, the node under the proof-of-work mechanism can obtain the reward f S k proportional to the size of the sample data, which in turn encourages the participants to contribute the data and the model. After joining the prediction federation, participants can obtain additional financial benefits by earning rewards while obtaining accurate prediction models. R e w a r d k = B l o c k R e w a r d k + f S k (47) B l o c k R e w a r d k = B a s e R e w a r d , successful mining 0 , others (48)

In summary, the pseudo-code of the decentralized asynchronous adaptive federated learning algorithmic process is shown in Table 1. Combining the advantages of federated learning and blockchain, the framework achieves privacy protection for local multi-source data while removing the dependence on a centralized central server. The approach improves training reliability and reduces communication costs, while ensuring the scalability of the edge node network and reducing the risk of single point of failure.

The proposed method is implemented on a high performance server equipped with Intel Xeon Gold 6136 and NVIDIA TITAN XP. The Python libraries, including TensorFlow and Keras are used to build our model. The experimental data was selected as a dataset of historical data from 1st January 2015 to 31st December 2018 for five regions of the Spanish electricity market, which consists of time-series data on electricity price, generation, load, weather, and date information, sampled every 1 h, where the price of electricity is used as a label for the prediction. Missing values and outliers in the series are filled and replaced using linear interpolation, normalization is used before prediction training to prevent the effect of different magnitudes, and inverse normalization is used for prediction output to restore the relevance of the output data.

First of all, to show the overall implementation effect of the proposed framework in this paper, the data set is divided into training set, validation set and test set for prediction training, the RMSE of the training set and validation set in the iterative process is recorded, and finally the prediction of the test set is performed and the result is compared with the real value, and the partial tariff prediction results and RMSE iteration curves of the nodes are shown in Figure 3.

To verify the improvement of the prediction performance of the proposed framework in this paper, the distributed regional tariff prediction model and the centralized tariff prediction model are compared and analyzed, and the prediction results of the two are shown in Figure 4.

As shown in Figure 4, the predicted and actual values of the two prediction models are relatively close to each other, and the predicted value of the distributed model is slightly different from the actual value, because the samples used for training in the centralized model are the data of all nodes, while the samples used for training in each node in the distributed model are only the local data, but the distributed model strengthens the correlation between nodes by protecting the transmission of the model parameters and overcomes the phenomenon of data islanding.

The prediction performances of the distributed and centralized modes are shown in Table 2, except for the prediction accuracy, the distributed mode mostly outperforms the centralized mode. In terms of running time, since the computational tasks are assigned to each edge node in the distributed mode, and only local data is used for single-point training, the computational volume of a single computational unit is greatly reduced, and the single-point training time in the distributed mode is less than that in the centralized mode. In terms of data transmission, only the model parameters are transmitted in the distributed mode, which protects the data of each node and does not reveal the private data such as power generation information and user habits, and “53601” represents the sum of the elements of the weight matrix and bias vector of the prediction model. The centralized mode requires the transmission of all sample data, which has the risk of data leakage, where “5 × 35064 × 33” means that there are 35,064 sample data in each of the five nodes, and each sample contains 33 dimensions of data. Therefore, the distributed model can effectively reduce the amount of data transmitted in a single transmission and provide a certain level of security to protect the privacy of electricity.

Choosing a more appropriate model for distributed prediction at the edge nodes can effectively improve the overall prediction accuracy. To improve the implementation effect of the method proposed in this paper, LSTM model, CNN model and CNN-LSTM model are used at the edge nodes to predict the electricity price for the same data set. The model parameters are adjusted based on the method proposed in the related literature, and the prediction performance under the three models is compared and analyzed, as shown in Figure 5. The RMSE is calculated again after converting the model output into real size, and the results of the RMSE of each model evaluation index are shown in Table 3.

From Figures 5, 6 and Table 3, it can be seen that the overall prediction value of CNN-LSTM is closest to the real value, and the efficiency of the training iteration is also higher. The main reason is that CNN-LSTM combines the advantages of CNN and LSTM, capturing both local and global temporal features at different time scales through sliding window and loop structure, while the feature of parameter sharing of convolutional layer reduces the number of parameters of LSTM network and improves the computational efficiency. Therefore, choosing CNN-LSTM model for prediction at edge nodes can have good accuracy and efficiency.

To verify that the DAAFed algorithm proposed in this paper has better prediction performance, FedAvg, FedProx (Li et al., 2020b), FedPer (Arivazhagan et al., 2019) and DAAFed are used to predict the electricity price on the same dataset, and the prediction performance of the above algorithms are compared and analysed. FedProx adds regularization terms to constrain the similarity between the global model and the local model to prevent overfitting (Li et al., 2020a). Each node model in FedPer has its own personalization layer that preserves the characteristics of the node data (Arivazhagan et al., 2019).

The RMSE and average RMSE for each node using the above federated learning algorithms with different iteration settings are shown in Table 4. FedProx adds a regularization term to limit the similarity between the global model and the local model and prevent overfitting; and each node has its own personalization layer in each node’s model in FedPer, which is used to preserve the node data’s own characteristics. Table 4 shows that the prediction performance of these two improved algorithms is similar to that of FedAvg. The DAAFed algorithm used in this paper has a smaller error compared to the other three algorithms, because the asynchronous mechanism used in the aggregation of the global model allows the nodes to update the model based on the latest global model, and at the same time, the step size is adaptively adjusted in the updating of the node’s local model, so the model learns to the better parameter faster, and thus the overall performance is better. Therefore, the overall performance is better.

From Table 4, it can be seen that setting different local iteration epochs and global aggregation rounds has different prediction effects, and choosing the appropriate settings can fully exploit the advantages of the federated learning framework.

Figure 7 shows the iteration curves under different settings. It shows that when epochs = 10 and rounds = 10 there is better iteration efficiency than the other two settings. When epochs = 5 there is a slower decrease in the iteration curve because when the number of local iterations is small, the local model of each node does not have enough performance and cannot make full use of the local data. When epochs = 20 there is a slow decrease in the iteration curve before global aggregation, which means that a higher number of local iterations leads to a slower decrease of the iteration curve before global aggregation. This means that if the number of local iterations is high, it can lead to overfitting, which affects the ability of global model generalization, so the setting of epochs = 10 and rounds = 10 can give better federated learning results.

Based on this setting, the prediction results of a node under each of the above algorithms are shown in Figures 8–11, from which it can be seen that the predicted value of DAAFed is closer to the real value compared to several other federated learning algorithms.