Considering that Nanjing is the fifth largest city in China and the amount of POI data is huge, this paper uses Mini Batch K-Means to cluster POI data. The Mini Batch K-Means [12] algorithm is an optimised variant of the K-Means [15] algorithm, which reduces the computation time by using a small subset of the data, and at the same time optimises the objective function, to reduce convergence time while maintaining similar results to the standard K-Means algorithm.

The algorithm flow is roughly as follows:

1) Randomly Extract Mini-Batch: a mini-batch dataset of fixed size is randomly extracted from the entire dataset.

2) Update Nearest Cluster Centre: For each extracted data point, calculate its distance from the current cluster centre and assign it to the cluster centre with the nearest distance.

3) Update Cluster Centres: For each new mini-batch data point, update the position of the cluster centre according to the cluster centre assigned to them. This is usually done using a moving average, i.e., a weighted average of the new sample points and the old cluster centres.

4) Convergence judgement: repeat the above steps until the conditions for stopping the iteration are met, e.g., the maximum number of iterations is reached, the change in the cluster centre is less than a certain threshold, etc.

Simulated Annealing is a global optimisation algorithm for searching for optimal solutions in the solution space. It simulates the principle of the solid annealing process to avoid falling into local optimal solutions by accepting the possibility of slightly worse solutions in the solution space.

The basic flow of the algorithm is as follows:

1) Select an initial solution and set an initial temperature (T), which is usually high enough to ensure that the algorithm accepts almost all solutions.

2) Randomly perturb the current solution to produce a new one. This perturbation can be small or large, depending on the temperature.

3) Calculate the cost difference (ΔE) between the new solution and the current solution, and if the new solution is better (ΔE<0), accept the new solution as the current solution.

4) If the new solution is worse (ΔE>0), accept the new solution with some probability. This probability is usually determined by the current temperature and is given by P(ΔE)=e−ΔE/kT, where k is the Boltzmann constant.

5) After each iteration or a certain number of iterations, the temperature T is gradually reduced.

6) The algorithm terminates when the temperature is low enough or a predetermined number of iterations is reached.

Since the initial clustering centre of a single Mini Batch K-Means can easily fall into the local optimum and lead to unsatisfactory clustering results, this paper uses the simulated annealing algorithm to optimise the initial clustering centre of Mini Batch K-Means, to improve the clustering effect. The simulated annealing algorithm mimics the thermodynamic principle of the physical annealing process, allowing a certain probability of a ‘backward step’ in the solution process, which helps to jump out of the local optimal solution, and ultimately find the global optimal solution.

The model in this paper uses the Silhouette Coefficient to determine the K-value for each small region of the study. The Silhouette Coefficient (SC) is a measure of the quality of the clustering results. It combines the similarity of each data point to other data points within the same cluster and to data points i within different clusters. The contour coefficient has a value range of [–1, 1], with larger values indicating better clustering results. For any data point i in the data set, a(i) denotes the average distance from the data point i to other data points in the same cluster. b(i) denotes the average distance from the data point i to all data points in the nearest cluster. The contour coefficient of a data point is defined as shown in (1): (1)s(i)=b(i)−a(i)max(a(i),b(i))

The contour coefficient for each study subregion is the average of the contour coefficients of all data points within the study subregion, as shown in (2): (2)S=1n∑i=1ns(i)

Initialise the clustering centre C={c1,c2,…,ck} for each study area based on the K-value of Mini Batch K-Means determined in the previous stage, where k is the number of clusters. Also initialise the initial temperature T0, the minimum temperature Tmin and the cooling rate α. Since Mini Batch K-Means reduces computation and accelerates convergence by using a small batch (mini-batch) of data to update the cluster centres, it is also necessary to initialise the batch size b.

Mini Batch K-Means randomly selects a small batch b of data points from the dataset and assigns the mini-batch data points to the clusters that belong to the closest cluster centres. The cluster centres are recalculated using the small batch of data points and the update formula is shown in (3) below:

(3)ci(t+1)=ci(t)+η(x−ci(t))where ci(t) is the value of the ith clustering centre at the tth iteration, x is a data point assigned to that clustering centre, and η is the learning rate.

The goal of Mini Batch K-Means is to minimise the objective function and the formula is shown in (4):

(4)J=∑i=1k∑x∈Ci‖x−ci‖2where is a data point, is the centre of the ith cluster, and is the set of data points contained in the ith cluster.

The simulated annealing is used to optimise the amount of change in the objective function, and the cost difference ΔE=Jnew−Jcurrent between the new solution and the current solution is calculated. From the cost difference ΔE, according to the Metropolis criterion, decide whether to accept the new solution or not, the formula is shown in (5):

(5)P=exp⁡(−ΔEkT)where k is Boltzmann’s constant and T is the temperature, determined from T=α⋅T, α∈(0,1). When ΔE<0, then the new solution is accepted as the current solution. When ΔE>0, then accept the new solution with probability P.

The complete steps of the fusion algorithm in this paper are shown in Fig. 2.

As shown in the figure, the fusion algorithm proposed in this paper contains three parts:

1) Initial Cluster Centre Generation: Mini Batch K-Means is used to generate the initial cluster centres.

2) Simulated annealing process:

Define the initial temperature, minimum temperature and cooling rate.

3) Final clustering: re-perform Mini Batch K-Means clustering using the optimised clustering centres.

In this experiment, an electric vehicle charging station layout optimisation model incorporating simulated annealing algorithm and Mini Batch K-Means is constructed, aiming at balancing the problem of irrational distribution of charging stations through layout optimisation, and then enhancing the utilisation rate of electric vehicle charging stations. Since the EV charging station layout problem is a maximum coverage problem, firstly, the POI data of important places in Nanjing are selected and the abnormal or incomplete data are deleted, and then the POI data are divided into 11 study subregions according to the administrative area of Nanjing, and the optimal K-value within each study subregion is determined using the contour coefficient. Next, a fusion simulated annealing algorithm and Mini Batch K-Means were used to find the clustering centres for each study subregion. Finally, the model results were compared to traditional K-Means and the results were analysed.

The experiments were written in Python in PyCharm 2023 and conducted on a computer equipped with 32 GB RAM and a 12th Gen Intel(R) Core(TM) i5-12500H processor. The POI data of Nanjing was obtained from Gaode Map API (Application Programming Interface), the vector file of the Nanjing administrative area was obtained from the “OpenStreetMap” platform, and the geographic coordinate system was WGS84 (World Geodetic System 1984).

To ensure the accuracy of the geographic location of EV charging stations obtained after clustering the POI data from a single source, the POI data are screened out of the POI data of important places by Table 1 and the abnormal error data are removed, and the processing results are shown in Fig. 3.

Different colours in the figure represent different types of POIs in Nanjing, e.g., cyan for shopping malls and blue for leisure centres in the dense Nanjing city centre in the figure.

In this paper, we first determine the K-value for each small region of the study using the Silhouette Coefficient, a metric used to assess the effectiveness of clustering, which reflects the closeness and separation of the data points in the clusters. It not only considers the degree of cohesion between a sample point and its cluster but also compares the degree of separation between a sample point and its nearest neighbour cluster. The contour coefficient has a value range of [−1, 1], and the larger the value, the better the clustering effect. Through several iterations, the contour coefficients of each region in Nanjing are obtained as shown in Fig. 4. According to the K-value corresponding to the highest value of the contour coefficient, the K-value of each study area in Nanjing is finally obtained as shown in Table 2.

The contour coefficients corresponding to different values of K in Fig. 4 are reacted as different points on the line graph, the size of the contour coefficients can be reacted as the size of the difference in the mean inter-cluster distance and the mean intra-cluster distance, when the contour coefficients are large it can be interpreted as a larger mean inter-cluster distance, a smaller mean intra-cluster distance, the sample points are closer to the other points of its cluster with good cohesion and at the same time, the sample points are separated from the other clusters with a higher degree of separation. As the value of K becomes progressively larger and the number of clusters clustered increases, the average distance between clusters decreases, leading to a decrease in the contour coefficient, which can be improved by a suitable value of K. Therefore, a sudden increase in the contour coefficient can be observed in some cases.

Based on the K-value of each region obtained from Table 2, the results are obtained by fusing Mini Batch K-Means and simulated annealing EV charging station layout optimisation algorithms as shown in Fig. 5. The red points in Fig. 5 represent the clustering centre, and the other coloured points represent the POI points belonging to different clusters respectively. Taking Xuanwu District in Fig. 5 as an example, the POI points are uniformly scattered around the clustering centre, and all POI points from the clusters belonging to the clustering centre to the current clustering centre are the shortest distance from the POI to all the clustering centres, which meets the need of the shortest distance for EV users to charge.

The model in this paper uses the Lorenz curve to assess the optimisation of the layout of electric vehicle charging stations in the current real-world environment. The Lorenz curve was introduced by American economist Max Lorenz in 1905 to graphically represent the degree of equality in the distribution of resources. Ideally, in the case of a perfectly equal distribution, resources would be distributed perfectly evenly to everyone, and the Lorenz curve would be a straight line from the origin (0,0) to (1,1).

The comparison table of the cumulative percentage of the fusion algorithm, and cumulative percentage of real charging stations are shown in Table 3, and the comparison graph of the fusion algorithm Lorenz curve, real charging station Lorenz curve and ideal Lorenz curve is shown in Fig. 6, and from the comparison in Fig. 6, it can be observed that the optimisation method of EV charging stations using the fusion of Mini Batch K-Means and simulated annealing algorithms is effective in adjusting the distribution pattern of charging stations. Not only does it smooth out the Lorenz curve, making it closer to the ideal Lorenz curve, but more importantly, it promotes the balanced distribution of charging station resources among different regions, significantly improving the problem of uneven distribution of charging station resources in the original layout. This improvement is undoubtedly of great practical significance for enhancing the fairness and efficiency of charging network services.

To compare the superiority of the proposed method, this paper evaluates the fusion clustering method using 2 datasets. Table 4 summarises the number of features, total number of samples and number of clusters for each dataset.

Comparison of the proposed fusion Mini Batch K-Means and Simulated Annealing algorithms with the standalone K-Means algorithm PCA (Principal Component Analysis) is a statistical technique that is mainly used for dimensionality reduction, data compression and feature extraction. It preserves the principal information of the data by identifying the direction with the highest variance in the dataset and projecting the high-dimensional data into a low-dimensional space. For all selected datasets, a fusion of Mini Batch K-Means and Simulated Annealing algorithms with independent K-Means algorithm obtained the PCA dimensionality reduced images as shown in Figs. 7 and 8.

Fig. 7a shows the PCA plot obtained by the K-Means algorithm. The figure also shows the cluster centres of each cluster found by the K-Means algorithm, indicated by the red forks. Fig. 7b shows the distribution of HTRU2 data between dimensions obtained by fusing Mini Batch K-Means and simulated annealing algorithms, it also shows the distribution of high dimensional data in two dimensions.

Fig. 8a shows the PCA plot obtained by the K-means algorithm. The figure also shows the cluster centres of each cluster found by the K-Means algorithm, indicated by red forks. Fig. 8b shows the distribution of the Wine data between dimensions obtained by fusing the Mini Batch K-Means and simulated annealing algorithms, and it also shows the distribution of the high-dimensional data in two dimensions.

To demonstrate the performance of the proposed fused Mini Batch K-Means and simulated annealing algorithms, the accuracy and lift of the combined Mini Batch K-Means and simulated annealing algorithms compared to the standalone K-Means algorithm are recorded in Table 5. The results show that the fused Mini Batch K-Means and simulated annealing algorithms produce more accurate results than K-Means. The possible reason for obtaining such results is that the fused Mini Batch K-Means and simulated annealing algorithm avoids early convergence and achieves a compromise between global and local searches, which improves the ability to identify nonlinear clusters.

Fig. 5 demonstrates the results of the EV charging station optimisation method based on the fusion of Mini Batch K-Means and simulated annealing algorithms for single-source POIs with superior EV charging station layouts in Nanjing. By carefully comparing and analysing Fig. 6, it can be observed that the fusion algorithm effectively improves the imbalance of the EV charging station layout in Nanjing, which makes the charging station resources more reasonably and evenly distributed in the city.

Although the experiments in this paper are based on the optimisation of charging station layout based on a wide range of POI types, it is important to recognise that in the real-world environment, there is a difference in the impact of the same POI type on the passenger flow of EV charging station construction in different areas, or different POIs in the same region. If the influence of POI attributes around EV charging stations on charging station passenger flow is further considered in future studies, new adjustment needs for the charging station layout may arise. For example, the distribution of charging stations may become denser in some areas and relatively sparse in others.

The optimisation strategy in this paper is mainly based on the type of POI in the surrounding environment of EV charging stations. However, in the actual operation process, the charging flow carrying capacity of charging stations is also an important factor that cannot be ignored. If this factor is further considered, there is a mismatch between the capacity of some charging stations at the time of design and the actual buildable capacity. This not only affects the operational efficiency of charging stations but may also hurt the charging experience of users.