In recent years, the global energy transition has accelerated significantly as many countries pursue clean and low-carbon development pathways. In China, the “dual carbon” targets have promoted a fundamental change in the way electricity is generated, delivered, and consumed. Within the transportation sector, electric vehicles (EVs) have become a powerful symbol of this evolution. Their low tailpipe emissions, high energy efficiency, and increasing consumer acceptance have led to a rapid expansion in the number of EVs on Chinese roads. The pace of EV adoption, however, has imposed new challenges on urban distribution networks. As the number of EVs continues to rise, so too does the demand for accessible and powerful charging infrastructure. The charging load introduced by these vehicles is inherently stochastic and highly concentrated in certain time windows, which can lead to severe impacts on local power grids.
The temporal coincidence of EV charging demand with existing residential or commercial peaks can push distribution transformers towards or even beyond their rated capacities. In addition, voltage violations at the end of radial feeders become more likely when high-power chargers are connected at locations far from the substation. For a long time, the standard response to these threats has been simply to increase the capacity of distribution transformers and feeders. Nevertheless, transformer upgrades are expensive, they take time to plan and construct, and they frequently lead to low loading factors during periods when the charging demand is low. A more intelligent solution is to employ energy storage as a flexible resource that can absorb energy during low-demand hours and release it during high-demand hours, thereby lowering the peak demand imposed on the grid. However, the energy storage capacity can not be specified arbitrarily; it must be coordinated with the installed transformer capacity and the daily operating schedule of the charging station. This thesis addresses the capacity configuration and operation regulation of an integrated photovoltaic (PV) plus energy storage plus charging (PV-storage-charging) station for EV applications. I developed a complete framework covering typical load extraction, distribution-network voltage-constrained capacity sizing, and a bi-level coordinated optimization strategy involving both the transformer and the battery energy storage system.
1. System Architecture of the PV-Storage-Charging Station
From an engineering point of view, integrated PV-storage-charging stations combine photovoltaic generation, battery energy storage, EV charging terminals, and grid-connected power conversion equipment. The architecture selected for a specific station determines how power flows between these units and influences the control complexity, conversion efficiency, and operating flexibility. I compare two mainstream architectures in this section, namely the AC-coupled configuration and the DC-coupled configuration.
In the AC-coupled system, the PV arrays feed into an AC bus through an inverter, the battery storage is connected through a bidirectional converter, and the EV chargers receive power from the same AC bus. The main advantage of this topology lies in its maturity and high compatibility with conventional AC distribution systems. However, the multiple conversion stages may slightly reduce the round-trip efficiency. In the DC-coupled system, the PV unit and the storage unit share a common DC bus, and charging terminals can be supplied directly from that bus through DC-DC converters. This configuration reduces the number of power conversion stages and can achieve higher energy efficiency, especially for fast-charging scenarios, but it introduces more strict requirements on control and protection.
| Comparative item | AC-coupled system | DC-coupled system |
|---|---|---|
| Number of conversion stages | Relatively high | Relatively low |
| Conversion efficiency | Lower | Higher |
| Front-end converter requirement | Not required | Required |
| Bus form | AC bus | DC bus |
| Protection complexity | Simple | Complex |
| Control complexity | Complex | Relatively simple |
The chargers used in such stations can be classified according to the output form and the power level. AC chargers are usually installed in residential communities or office parking lots because their output power is moderate. DC chargers are preferred in public fast-charging stations and highway service areas because they can provide tens or even hundreds of kilowatts directly to the EV battery, bypassing the vehicle on-board charger. Table below compares typical charger types.
| Type | Output form | Typical power | Charging speed | Application scenario |
|---|---|---|---|---|
| AC slow charging | AC | 7 kW, 11 kW, 21 kW | Slow | Residential, office |
| DC fast charging | DC | 30 kW, 60 kW, 120 kW | Medium-fast | Commercial, public station |
| DC ultra-fast charging | DC | 160 kW, 240 kW, 320 kW | Very fast | Highway service area, urban hub |
The distinction among charger types is significant because the station-level charging load outcome is strongly influenced by the mix of chargers in the station. The simultaneous activation of several high-power DC terminals in a particular time window can produce a sharp instantaneous power demand and, consequently, a peak that is much higher than the daily average. As will be shown in the rest of this thesis, such features make an optimized sizing solution necessary.
2. Improved K-Means Clustering for EV Charging Load Construction
2.1 Limitations of conventional load characterization
The charging load of EVs in a station is affected by several mechanisms, including vehicle arrival and departure times, driver behavior, calendar effects, and weather conditions. As a result, the load curves observed on different days are rarely identical. In Figure-like visual inspections performed during the data analysis, I noticed that certain days had long plateau-shaped high-load intervals while other days had narrow but extremely tall load spikes. These differences imply that a single average load curve is not sufficient to represent the diversity of operational conditions that a station will face. The integration of recorded load patterns into a limited set of typical daily scenarios is therefore necessary for realistic capacity planning.
K-means clustering is a well-known unsupervised learning approach that groups samples according to the Euclidean distance between each sample and the cluster centroids. The algorithm minimizes the sum of squared errors inside every cluster. This objective can be written as:
$$
J = \sum_{k=1}^{K} \sum_{x_i \in C_k} \lVert x_i – \mu_k \rVert^2
$$
During every iteration, each sample is assigned to the closest centroid:
$$
\lVert x_i – \mu_k \rVert^2 \le \lVert x_i – \mu_j \rVert^2, \quad j = 1, 2, \ldots, K
$$
and the centroid of class \( k \) is updated as the mean of all samples belonging to that class:
$$
\mu_k = \frac{1}{\lvert C_k \rvert} \sum_{x_i \in C_k} x_i
$$
Although K-means is simple and computationally attractive, it suffers from three main weaknesses in the context of EV load curve classification. First, random initialization of the centroids often leads to unstable results and may trap the algorithm in a local minimum. Second, the Euclidean distance metric is based only on the shape of the chronological curves, so important statistical attributes such as total charging energy, the peak load, and the peak-valley difference are ignored. Third, outliers and noisy data can pull the centroids away from the true center of a cluster.
2.2 Proposed improvements to K-means
To overcome these issues, I introduced three modifications to the standard K-means algorithm. The first modification is the use of K-means++ initialization. In K-means++, the initial centroids are positioned such that they are far away from one another. The distance from each sample to the nearest already-selected centroid is computed as:
$$
D(x_i)^{2} = \min_{\mu \in M} \lVert x_i – \mu \rVert^{2}
$$
and the next centroid is selected with a probability proportional to that squared distance:
$$
p(x_i) = \frac{D(x_i)^{2}}{\sum_{j=1}^{n} D(x_j)^{2}}
$$
Because samples located far from the current centroid set have a greater chance of being chosen, the initial centers cover the sample space much more uniformly. This reduces the dependence of the final solution on the initial random seed and increases the stability of the clustering output.
The second modification concerns the feature space. I retained the normal load curve as the primary input, but I also extracted several statistical features that are relevant to power system planning. The daily total charging energy of the \( i \)-th sample is:
$$
E_i = \sum_{t=1}^{T} P_{i,t} \Delta t
$$
The peak load and the valley load are:
$$
P_i^{\max} = \max_{1 \le t \le T} P_{i,t}, \quad P_i^{\min} = \min_{1 \le t \le T} P_{i,t}
$$
from which the peak-valley difference is obtained:
$$
D_i = P_i^{\max} – P_i^{\min}
$$
The average load and the time point at which the daily peak occurs are:
$$
\bar{P}_i = \frac{1}{T} \sum_{t=1}^{T} P_{i,t}
$$
$$
T_i^{\max} = \arg \max_{1 \le t \le T} P_{i,t}
$$
After extracting these features, min-max normalization is applied to both the raw curves and the statistical features, so that variables with significantly different magnitudes do not dominate the distance calculation. The normalized statistical features are then weighted and fused with the normalized curves to form a comprehensive clustering input matrix. The weight vector used for the five statistical features is \( w = [0.30, 0.35, 0.25, 0.05, 0.05] \), which places the strongest emphasis on peak load, total charging energy, and peak-valley difference. In the final input vector, a weight of \( \alpha = 0.6 \) is assigned to the normalized curve and a weight of \( \beta = 0.4 \) is assigned to the statistical features, since the curve itself contains the main temporal shape information while the statistical features supplement the differences that are most important for capacity sizing.
The third modification is the introduction of a data preprocessing stage. Duplicate records are removed, missing values are interpolated with Lagrange interpolation, and abnormal pulses caused by measurement errors are first detected, then smoothed with a Gaussian filter, and finally replaced. These steps ensure that the clustering algorithm operates on a clean and consistent data set.
2.3 Selection of the number of clusters
The number of clusters is selected using both the elbow method and the silhouette coefficient. The sum of squared errors for a given clustering result is:
$$
\mathrm{SSE} = \sum_{j=1}^{K} \sum_{p \in C_j} \lVert p – m_j \rVert^{2}
$$
in which \( m_j \) is the centroid of cluster \( C_j \). The silhouette coefficient for a sample \( x_i \) is:
$$
s(i) = \frac{b(i) – a(i)}{\max\{a(i), b(i)\}}
$$
where \( a(i) \) measures the mean distance to all other samples in the same cluster and \( b(i) \) is the mean distance to samples in the nearest neighboring cluster. The overall silhouette value is obtained by averaging \( s(i) \) over all samples. A larger silhouette value indicates a more compact and well-separated clustering result.
For the industrial-area dataset, the elbow curve indicates that the SSE drops sharply when \( K \) changes from 1 to 2, while the silhouette coefficient reaches its maximum value at \( K = 2 \). Similar evaluations are carried out for the commercial area, the residential area, and the highway service area. After combining the numerical indicators with engineering interpretability, the optimal number of clusters selected for each scenario is two or three.
2.4 Typical-day clustering results
I applied the developed method to load data from four representative locations: an industrial zone, a commercial district, a residential district, and a highway service area. The derived typical-day profiles are physically meaningful in each scenario. For the industrial zone, two clusters are obtained. The first cluster has a stronger load peak that appears around noon, while the second cluster has a somewhat lower but more sustained peak during the morning and midday hours. For the commercial district, the first cluster corresponds to working days and has two visible peaks, one in the morning and another in the afternoon; the second cluster corresponds to weekends and holidays, when customers tend to arrive later and generate a more concentrated load peak around the afternoon. The residential area is represented by three typical days. Their curves all have an evening peak that lasts into the night, but the absolute values differ according to different seasonal or social factors. For the highway service area, the first cluster represents normal weekdays with lower but continuous traffic flow, whilst the second cluster represents holidays when the traffic flow and, consequently, the charging demand are much higher.
To quantify the advantage of the improved algorithm, I compare it with standard K-means and with ordinary K-means++ using the industrial-area data. The comparison is based on three internal indicators, namely SSE, silhouette coefficient, and the Calinski-Harabasz index. The Calinski-Harabasz index is calculated as:
$$
\mathrm{CHI} = \frac{\operatorname{tr}(B_K) / (K-1)}{\operatorname{tr}(W_K) / (n-K)}
$$
in which \( B_K \) is the between-cluster scatter matrix and \( W_K \) is the within-cluster scatter matrix. A higher CHI value means that the clusters are more dense and more separate. The results obtained by the developed method are better than those of the two reference methods under all three metrics. The improved procedure yields a lower SSE, a higher silhouette coefficient, and a higher CHI value, confirming that the integration of statistical features improves the quality of the cluster partitions.
3. Storage Configuration with Distribution Network Voltage Constraints
3.1 Problem description and modeling of the network
After obtaining typical daily load curves, I used them as inputs to a distribution network model to evaluate the impact of the charging station on node voltages. The test network is a 7-node radial distribution system with node 1 acting as the slack node and the charging station connected to the feeder-end node 7. The per-unit branch parameters used for the network are summarized below.
| Branch No. | From bus | To bus | R (p.u.) | X (p.u.) | Base load (p.u.) |
|---|---|---|---|---|---|
| 1 | 1 | 2 | 0.010 | 0.030 | 0.30 |
| 2 | 2 | 3 | 0.012 | 0.045 | 0.25 |
| 3 | 3 | 4 | 0.010 | 0.040 | 0.30 |
| 4 | 4 | 5 | 0.011 | 0.040 | 0.30 |
| 5 | 5 | 6 | 0.012 | 0.042 | 0.25 |
| 6 | 6 | 7 | 0.013 | 0.035 | 0.10 |
For the power-flow calculation, I used the DistFlow model, which is suitable for radial distribution networks. The general power flow equations are:
$$
\sum_{k: j \to k} P_{jk}(t) = P_{ij}(t) + P_{G,j}(t) – P_{L,j}(t) – R_{ij} I_{ij}^{2}(t)
$$
$$
\sum_{k: j \to k} Q_{jk}(t) = Q_{ij}(t) + Q_{G,j}(t) – Q_{L,j}(t) – X_{ij} I_{ij}^{2}(t)
$$
$$
V_j^{2}(t) = V_i^{2}(t) – 2\left(R_{ij}P_{ij}(t) + X_{ij}Q_{ij}(t)\right) + \left(R_{ij}^{2} + X_{ij}^{2}\right) I_{ij}^{2}(t)
$$
By neglecting the higher-order loss terms, the voltage relation becomes:
$$
V_j(t) \approx V_i(t) – \frac{R_{ij}P_{ij}(t) + X_{ij}Q_{ij}(t)}{V_0}
$$
This linearized relation is sufficiently accurate for planning studies under normal operating conditions. The node voltage must also be restricted by upper and lower bounds:
$$
V_i^{\min} \le V_i(t) \le V_i^{\max}
$$
with \( V_i^{\min} = 0.93 \) p.u. and \( V_i^{\max} = 1.07 \) p.u. for the network considered in this thesis.
3.2 PV and energy storage models
The PV output is obtained from the normalized irradiation curve of a typical sunny day in the Anhui region. For a given nominal PV capacity \( P_{\rm PV,nom} \), the actual output is:
$$
P_{\rm PV}(t) = \eta_{\rm PV} P_{\rm PV,nom} \frac{G(t)}{G_{\rm STC}}
$$
in which \( G(t) \) is the actual in-plane irradiance, \( G_{\rm STC} = 1000~\text{W/m}^2 \) is the standard test irradiance, and \( \eta_{\rm PV} = 0.90 \) is the conversion efficiency. The resulting PV profile approximately follows a bell-shaped curve, rising after sunrise, peaking at noon, and falling to zero after sunset.
For the battery energy storage system, I modeled the state of charge evolution as:
$$
SOC(t+1) = SOC(t) + \frac{P_{\rm es}^{\rm ch}(t)\Delta t}{E_{\rm es}} – \frac{P_{\rm es}^{\rm dis}(t)\Delta t}{\eta_{\rm es} E_{\rm es}}
$$
where \( P_{\rm es}^{\rm ch}(t) \) and \( P_{\rm es}^{\rm dis}(t) \) represent the charging and discharging power, \( \eta_{\rm es} \) is the round-trip efficiency equal to 0.95, and \( E_{\rm es} \) is the rated capacity. The SOC is limited to the interval \( [SOC_{\min}, SOC_{\max}] = [0.10, 0.90] \), and the final SOC must be equal to the initial value to guarantee a daily cyclic operation:
$$
SOC(1) = SOC(T+1)
$$
3.3 Capacity sizing model with voltage constraints
I formulated a mixed-integer linear programming model that minimizes the total equivalent daily cost of the charging station. The objective function is:
$$
F_{\rm day} = C_{\rm inv}^{\rm day} + C_{\rm op}^{\rm day} + C_q^{\rm day}
$$
The equivalent daily storage investment cost is:
$$
C_{\rm inv}^{\rm day} = \frac{r(1+r)^{N_{\rm es}}}{365\left[(1+r)^{N_{\rm es}} – 1\right]} \left(c_E E_{\rm es} + c_P P_{\rm es}\right)
$$
The operating cost includes the grid purchase cost and the maintenance cost:
$$
C_{\rm op}^{\rm day} = \sum_{t=1}^{T} \lambda(t) P_{\rm grid}(t) \Delta t + \sum_{t=1}^{T} c_{\rm om}^{\rm es} \left(P_{\rm es}^{\rm ch}(t) + P_{\rm es}^{\rm dis}(t)\right) \Delta t
$$
The reactive power regulation cost is incorporated to avoid excessive usage of inverter reactive power for voltage support:
$$
C_{q}^{\rm day} = \sum_{t=1}^{T} \left(c_{Q,\rm es} |Q_{\rm es}(t)| + c_{Q,\rm PV} |Q_{\rm PV}(t)|\right)\Delta t
$$
The constraints include the active power balance equation:
$$
P_{\rm grid}(t) + P_{\rm PV}(t) + P_{\rm es}^{\rm dis}(t) = P_{\rm base}(t) + P_{\rm ev}(t) + P_{\rm es}^{\rm ch}(t)
$$
the charging/discharging mutual exclusion constraint:
$$
0 \le P_{\rm es}^{\rm ch}(t) \le u(t) P_{\rm es}
$$
$$
0 \le P_{\rm es}^{\rm dis}(t) \le (1-u(t)) P_{\rm es}
$$
$$
u(t) \in \{0,1\}
$$
the storage boundary constraints:
$$
E_{\rm es}^{\min} \le E_{\rm es} \le E_{\rm es}^{\max}
$$
$$
P_{\rm es}^{\min} \le P_{\rm es} \le P_{\rm es}^{\max}
$$
and the inverter capacity constraint:
$$
P_{\rm es}^{\rm ch}(t) + P_{\rm es}^{\rm dis}(t) + |Q_{\rm es}(t)| \le \alpha_{\rm es} P_{\rm es}
$$
The reactive output of the PV inverter is limited by its minimum power factor:
$$
|Q_{\rm PV}(t)| \le \tan(\arccos(pf_{\rm PV,min})) P_{\rm PV}(t)
$$
3.4 Numerical results and discussion
The relevant economic parameters are summarized in the table below.
| Parameter | Symbol | Value |
|---|---|---|
| Storage energy-specific cost | \( c_E \) | 930 yuan/kWh |
| Storage power-specific cost | \( c_P \) | 1000 yuan/kW |
| Discount rate | \( r \) | 0.08 |
| Storage lifetime | \( N_{\rm es} \) | 10 years |
| Storage O&M coefficient | \( c_{\rm om}^{\rm es} \) | 0.02 yuan/kWh |
| Lower / upper SOC | \( SOC_{\min} / SOC_{\max} \) | 0.10 / 0.90 |
| Base load power factor | \( pf_{\rm base} \) | 0.95 |
| Charging load power factor | \( pf_{\rm ev} \) | 0.98 |
| PV minimum power factor | \( pf_{\rm PV,min} \) | 0.95 |
The time-of-use electricity price uses standard peak, flat, and valley periods.
| Period type | Time interval | Price (yuan/kWh) |
|---|---|---|
| Valley | 0:00–7:00, 22:00–24:00 | 0.35 |
| Flat | 10:00–17:00 | 0.50 |
| Peak | 7:00–10:00, 17:00–22:00 | 0.88 |
Using the two industrial-area typical days presented earlier, I solved the storage sizing model with and without node voltage constraints. The results are listed in the table below.
| Scenario | Storage capacity (kWh) | Storage power (kW) | Operating cost (yuan) | Reactive cost (yuan) |
|---|---|---|---|---|
| Typical day 1 with voltage constraint | 4439 | 1915 | 19464 | 6.3 |
| Typical day 2 with voltage constraint | 4736 | 2000 | 17996 | 5.7 |
| Typical day 1 without voltage constraint | 663 | 244 | 6978 | – |
| Typical day 2 without voltage constraint | 1485 | 547 | 5309 | – |
A comparison of these cases reveals a striking difference caused by the voltage constraint. When the voltage safety limit is neglected, the optimizer selects a small storage unit, because storage is only used for peak-load shaving under the time-of-use tariff. The rated capacity is only 663 kWh in typical day 1 and 1485 kWh in typical day 2. When the voltage constraint is included, the required capacity increases by a large margin. At the feeder-end node, the storage system is expected not only to realize economic arbitrage but also to suppress the voltage sag by reducing the current drawn from the upstream network during peak periods. In the case of typical day 1, the capacity must reach 4439 kWh to keep the voltage at node 7 above the 0.93 p.u. lower limit. The storage output must also have sufficient power capability. The required power increases correspondingly to almost the upper bound.
An additional consequence is that the operating and maintenance cost increases when the voltage constraint is imposed, because the battery experiences more daily charge-discharge cycles. The voltage at node 7 remains within the permissible range for the whole 24 h horizon, while in the unconstrained case the same node voltage drops below the lower bound during several peak-load hours. This indicates that, if the planning objective only considers economical aspects, the resulting configuration may be technically infeasible.
The power dispatch under the voltage-constrained configuration follows, to a large extent, the time-of-use tariff. In particular, the battery is charged during the valley price at night, and then discharged in the morning, noon, and evening peak windows. When the voltage constraint is active, some charging actions are introduced in low-price periods even if they do not immediately reduce the peak demand, because they allow the storage to provide voltage support later in the day. The daily power curves show that the grid purchase power is effectively capped during the noon hours and the storage output fills the gap between the EV demand and the transformer limit. These observations confirm that the inclusion of voltage limits in the storage sizing formulation is essential for guaranteeing safe and reliable operation of the charging station connected to a weak radial feeder.
4. Bi-Level Coordinated Transformer and Storage Optimization
4.1 Motivation and overall framework
In the previous chapter, I assumed that the transformer capacity was known in advance and optimized only the storage configuration. In engineering practice, the transformer and the storage system should be designed together, because they are economically substitutable resources. A larger transformer can directly reduce the need for storage, while a well-sized storage system can lower the peak demand and therefore allow the installation of a smaller transformer. In this chapter, I therefore consider a bi-level optimization framework in which the upper level optimizes the transformer capacity and the storage rated parameters, while the lower level computes the optimal day-ahead operating schedule. The overall framework is summarized in Fig.-style representation: the upper level determines a candidate configuration defined as:
$$
x_{\rm up} = [S_T, E_{\rm es}, P_{\rm es}]
$$
The lower level receives that configuration and performs a complete optimal dispatch over the 24 h time horizon. The resulting optimal operating cost is returned to the upper level. The upper level uses this feedback to evaluate the fitness of each candidate solution and iteratively improves the configuration until convergence is achieved.
4.2 Upper-level configuration model
The upper-level objective is to minimize the sum of the transformer investment cost, the storage equivalent daily investment cost, and the lower-level operating cost:
$$
\min \; F_{\rm up} = C_{\rm inv}^{T} + C_{\rm inv}^{\rm day} + C_{\rm op}^{*}
$$
The transformer investment cost is described by a linear relation:
$$
C_{\rm inv}^{T} = c_T S_T
$$
The storage investment cost form is identical to that of the voltage-constrained model. The decision variables include the rated transformer capacity, the storage rated capacity \( E_{\rm es} \), and the storage rated power \( P_{\rm es} \). The upper-level constraints specify the allowable search intervals:
$$
S_T^{\min} \le S_T \le S_T^{\max}
$$
$$
E_{\rm es}^{\min} \le E_{\rm es} \le E_{\rm es}^{\max}
$$
$$
P_{\rm es}^{\min} \le P_{\rm es} \le P_{\rm es}^{\max}
$$
4.3 Lower-level dispatch model
For a given transformer and storage configuration, the lower-level model seeks the optimal operating schedule. Its objective comprises three components: the grid purchase cost, the transformer overload penalty, and the net-load fluctuation suppression cost:
$$
\min \; F_{\rm low} = C_{\rm grid}^{\rm day} + C_{\rm pen} + C_{pv}
$$
The grid purchase cost is:
$$
C_{\rm grid}^{\rm day} = \sum_{t=1}^{T} \lambda(t) P_{\rm grid}(t) \Delta t
$$
The overload penalty is proportional to the portion of the purchased power that exceeds the transformer capacity:
$$
C_{\rm pen} = \sum_{t=1}^{T} \beta P_{ov}(t) \Delta t
$$
The last term reflects the peak-valley difference of the net load, defined as:
$$
C_{pv} = \gamma \left( P_{\rm net}^{\max} – P_{\rm net}^{\min} \right)
$$
where \( P_{\rm net}(t) \) is the net load seen at the grid connection point. This term encourages the optimizer to smooth the power profile. The transformer capacity constraint is written as:
$$
P_{\rm grid}(t) \le S_T + P_{ov}(t)
$$
$$
P_{ov}(t) \ge 0
$$
All other constraints, including power balance, energy storage dynamics, SOC limits, and node voltage limits, are the same as in the voltage-constrained model of the previous chapter.
4.4 Improved sparrow search algorithm
Because the upper-level objective is non-linear and not convex, and each fitness evaluation requires an exact lower-level solution, I used a mixed solution strategy that combines an improved sparrow search algorithm for the upper level with a CPLEX solver for the lower level. The standard sparrow search algorithm mimics three types of individuals in a sparrow flock: producers, scroungers, and sentinels. The producer updates its position according to the presence of predators, while the scrounger follows the best-performing producers and searches for food nearby. The standard position updating rules can be expressed as:
$$
x_{i,j}^{t+1} =
\begin{cases}
x_{i,j}^{t} \exp\left(-\dfrac{i}{\alpha T_{\max}}\right), & R_2 < ST \\[6pt]
x_{i,j}^{t} + Q L, & R_2 \ge ST
\end{cases}
$$
for producers, and
$$
x_{i,j}^{t+1} =
\begin{cases}
Q \exp\left(\dfrac{x_{worst}^{t} – x_{i,j}^{t}}{i^2}\right), & i > N/2 \\[8pt]
x_{p}^{t+1} + |x_{i,j}^{t} – x_{p}^{t+1}| A^{+} L, & i \le N/2
\end{cases}
$$
for scroungers. However, I identify several shortcomings of the original algorithm. The random initialization may cause poor coverage; the rapid loss of population diversity in the later iterations often leads to premature convergence; and the final solution quality may be affected by local trapping.
To strengthen the search ability, I incorporated four modifications into the original sparrow search algorithm. First, the Tent chaotic map is used for population initialization, generating a uniform distribution of individuals over the solution space:
$$
z_{k+1} =
\begin{cases}
z_k / 0.5, & 0 \le z_k < 0.5 \\
(1-z_k) / 0.5, & 0.5 \le z_k \le 1
\end{cases}
$$
The chaotic values are linearly mapped into the decision-variable bounds:
$$
x_{i,j}^{0} = x_j^{\min} + z_{i,j} \left( x_j^{\max} – x_j^{\min} \right)
$$
Second, a nonlinear adaptive weight is introduced into the producer update, allowing a large step in the early iterations and a finer step in the late iterations:
$$
w(t) = w_{\min} + (w_{\max} – w_{\min}) \left(1 – \frac{t}{T_{\max}}\right)^{2}
$$
Third, Levy flight perturbations are added to the scrounger update:
$$
\mathrm{Levy}(\lambda) = \frac{\mu}{|\nu|^{1/\lambda}}
$$
The inclusion of an occasional long-distance jump helps the algorithm escape from local minima.
Fourth, an elitism strategy is used, in which the globally best individual generated so far always survives to the next generation, preventing the loss of high-quality solutions due to stochastic disturbance. After the position update and boundary correction, the worst individual in the new generation is replaced by the best individual \( X_{\rm best}^{t} \):
$$
X_{worst}^{t+1} = X_{\rm best}^{t}
$$
4.5 Solution procedure
The whole bi-level procedure is executed in the following steps.
Step 1: Input the original data, including the base load, the EV charging load profile, the PV generation curve, and the time-of-use tariff. Set the improved sparrow search algorithm parameters, such as population size, maximum iteration count, and search bounds.
Step 2: Generate the initial population with the Tent chaotic map. Each individual is a candidate combination \( [S_T, E_{\rm es}, P_{\rm es}] \).
Step 3: For every candidate solution, solve the lower-level mixed-integer linear programming model with CPLEX. The returned optimal operating cost \( C_{\rm op}^{*} \) is used as a part of the upper-level fitness.
Step 4: Evaluate the fitness of every sparrow, then update the population according to the modified producer, scrounger, and sentinel formulas.
Step 5: Apply the elitism strategy, preserving the current best solution and removing the worst individual if necessary.
Step 6: Check whether the maximum iteration number is reached. If yes, output the optimal \( [S_T, E_{\rm es}, P_{\rm es}] \) and the corresponding operating schedule; otherwise, return to Step 3.
4.6 Simulation results
The simulation uses the same 7-node network and the industrial-area typical-day data as before. The transformer parameters and penalty coefficients used in the bi-level model are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| Transformer cost coefficient | \( c_T \) | 1000 yuan/kW |
| Transformer lifetime | \( N_T \) | 15 years |
| Overload penalty coefficient | \( \beta \) | 2.5 |
| Peak-valley weight coefficient | \( \gamma \) | 0.1 |
The optimized capacities obtained for the two industrial typical days are presented below.
| Scenario | Transformer capacity (kVA or kW) | Storage capacity (kWh) | Storage power (kW) |
|---|---|---|---|
| Industrial typical day 1 | 3903 | 3340 | 1254 |
| Industrial typical day 2 | 2984 | 4610 | 1702 |
For comparison, the transformer capacities required when no storage is installed are 4819 kW for typical day 1 and 3805 kW for typical day 2. The advantage of the coordinated design is clearly visible: by adding storage with moderate capacity, the transformer rating can be reduced by approximately 19% for typical day 1 and by 22% for typical day 2. This reduction significantly lowers the fixed-cost investment of the station.
The cost breakdown is summarized in the following table.
| Scenario | Transformer daily investment (yuan) | Storage daily investment (yuan) | Operating cost (yuan) | Total daily cost (yuan) |
|---|---|---|---|---|
| Typical day 1 | 1185 | 1678 | 20745 | 23609 |
| Typical day 2 | 923 | 2306 | 18597 | 21827 |
On typical day 1, the situation is more demanding because the charging load peak is higher and more concentrated; therefore, the model selects a larger transformer and a moderate storage system. On typical day 2, the transformer capacity can be decreased while the storage capacity is increased, because the storage can then carry the load over an extended period by shifting cheap valley energy to the daytime peak. In both cases, the transformer loading profile is flattened, and the grid purchase never exceeds the rated transformer capacity. The charging power supplied by the storage closely follows the daily price structure, confirming that the lower-level model successfully balances economics and security constraints.
In the storage operation results, both typical days show that the charging operation occurs mainly in the valley period and, to some extent, during the noon PV-abundant period. Discharging is arranged in the morning and evening peak windows. For typical day 2, the SOC curve changes more gently over time, revealing that the battery is designed to provide a longer-lasting discharge plateau rather than a short pulse of high power. For typical day 1, the SOC variation is more abrupt, reflecting a stronger focus on peak shaving during the very sharp noon spike. This confirms that the configuration outcome adapts itself to the statistical characteristics of each representative day.
4.7 Algorithm comparison
To validate the solution quality and convergence performance of the improved sparrow search algorithm, I also solved the same bi-level optimization problem with the standard sparrow search algorithm, the particle swarm optimization algorithm, the grey wolf optimizer, and the whale optimization algorithm. The convergence trajectories are evaluated in terms of the fitness value, where a smaller fitness represents a better combination of the transformer and energy storage capacity.
The improved sparrow search algorithm converges faster than the other four methods in both typical-day scenarios. Its fitness decreases sharply during the initial iterations and reaches the lowest final value in a small number of iterations. The standard sparrow search algorithm is also able to converge, but its final fitness is slightly higher, proving that it has found a slightly worse configuration. The particle swarm optimizer and grey wolf optimizer demonstrate a slower convergence speed; in particular, the grey wolf optimizer requires more iterations to stabilize. The whale optimization algorithm tends to converge prematurely and is less accurate. These results confirm the advantage of integrating chaotic mapping, adaptive weights, Levy flight, and elitism into the sparrow search framework.
5. Conclusion
In this thesis, I have carried out a systematic investigation on the capacity configuration and operation regulation of an integrated PV-storage-charging station for electric vehicle charging applications. The study is structured in three complementary layers, ranging from the typical charging load reconstruction to the voltage-constrained storage sizing and then to the transformer-storage coordinated bi-level design.
In the first step, I proposed an improved K-means clustering method based on K-means++ initialization and multidimensional statistical features. Unlike the conventional K-means that exclusively compares the chronological shape of the load curves, the improved method also takes into account the daily total energy, the peak load, the peak-valley difference, the mean load, and the time of the daily peak. The proposed method was validated on charging data from industrial areas, commercial areas, residential areas, and highway service areas. The extracted typical daily profiles reproduce the physical charging behavior of each scenario and provide compact but representative inputs to the planning models.
In the second step, I established a storage capacity optimization model that incorporates the node voltage security constraints of the distribution network. The node voltage is calculated using a linearized DistFlow model, and the storage operation is optimized together with the storage sizing under a time-of-use tariff. The results demonstrate that voltage-constrained storage sizing produces much larger storage capacities than unconstrained sizing. With the proposed configuration, the feeder-end voltage remains above the lower bound during the whole day, while, in the unconstrained case, the voltage is violated during the peak charging hours. Therefore, including the voltage constraint in the design procedure is indispensable for the safe integration of large-scale EV charging stations.
In the third step, I constructed a bi-level coordinated optimization model to handle the relationship between transformer capacity and the energy storage system. The upper level minimizes the total daily equivalent cost, which includes the transformer investment cost, the storage investment cost, and the operating cost fed back from the lower level. The lower level minimizes the sum of the grid purchase cost, the overload penalty, and the net-load peak-valley cost. To solve this model, I developed an improved sparrow search algorithm that integrates the Tent chaotic map, a nonlinear adaptive weight, Levy flight perturbation, and an elitist preservation strategy. The lower level is solved with CPLEX. The numerical results show that the coordinated approach can reduce the transformer rating by about one fifth relative to the case without storage while satisfying the daily charging demand. The storage system generally operates according to the valley-charging and peak-discharging principle, lowering the operating cost as well as the pressure imposed on the transformer. The comparison with four other heuristic algorithms reveals that the improved sparrow search algorithm provides a better balance between convergence speed and solution accuracy.
In future work, I will extend the model to include uncertainties in PV generation, EV charging demand, and market prices. The effects of battery degradation and thermal characteristics could also be represented more accurately. In addition, the coordination between multiple charging stations and the upper-level distribution network will be an important direction when the penetration of EV charging load continues to grow.