As the global energy landscape shifts toward sustainability, the integration of renewable energy sources such as wind and solar has become a key strategy for reducing carbon emissions and mitigating the depletion of fossil fuels. In recent years, distributed generation (DG) has grown rapidly, and the installed capacity of wind and photovoltaic power has increased significantly. However, the inherent intermittency and randomness of these renewable sources pose significant challenges to the secure and economic operation of distribution networks. In my research, I focus on the optimal allocation of an energy storage system in a distribution network with high penetration of wind and solar power. An energy storage system can effectively smooth the fluctuations of DG, shift load peaks, and improve voltage profiles. Nevertheless, the investment cost of an energy storage system remains high, and its benefits strongly depend on the placement and capacity design. Therefore, it is crucial to establish a comprehensive optimization model that balances economic efficiency and network voltage stability, and to employ a robust algorithm for solving this high-dimensional, nonlinear problem.
1. Introduction and Background
Energy is the foundation of modern society, driving industrial production, transportation, and daily life. With the depletion of traditional fossil fuels and the increasing attention to environmental protection, renewable energy technologies such as wind and solar power have developed rapidly. According to statistics from the National Energy Administration, by the end of October 2023, the total installed capacity of renewable energy in China reached approximately 1.404 billion kW, accounting for 49.9% of the total installed capacity. Among these, wind and solar power accounted for more than 1 billion kW. The generation from renewable sources reached 2.33 trillion kWh in the first ten months of 2023, representing 31.8% of the total electricity generation. This trend demonstrates the expanding role of distributed generation in the modern power system.
However, the integration of DG into the distribution network brings not only benefits but also technical challenges. The variability and uncertainty of wind and solar power can cause voltage fluctuations, reverse power flows, and increased network losses. In this context, an energy storage system (ESS) offers an effective solution. By storing excess energy during low-demand periods and releasing it during high-demand periods, an ESS can mitigate the negative impacts of DG and improve the flexibility of the distribution network. Moreover, an energy storage system can provide ancillary services such as frequency regulation and voltage support. In particular, the distributed energy storage system (DESS), which is deployed at multiple locations within the distribution network, has attracted considerable attention because of its flexible installation and rapid response capability.
Despite these advantages, the optimal configuration of an energy storage system in distribution networks remains a challenging task. The location, capacity, and operation strategy of ESS directly affect the technical and economic performance of the network. Therefore, my study aims to develop a multi-objective optimization model for DESS allocation, considering both the annual comprehensive cost of the energy storage system and the daily voltage deviation of the distribution network. To solve this problem, I improve the NSGA-II algorithm by incorporating an opposition-based learning strategy, an adaptive crossover and mutation mechanism, and a novel elite preservation scheme. The proposed method is validated on the IEEE-33 node distribution system with integrated wind and solar generation.
2. Development Status and Classification of Energy Storage Technologies
Energy storage technologies can be classified according to the form of energy stored, including mechanical, electrochemical, electromagnetic, thermal, and chemical energy storage. Figure below shows an overview of the energy storage system classification and some typical technologies. From the perspective of application, an energy storage system can participate in the generation side, grid side, and user side. In my research, I focus on the distribution network side, where the energy storage system serves as a flexible resource to improve voltage quality and reduce costs.
Among the various types, electrochemical energy storage, especially lithium-ion batteries, has become the dominant technology for distributed applications due to its high energy density, high efficiency, long cycle life, and declining cost. Table 1 provides a comparison of several common battery types used in distribution network energy storage systems.
| Type | Energy density (Wh/kg) | Cycle life (times) | Efficiency (%) | Advantages | Disadvantages |
|---|---|---|---|---|---|
| Lead-acid | 30–50 | 500 | 80–90 | Low cost, mature | Low density, pollution |
| Lithium-ion | 120–250 | 2000–5000 | 90–95 | High density, fast response | Safety, cost |
| Sodium-sulfur | 150–240 | 4500 | 75–85 | High density, long life | High operating temperature |
| Vanadium redox flow | 15–25 | 10000 | 70–80 | Very long life, deep discharge | Low density, high cost |
| Sodium-ion | 100–150 | 2000–4000 | 85–90 | Abundant resources, low cost | Immature technology |
In my work, I select lithium-ion batteries as the storage units for the following reasons: they offer excellent cycle life, high round-trip efficiency, and modularity, which are essential for distributed installation in the distribution network. The basic parameters adopted in the simulation are summarized in Table 2.
| Parameter | Value |
|---|---|
| Unit capacity cost (RMB/kWh) | 3220 |
| Unit power cost (RMB/kW) | 1080 |
| Project lifetime (years) | 10 |
| Auxiliary equipment unit capacity cost (RMB/kWh) | 32 |
| Annual fixed O&M cost (RMB/kW) | 155 |
| Variable O&M cost (RMB/kWh) | 0.03 |
| Self-discharge rate (%/month) | 0.1 |
| Charging/discharging efficiency (%) | 90 |
| SOC range | 0.1–0.9 |
3. Operation Characteristics of Wind, Solar, and Energy Storage System
3.1 Wind Power Output Model
Wind power generation depends primarily on the wind speed. The stochastic behavior of wind speed is often described by the Weibull distribution. The probability density function of wind speed \(v\) is given by
$$
f(v)=\frac{k}{\lambda}\left(\frac{v}{\lambda}\right)^{k-1}\exp\left[-\left(\frac{v}{\lambda}\right)^k\right]
$$
where \(k\) is the shape parameter and \(\lambda\) is the scale parameter. The output power of a wind turbine can be expressed as a piecewise function of wind speed:
$$
P_w(v)=\begin{cases}
0, & v \le v_{ci} \ \text{or}\ v \ge v_{co} \\
P_r \dfrac{v-v_{ci}}{v_r-v_{ci}}, & v_{ci} < v < v_r \\
P_r, & v_r \le v < v_{co}
\end{cases}
$$
where \(P_r\) is the rated power, \(v_{ci}\), \(v_r\), and \(v_{co}\) are the cut-in, rated, and cut-out wind speeds, respectively. Figure 2 shows the typical daily wind power output curve, which is characterized by higher output during the night and lower output at noon.
In order to model the wind generation in the distribution network, I use the normalized typical daily output curve for the selected region. The wind turbine output power in per unit at time \(t\) is denoted as \(p_w(t)\).
3.2 Photovoltaic Output Model
Photovoltaic (PV) power generation depends on solar irradiance and temperature. The solar irradiance often follows a Beta distribution. The probability density function of critical solar irradiance \(s\) is
$$
f(s)=\frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)}s^{\alpha-1}(1-s)^{\beta-1}, \quad 0\le s \le 1,\ \alpha,\beta > 0
$$
The relationship between PV output power and irradiance can be simplified as
$$
P_{pv}(t) = \eta A G(t)
$$
where \(\eta\) is the conversion efficiency, \(A\) is the total area of the PV panels, and \(G(t)\) is the solar irradiance at time \(t\). The typical daily PV output curve is approximately bell-shaped, with zero output at night and the peak around noon.
3.3 Energy Storage System Model and Operation Characteristics
An energy storage system has two main operating states: charging and discharging. The state of charge (SOC) is a key parameter that represents the remaining capacity of the battery. The SOC at time \(t+1\) can be expressed by the following discrete-time model:
When charging:
$$
E(t+1) = E(t) + \eta_{ch} P_{ch}(t) \Delta t – \alpha E(t)
$$
When discharging:
$$
E(t+1) = E(t) – \frac{P_{dis}(t)}{\eta_{dis}} \Delta t – \alpha E(t)
$$
Here, \(E(t)\) is the stored energy at time \(t\); \(P_{ch}(t)\) and \(P_{dis}(t)\) are the charging and discharging powers; \(\eta_{ch}\) and \(\eta_{dis}\) are the charging and discharging efficiencies; \(\alpha\) is the self-discharge rate; and \(\Delta t\) is the time step. The SOC is defined as
$$
SOC(t) = \frac{E(t)}{E_N} \times 100\%
$$
where \(E_N\) is the rated capacity of the energy storage system. In my model, the SOC is restricted to the range [0.1, 0.9] to avoid overcharging and over-discharging. Moreover, the daily energy balance constraint is imposed to ensure that the total charging energy equals the total discharging energy over a day, which can be written as
$$
\sum_{t=1}^{T} \left[ \eta_{ch} P_{ch}(t) – \frac{P_{dis}(t)}{\eta_{dis}} \right] \Delta t = 0
$$
3.4 Load Characteristics
The daily load curve is another important input for the optimization. In a typical residential and commercial area, the load has two peaks: one around noon and one in the evening. The typical daily load curve used in my simulation is normalized and shown in Figure 4. It can be approximated by a probability density function, but for the deterministic optimization, I use the average daily load profile.
4. Impact of DG and Energy Storage System on Distribution Network
4.1 Impact on Power Losses
When DG is connected, the power flow in the distribution network becomes bidirectional, and the network losses may increase or decrease depending on the DG location and capacity. Similarly, the energy storage system can act as a load or a generator, and its influence on network losses depends on its location and power output. Based on the simplified two-node model, the incremental change in network loss caused by an energy storage system can be derived. For a typical radial feeder, the total power loss after connecting a DG and an energy storage system can be expressed as
$$
P_{loss,ESS} = \frac{R}{V^2}\left[ P_L^2 + Q_L^2 + P_{ESS}^2 + Q_{ESS}^2 – 2P_L P_{ESS} – 2Q_L Q_{ESS} \right]
$$
where \(R\) is the total line resistance, \(V\) is the nominal voltage, \(P_L\) and \(Q_L\) are the active and reactive power demands, and \(P_{ESS}\) and \(Q_{ESS}\) are the active and reactive power outputs of the energy storage system. When the term is negative, the ESS reduces network losses; when positive, it increases them. Therefore, the appropriate placement and sizing of an energy storage system are crucial.
4.2 Impact on Voltage Profile
In a radial distribution network, the voltage drop along a feeder can be approximated by
$$
\Delta U = \frac{PR + QX}{U}
$$
When DG is connected, the active power flows from the DG to the load, reducing the voltage drop and possibly causing overvoltage if the DG output is high. An energy storage system can absorb excess power during high DG output and inject power during high demand, thereby mitigating voltage rise and supporting voltage stability. For a multi-node network, the voltage at node \(m\) can be expressed as a function of the line impedances and the nodal power injections. The daily voltage deviation is defined in my objective function as
$$
f_3 = \sum_{i=1}^{N}\sum_{t=1}^{T} \left| V_{i,t} – V_{i,N} \right|
$$
where \(V_{i,t}\) is the actual voltage magnitude at node \(i\) at time \(t\), and \(V_{i,N}\) is the rated voltage (1.0 p.u.).
5. Multi-objective Optimization Algorithm
5.1 Basic Concepts of Multi-objective Optimization
A general multi-objective optimization problem can be formulated as
$$
\min \mathbf{F}(x) = \left[ f_1(x), f_2(x), \ldots, f_m(x) \right]^T
$$
subject to equality and inequality constraints. Since multiple objectives conflict with each other, there is no single solution that optimizes all objectives simultaneously. Instead, a set of Pareto-optimal solutions is obtained. A solution \(x_1\) dominates another solution \(x_2\) if \(f_i(x_1) \le f_i(x_2)\) for all \(i=1,\ldots,m\) and \(f_j(x_1) < f_j(x_2)\) for at least one \(j\). The set of all non-dominated solutions is called the Pareto optimal set, and its image in the objective space is the Pareto front.
5.2 Classical NSGA-II
Non-dominated Sorting Genetic Algorithm II (NSGA-II) is a popular evolutionary algorithm for solving multi-objective optimization problems. It employs a fast non-dominated sorting procedure, a crowding distance assignment, and an elitist selection mechanism. The basic flow can be summarized as follows:
- Initialize a population of size \(N\).
- Evaluate the objective functions for each individual.
- Perform fast non-dominated sorting and compute crowding distance.
- Apply selection, crossover, and mutation to create offspring.
- Combine the parent and offspring populations, then perform non-dominated sorting and crowding distance-based selection to create the next generation.
- Repeat until the maximum number of generations is reached.
However, for high-dimensional and highly constrained problems such as the optimal allocation of an energy storage system, the classical NSGA-II may suffer from premature convergence and insufficient population diversity. In my research, I propose several improvements to enhance its performance.
5.3 Improvements to NSGA-II
5.3.1 Opposition-based Learning for Initialization
To enhance the diversity of the initial population, I adopt the opposition-based learning (OBL) strategy. For an individual \(Y_{i,j}\) in the population, its opposition value is calculated as
$$
Y_{i,j}^{op} = a_j + b_j – Y_{i,j}
$$
where \(a_j\) and \(b_j\) are the lower and upper bounds of the \(j\)-th dimension. By evaluating the fitness of both the original and opposition-based individuals, the best \(N\) individuals are selected as the initial population. This approach helps the algorithm avoid local optima from the beginning.
5.3.2 Adaptive Crossover and Mutation
In the standard NSGA-II, the crossover and mutation probabilities are fixed, which may not be suitable for different stages of the search process. I introduce an adaptive crossover probability \(P_c\) and an adaptive mutation probability \(P_m\). The crossover probability is defined as
$$
P_c(k) = h_1 + (h_2 – h_1) \cdot \frac{k}{k_{max}}
$$
where \(h_1\) and \(h_2\) are the minimum and maximum crossover rates, \(k\) is the current generation, and \(k_{max}\) is the maximum generation. The mutation probability is defined as
$$
P_m(k) = v_1 + v_2 \cdot \frac{k}{k_{max}}
$$
where \(v_1\) and \(v_2\) are the minimum and maximum mutation rates. This adaptive mechanism allows the algorithm to maintain diverse individuals in the early stage and to enhance exploitation in the later stage.
5.3.3 Improved Elite Preservation Strategy
The original NSGA-II preserves elites in an implicit manner. In my improved version, I use a dynamic elite preservation mechanism. At the beginning of the iteration, if the number of individuals with non-dominated rank 1 is less than 50% of the population size, I select a proportion \(h_1\) of individuals from each non-dominated layer to form the next generation. In the later stage, when the non-dominated rank 1 individuals become more numerous, I use a proportion \(h_2\) (e.g., 0.7) to select individuals from all layers, thereby maintaining diversity and preventing premature convergence. The improved strategy can be expressed as
$$
P_{t+1} = \begin{cases}
Z_1 + h_1 \sum_{k=2}^{n} Z_k, & \text{if } |Z_1| < 0.5N \\
h_2 \sum_{k=1}^{n} Z_k, & \text{otherwise}
\end{cases}
$$
where \(Z_k\) is the \(k\)-th non-dominated layer and \(n\) is the total number of layers.
6. Mathematical Model for Optimal Allocation of Energy Storage System
6.1 Cost Analysis of the Energy Storage System
The annual comprehensive cost of the energy storage system consists of the initial investment cost, auxiliary equipment cost, operation and maintenance cost, and salvage value. The initial battery investment cost is annualized using a capital recovery factor \(\mu\):
$$
\mu = \frac{r(1+r)^y}{(1+r)^y – 1}
$$
where \(r\) is the discount rate (0.05) and \(y\) is the project lifetime (10 years). The total annual cost components are calculated as follows:
(1) Initial battery investment cost:
$$
C_1 = \mu \sum_{a=1}^{N_a} \left( n_p P_a + n_e E_a \right)
$$
where \(n_p\) and \(n_e\) are the unit price of power and capacity of the battery.
(2) Auxiliary equipment investment cost:
$$
C_2 = \mu \sum_{a=1}^{N_a} n_{E}^{sup} E_a
$$
where \(n_{E}^{sup}\) is the unit capacity price of auxiliary equipment.
(3) Annual operation and maintenance cost:
$$
C_3 = C_{fix} + C_{var}
$$
where \(C_{fix}\) is the fixed O&M cost proportional to the power rating, and \(C_{var}\) is the variable O&M cost proportional to the discharged energy.
(4) Salvage value:
$$
C_4 = k_s C_1
$$
with \(k_s = 0.1\) in this study.
6.2 Objective Functions
In my optimization model, I consider two objectives. The first objective \(F_1\) is the sum of the annual comprehensive cost of the energy storage system and the annual network loss cost. The network loss cost is calculated by converting the energy losses to a monetary value using the time-of-use (TOU) electricity price. The network loss cost is given by
$$
C_{loss} = \sum_{d=1}^{D}\sum_{t=1}^{T} \lambda_t \sum_{(i,j)\in B_L} \frac{r_{ij}(P_{ij,t}^2 + Q_{ij,t}^2)}{U_{i,t}^2}
$$
where \(\lambda_t\) is the TOU price at time \(t\), \(r_{ij}\) is the resistance of branch \(ij\), \(B_L\) is the set of branches, and \(U_{i,t}\) is the voltage magnitude. Thus, the first objective is
$$
F_1 = C_1 + C_2 + C_3 – C_4 + C_{loss}
$$
The second objective is the total daily voltage deviation of all nodes:
$$
F_2 = \sum_{i=1}^{N}\sum_{t=1}^{T} \left| V_{i,t} – 1.0 \right|
$$
Therefore, the multi-objective optimization model is formulated as
$$
\min \ \mathbf{F} = (F_1, F_2)
$$
6.3 Constraints
The following constraints are imposed on the system and the energy storage system:
(1) Power flow equations:
$$
P_{G,i} – P_{D,i} – \sum_{j=1}^{N} V_i V_j (G_{ij}\cos\theta_{ij} + B_{ij}\sin\theta_{ij}) = 0
$$
$$
Q_{G,i} – Q_{D,i} – \sum_{j=1}^{N} V_i V_j (G_{ij}\sin\theta_{ij} – B_{ij}\cos\theta_{ij}) = 0
$$
(2) Node voltage limits:
$$
V_{\min} \le V_i \le V_{\max}
$$
(3) Branch current limits:
$$
I_{ij,\min} \le I_{ij} \le I_{ij,\max}
$$
(4) ESS charging/discharging power limits:
$$
0 \le P_{ch,i}(t) \le \eta_{ch} P_{i,\max}
$$
$$
-P_{i,\max} \le \frac{P_{dis,i}(t)}{\eta_{dis}} \le 0
$$
(5) SOC limits:
$$
SOC_{\min} \le SOC_i(t) \le SOC_{\max}
$$
(6) Capacity and power limits of the installed energy storage system:
$$
E_{i,\min} \le E_i \le E_{i,\max}
$$
$$
P_{i,\min} \le P_i \le P_{i,\max}
$$
(7) Daily energy balance constraint for each ESS:
$$
\sum_{t=1}^{T} P_{c,d,i}(t) \Delta t = 0
$$
7. Solution Procedure
The optimization problem is solved using the improved NSGA-II algorithm. The chromosome encoding combines the installation location and the hourly power scheduling of the energy storage system. A simple integer coding is used for the location, while real coding is used for the power variables. The decision vector is
$$
X = \left[ x_1, x_2, \ldots, x_M, y_{1,1}, y_{1,2}, \ldots, y_{M,T} \right]
$$
where \(x_m\) is the node number for the \(m\)-th ESS, and \(y_{m,t}\) is the charging/discharging power of the \(m\)-th ESS at time \(t\).
The overall solution procedure is as follows:
- Load the system data (wind power, PV power, load profile, network parameters) and initialize the algorithm parameters.
- Generate the initial population using the opposition-based learning strategy.
- Run the power flow calculation using MATPOWER for each individual and compute the objective functions.
- Apply non-dominated sorting and crowding distance assignment.
- Generate offspring using the adaptive crossover and mutation operators.
- Combine parent and offspring populations and apply the improved elite preservation strategy to select the next generation.
- Repeat steps 3–6 until the maximum number of generations is reached.
- Obtain the Pareto-optimal solution set and use the improved ideal point decision (IIPBD) method to select the final solution.
The IIPBD method normalizes the objective values and computes the Euclidean distance to the ideal point (0,0) for each Pareto solution. The solution with the smallest distance is chosen as the best compromise. The normalization and distance are given by
$$
y_s(x_i) = \frac{F_s(x_i) – F_{s,\min}}{F_{s,\max} – F_{s,\min}}
$$
$$
D(x_i) = \sqrt{ \sum_{s=1}^{2} \omega_s \left( y_s(x_i) \right)^2 }
$$
where \(\omega_s\) is the weight for the \(s\)-th objective, and I set \(\omega_1 = \omega_2 = 0.5\) in the default case.
8. Case Study and Simulation Results
8.1 Test System and Parameter Settings
To verify the effectiveness of the proposed model and algorithm, I use the IEEE-33 node distribution system as the test network. The nominal voltage is 10 kV, and the total active and reactive loads are 3715 kW and 2300 kvar, respectively. The network topology consists of 33 nodes and 32 branches. The potential installation nodes for the energy storage system are nodes 2 to 33. Distributed wind power generators are placed at nodes 9 and 30 with a rated power of 200 kW each. Distributed photovoltaic generators are placed at nodes 14 and 20 with a rated power of 200 kW each. The parameters of the improved NSGA-II are listed in Table 3.
| Parameter | Value |
|---|---|
| Initial population size | 100 |
| Maximum number of generations | 200 |
| Minimum crossover rate | 0.4 |
| Maximum crossover rate | 0.8 |
| Minimum mutation rate | 0.1 |
| Maximum mutation rate | 0.8 |
The time-of-use electricity prices used for calculating the network loss cost are shown in Table 4.
| Load period | Time interval | Price (RMB/kWh) |
|---|---|---|
| Valley | 23:00–8:00 | 0.29 |
| Flat | 11:30–18:30 | 0.49 |
| Peak | 8:00–11:30, 18:30–23:00 | 0.54 |
8.2 Scenario Definitions
In order to comprehensively analyze the effect of the energy storage system and the performance of the improved algorithm, I define three scenarios:
- Scenario 1: Distribution network with DG but without any energy storage system.
- Scenario 2: Distribution network with DG and an energy storage system optimized by the standard NSGA-II.
- Scenario 3: Distribution network with DG and an energy storage system optimized by the improved NSGA-II.
8.3 Optimization Results
Table 5 presents the optimal results for the three scenarios. It can be observed that in Scenario 1, the total daily voltage deviation is 23.76 p.u., which indicates poor voltage quality. In Scenario 2, the installation of an energy storage system significantly reduces the voltage deviation to 7.46 p.u. In Scenario 3, the improved algorithm further reduces both the total cost and voltage deviation. The daily voltage deviation is decreased to 7.24 p.u., and the comprehensive cost is reduced by about 4% compared with Scenario 2. The optimal installation locations and capacities are identified as node 3 with 1.648 MWh and 140.1165 kW, and node 13 with 1.471 MWh and 138.3941 kW.
| Scenario | Total cost (RMB) | Daily voltage deviation (p.u.) | ESS location | ESS capacity (MWh) | ESS power (kW) |
|---|---|---|---|---|---|
| 1 | – | 23.76 | – | – | – |
| 2 | 1,169,786 | 7.46 | 7, 18 | 1.719, 1.514 | 142.43, 139.27 |
| 3 | 1,124,235 | 7.24 | 3, 13 | 1.648, 1.471 | 140.12, 138.39 |
To further demonstrate the superiority of the improved NSGA-II, I compare it with two other widely used multi-objective algorithms: the multi-objective differential evolution (MODE) and the multi-objective particle swarm optimization (MOPSO). The results in Table 6 show that the improved NSGA-II achieves the lowest comprehensive cost and voltage deviation among all three algorithms, proving its excellent search capability for the energy storage system allocation problem.
| Algorithm | Cost (RMB) | Voltage deviation (p.u.) | ESS location | ESS capacity (MWh) | ESS power (kW) |
|---|---|---|---|---|---|
| MODE | 1,186,246 | 7.27 | 2, 21 | 1.741, 1.525 | 150.15, 148.52 |
| MOPSO | 1,178,661 | 7.25 | 24, 33 | 1.727, 1.504 | 141.18, 143.28 |
| Improved NSGA-II | 1,124,235 | 7.24 | 3, 13 | 1.648, 1.471 | 140.12, 138.39 |
8.4 Voltage Profile Analysis
Figures 5, 6, and 7 show the voltage distribution of all nodes during 24 hours for the three scenarios, respectively. In Scenario 1 (without ESS), the voltage fluctuates significantly, and some nodes exceed the acceptable limits. In Scenario 2 (with ESS), the voltage profile is improved, and all nodes remain within the safe range. In Scenario 3 (with improved algorithm), the voltage profile is more stable and closer to the rated value, demonstrating the effectiveness of the proposed method in mitigating voltage deviations caused by DG integration.
8.5 Operation Strategy of the Energy Storage System
Figures 8 and 9 illustrate the daily charging/discharging curves and SOC variations of the two energy storage systems installed at nodes 3 and 13, respectively. Both energy storage systems operate in a similar pattern: they charge during the valley period (0:00–8:00) when wind power is abundant, and also during the midday period (14:00–17:00) when PV output is high. They discharge during the peak periods (8:00–12:00 and 18:00–23:00) to support the grid and reduce peak load. The SOC remains within the specified bounds of 0.1–0.9, and the daily energy balance is maintained.
The load profile before and after the installation of the energy storage system is shown in Figure 10. It is evident that the energy storage system effectively reduces the peak-valley difference. The daily load peak-valley difference before installation is 0.3732 p.u., while after installation it is reduced to 0.2478 p.u. This confirms the significant role of the energy storage system in peak shaving and valley filling.
8.6 Convergence Performance Comparison
To further verify the improvement of the algorithm, I compare the convergence curves of the standard NSGA-II and the improved NSGA-II over 20 independent runs. Figures 11 and 12 show the best convergence curves for the comprehensive cost and the voltage deviation, respectively. It can be observed that the improved NSGA-II converges faster than the standard NSGA-II. The improved algorithm reaches a stable solution at about generation 60 for the cost objective and about generation 40 for the voltage deviation objective. The final optimal values obtained by the improved algorithm are also lower than those of the standard algorithm, confirming its superiority in terms of both convergence speed and solution quality.
9. Conclusion and Future Work
In this research, I have investigated the optimal allocation of an energy storage system in a distribution network with high penetration of wind and solar power. The main contributions and conclusions are summarized as follows:
(1) I analyzed the operation characteristics of distributed wind, photovoltaic, load, and energy storage systems. The impact of DG and energy storage on the distribution network losses and voltage profiles is studied. It is shown that the location and capacity of an energy storage system significantly influence the network performance.
(2) I established a multi-objective optimization model for the distributed energy storage system, considering the annual comprehensive cost of the energy storage system and the daily voltage deviation of all nodes. The network loss cost is incorporated into the cost objective using a time-of-use electricity price mechanism.
(3) I improved the NSGA-II algorithm by using an opposition-based learning initialization, an adaptive crossover and mutation strategy, and an improved elite preservation mechanism. Simulation results on the IEEE-33 node system demonstrate that the improved algorithm outperforms the standard NSGA-II, MODE, and MOPSO in both solution quality and convergence speed.
(4) The optimal configuration results show that the installation of an energy storage system can effectively reduce voltage deviations, lower the comprehensive cost, and smooth the load curve. The proposed method provides a practical reference for planning and operating an energy storage system in active distribution networks.
In the future, I plan to extend this research in several directions. First, different types of battery technologies, such as vanadium redox flow batteries or sodium-ion batteries, can be modeled and compared in the optimization framework. Second, the uncertainty of renewable generation and load can be addressed by using stochastic programming or robust optimization methods. Third, the proposed method can be applied to larger distribution networks and integrated with other flexible resources, such as electric vehicles and demand response. Finally, I will consider the optimal allocation of an energy storage system on both the generation side and user side to maximize the overall social welfare.
In summary, the optimal allocation of an energy storage system is of great significance for the safe and economic operation of distribution networks with high penetration of renewable energy. My research demonstrates the effectiveness of the proposed model and algorithm, and I believe it can provide useful insights for real-world engineering applications.