The integration of Renewable Energy Sources (RES), such as wind and solar photovoltaic (PV) systems, into distribution networks is accelerating globally. While this transition is crucial for decarbonization, it introduces significant operational challenges due to the inherent intermittency, randomness, and volatility of RES power generation. The mismatch between stochastic RES output and time-varying load demand can lead to voltage fluctuations, branch congestion, and compromised system stability. More critically, it often results in the curtailment of renewable energy—wind and solar power must be deliberately reduced when generation exceeds local demand and grid absorption capacity, representing a direct economic and environmental loss.
A battery energy storage system (BESS) is widely recognized as a pivotal flexible resource to mitigate these challenges. By strategically charging during periods of RES surplus and discharging during periods of deficit or peak demand, a BESS can “firm” renewable generation, enhance local consumption, defer grid infrastructure upgrades, and provide various ancillary services. However, the economic viability and technical effectiveness of a BESS are profoundly dependent on its appropriately sized power (kW) and energy (kWh) capacities. An undersized system fails to capture sufficient value, while an oversized one leads to prohibitive capital costs and underutilization. Therefore, developing an optimal capacity configuration strategy for battery energy storage systems within active distribution networks (ADNs) is a critical research and engineering problem.
This article delves into the problem of optimal BESS sizing for renewable energy integration. We begin by quantitatively modeling the key uncertainties in an ADN. Subsequently, we propose a rule-based operational strategy for the BESS aimed explicitly at maximizing renewable consumption. Building upon this, we formulate a comprehensive economic optimization model that balances the costs and benefits of the battery energy storage system. Finally, we demonstrate the effectiveness of the proposed strategy through numerical simulation on a standard test network.
1. Modeling Uncertainties in Active Distribution Networks
The core challenge in planning and operating an ADN stems from uncertainty. This uncertainty originates from both the supply side (variable renewable generation) and the demand side (fluctuating load). Effective modeling of these uncertainties is the foundation for robust BESS sizing.
1.1 Uncertainty in Renewable Generation
The actual power output from wind turbines and PV panels deviates from forecasted values due to the unpredictable nature of wind speed and solar irradiance. The actual power at time \( t \) can be expressed as the sum of a forecasted value and a forecast error:
$$
P_{wind}(t) = P_{wind}^{s}(t) + P_{wind}^{\epsilon}(t)
$$
$$
P_{pv}(t) = P_{pv}^{s}(t) + P_{pv}^{\epsilon}(t)
$$
where \( P_{wind}^{s}(t) \) and \( P_{pv}^{s}(t) \) are the forecasted wind and PV power outputs, respectively. The forecast errors, \( P_{wind}^{\epsilon}(t) \) and \( P_{pv}^{\epsilon}(t) \), are typically modeled as random variables following a zero-mean normal distribution. The standard deviation of this distribution is often proportional to the forecasted power level, reflecting that forecast uncertainty is relative to the generation scale.
$$
P_{wind}^{\epsilon}(t) \sim \mathcal{N}(0, \sigma_{w}^{t}), \quad \sigma_{w}^{t} = (\alpha_w + \beta_w \times P_{wind}^{s}(t)) S_w
$$
$$
P_{pv}^{\epsilon}(t) \sim \mathcal{N}(0, \sigma_{pv}^{t}), \quad \sigma_{pv}^{t} = (\alpha_{pv} + \beta_{pv} \times P_{pv}^{s}(t)) S_{pv}
$$
In these equations:
- \( \sigma_{w}^{t} \) and \( \sigma_{pv}^{t} \) are the time-dependent standard deviations for wind and PV forecast errors.
- \( \alpha_w, \alpha_{pv}, \beta_w, \beta_{pv} \) are technology-specific forecast accuracy parameters.
- \( S_w \) and \( S_{pv} \) represent the installed capacity of the wind farm and PV plant, respectively.
1.2 Uncertainty in Load Demand
Similarly, the active power demand at a network node \( k \) is not perfectly predictable. It can be modeled as a forecasted load profile plus a random error term:
$$
P_{L_k}(t) = g_k(t) + \epsilon_k, \quad \epsilon_k \sim \mathcal{N}(\mu_k, \sigma_k)
$$
Here, \( g_k(t) \) is the forecasted load at node \( k \), and \( \epsilon_k \) is the load forecast error, assumed to follow a normal distribution with mean \( \mu_k \) and standard deviation \( \sigma_k \). The parameters \( \mu_k \) and \( \sigma_k \) are related to the load forecasting methodology and its inherent accuracy.
The collective impact of these uncertainties creates a highly variable net load profile (\( P_{load}(t) – P_{wind}(t) – P_{pv}(t) \)), which the distribution grid and any ancillary resources like a battery energy storage system must manage.
2. BESS Operational Strategy for Renewable Energy Integration
The primary objective for configuring the battery energy storage system in this context is to enhance the utilization rate of local renewable generation. We propose a rule-based, renewable-following operational strategy. This strategy dictates the charging and discharging behavior of the BESS based on the instantaneous balance between local RES generation, load demand, and the state of charge (SOC) of the battery. The logic is designed to prioritize using renewable energy directly or storing it for later use, minimizing the export of excess generation or the need for power from the main grid when renewables are insufficient.
The core operational rules are defined as follows:
- Charging Mode: If the total renewable generation exceeds the local load demand AND the BESS is not fully charged, the surplus energy is used to charge the battery.
Condition: \( P_{wind}(t) + P_{pv}(t) > P_{load}(t) \) AND \( SOC(t) < SOC_{max} \)
Action: \( P_{battery}(t) < 0 \) (power into the battery). - Idle Mode (Full): If renewables exceed demand but the BESS is fully charged, the battery cannot absorb more energy and remains idle. This scenario may lead to curtailment.
Condition: \( P_{wind}(t) + P_{pv}(t) > P_{load}(t) \) AND \( SOC(t) = SOC_{max} \)
Action: \( P_{battery}(t) = 0 \). - Discharging Mode: If the local load demand exceeds renewable generation AND the BESS has sufficient stored energy, the battery discharges to cover the deficit.
Condition: \( P_{wind}(t) + P_{pv}(t) < P_{load}(t) \) AND \( SOC(t) > SOC_{min} \)
Action: \( P_{battery}(t) > 0 \) (power out of the battery). - Idle Mode (Empty): If load exceeds renewables but the BESS is depleted, it cannot provide support.
Condition: \( P_{wind}(t) + P_{pv}(t) < P_{load}(t) \) AND \( SOC(t) = SOC_{min} \)
Action: \( P_{battery}(t) = 0 \).
For this study, typical SOC limits are set to preserve battery health: \( SOC_{max} = 0.8 \) and \( SOC_{min} = 0.2 \). The actual power exchange of the battery energy storage system is also constrained by its maximum charge/discharge power rating \( P_{ab} \) and its round-trip efficiency \( \eta_{rt} \). The power setpoint can be modeled as:
$$
P_{battery}(t) =
\begin{cases}
-P_{ab}, & \text{if } P_{load}(t) – P_{wind}(t) – P_{pv}(t) – P_{grid}(t) < -P_{ab} \\
\eta_{ch}^{-1} \cdot (P_{load}(t) – P_{wind}(t) – P_{pv}(t) – P_{grid}(t)), & \text{if } -P_{ab} \leq P_{load}(t) – \sum P_{RES}(t) – P_{grid}(t) < 0 \quad \text{(Charging)} \\
0, & \text{if } P_{load}(t) – \sum P_{RES}(t) – P_{grid}(t) = 0 \\
\eta_{dis} \cdot (P_{load}(t) – P_{wind}(t) – P_{pv}(t) – P_{grid}(t)), & \text{if } 0 < P_{load}(t) – \sum P_{RES}(t) – P_{grid}(t) \leq P_{ab} \quad \text{(Discharging)} \\
P_{ab}, & \text{if } P_{load}(t) – P_{wind}(t) – P_{pv}(t) – P_{grid}(t) > P_{ab}
\end{cases}
$$
Where \( \eta_{ch} \) and \( \eta_{dis} \) are the charging and discharging efficiencies, often combined into a single round-trip efficiency \( \eta_{rt} = \eta_{ch} \cdot \eta_{dis} \). The grid power \( P_{grid}(t) \) is the net import (positive) or export (negative) from the main transmission network.
The operational logic is summarized in the following truth table:
| Net Renewable Surplus/Deficit | BESS SOC Status | BESS Action | Power \( P_{battery} \)** |
|---|---|---|---|
| Surplus (\( \sum P_{RES} > P_{load} \)) | Not Full (\( SOC < 0.8 \)) | Charge | Negative |
| Surplus (\( \sum P_{RES} > P_{load} \)) | Full (\( SOC = 0.8 \)) | Idle | Zero |
| Deficit (\( \sum P_{RES} < P_{load} \)) | Not Empty (\( SOC > 0.2 \)) | Discharge | Positive |
| Deficit (\( \sum P_{RES} < P_{load} \)) | Empty (\( SOC = 0.2 \)) | Idle | Zero |
3. Economic Optimization Model for BESS Capacity Configuration
The optimal sizing of the battery energy storage system is framed as an economic optimization problem. The goal is to find the combination of rated power (\( P_{rated}^{bess} \)) and rated energy capacity (\( E_{rated}^{bess} \)) that minimizes the net daily cost, which includes the BESS’s own costs, its operational revenue, and penalties for unmet objectives.
3.1 Cost-Benefit Analysis of the Battery Energy Storage System
1. Daily Amortized Cost of the BESS (\( K_c \)):
The total capital expenditure (CAPEX) for a BESS, such as a Vanadium Redox Flow Battery (VRFB) used in this example, is often linear with respect to its power and energy ratings. The daily cost can be expressed as:
$$
K_c = \frac{k_1 \cdot P_{rated}^{bess} + k_2 \cdot E_{rated}^{bess} + a_0}{365 \cdot Y}
$$
For daily optimization in a single-year snapshot, this is often simplified to:
$$
K_c = k_1 \cdot P_{rated}^{bess} + k_2 \cdot E_{rated}^{bess} + a_0
$$
where:
- \( k_1 \) is the cost coefficient related to power conversion system (PCS) and power-related balance of plant (€/kW).
- \( k_2 \) is the cost coefficient related to energy storage media (e.g., electrolytes for VRFB) (€/kWh).
- \( a_0 \) represents fixed costs like installation and commissioning, amortized daily.
- \( Y \) is the project lifetime in years (used in the full amortization formula).
2. Daily Operational Revenue of the BESS (\( K_b \)):
The battery energy storage system generates revenue primarily through energy arbitrage. It stores energy when electricity prices (or the implicit value of renewable energy) are low and discharges when they are high. In our renewable-integration context, the revenue comes from displacing more expensive or carbon-intensive grid power. The daily revenue can be modeled as:
$$
K_b = \int_{t_{dis}}^{} \left( P_{battery}(t) \cdot \eta_{dis} \cdot P_{sell}(t) \right) dt – \int_{t_{ch}}^{} \left( \frac{P_{battery}(t)}{\eta_{ch}} \cdot P_{buy}(t) \right) dt
$$
Where \( P_{sell}(t) \) and \( P_{buy}(t) \) are the selling and buying electricity prices at time \( t \), and \( t_{dis} \) and \( t_{ch} \) are the sets of time intervals for discharging and charging, respectively. In a simplified model focusing on displacing fossil-fueled grid generation, this can be approximated as:
$$
K_b \approx \sum_{t \in T_{dis}} \left( P_{battery}(t) \cdot \eta_{dis} \cdot P_e \right) – \sum_{t \in T_{ch}} \left( \frac{P_{grid, displaced}(t)}{\eta_g} \cdot P_f \right)
$$
Here, \( P_e \) is a fixed feed-in tariff or market price, \( P_f \) is the fuel cost for conventional generation, and \( \eta_g \) is the average efficiency of the displaced grid generation.
3. Penalty Cost for Renewable Energy Curtailment (\( P_c \)):
A key performance indicator is the amount of renewable energy that must be curtailed because it cannot be consumed locally or stored. To incentivize the optimization algorithm to minimize this, a penalty cost is introduced:
$$
P_c = c_a \cdot \int_{0}^{24} \max(0, P_{wind}(t) + P_{pv}(t) + P_{grid}(t) + P_{battery}(t) – P_{load}(t)) \, dt
$$
The term inside the integral represents the net power injection into the grid that exceeds what the local network can absorb (leading to curtailment). \( c_a \) is a penalty coefficient (€/kWh) representing the economic and environmental cost of wasted renewable energy.
3.2 Optimization Problem Formulation
The objective is to find the BESS power and energy capacity that minimizes the total net daily cost:
$$
\min_{P_{rated}^{bess}, E_{rated}^{bess}} f = K_c – K_b + P_c
$$
Subject to the following constraints:
1. BESS Capacity Bounds:
$$
P_{rated, min}^{bess} \leq P_{rated}^{bess} \leq P_{rated, max}^{bess}
$$
$$
E_{rated, min}^{bess} \leq E_{rated}^{bess} \leq E_{rated, max}^{bess}
$$
2. BESS Power and SOC Dynamics:
$$
| P_{battery}(t) | \leq P_{ab}(SOC(t))
$$
$$
SOC_{min} \leq SOC(t) \leq SOC_{max}
$$
$$
SOC(t+1) = SOC(t) – \frac{P_{battery}(t) \cdot \Delta t}{E_{rated}^{bess}} \quad \text{(considering efficiency)}
$$
3. Distribution Network Operational Constraints:
a) Power Flow Equations (DistFlow or AC-Optimal Power Flow):
For each node \( i \):
$$
P_i^{injection} = P_{load,i} + U_i \sum_{j=1}^{N} U_j (G_{ij}\cos\theta_{ij} + B_{ij}\sin\theta_{ij})
$$
$$
Q_i^{injection} = Q_{load,i} + U_i \sum_{j=1}^{N} U_j (G_{ij}\sin\theta_{ij} – B_{ij}\cos\theta_{ij})
$$
Where \( P_i^{injection} = P_{grid,i} + P_{wind,i} + P_{pv,i} + P_{battery,i} \).
b) Voltage and Current Limits:
$$
U_{i, min} \leq U_i \leq U_{i, max}, \quad \forall i \in \text{Nodes}
$$
$$
|I_{l}| \leq I_{l, max}, \quad \forall l \in \text{Branches}
$$
This forms a constrained, non-linear optimization problem. Heuristic algorithms like Genetic Algorithms (GA), Particle Swarm Optimization (PSO), or mixed-integer linear programming (MILP) approximations are commonly employed to find near-optimal solutions.
4. Case Study and Simulation Analysis
To validate the proposed optimal configuration strategy for the battery energy storage system, a simulation was conducted on the modified IEEE 33-bus radial distribution test system.
4.1 Simulation Setup
The network parameters and resource placements are configured as follows:
| Component | Connection Bus | Capacity / Parameter |
|---|---|---|
| PV Plant | Bus 17 | 1.0 MW |
| Wind Farm | Bus 32 | 3.6 MW |
| BESS | Bus 1 (Slack bus area) | To be optimized |
| Grid Connection | Bus 0 (Slack) | Unlimited (with cost) |
The forecast error and load uncertainty parameters are set based on typical values:
| Parameter | Description | Value |
|---|---|---|
| \( \alpha_w, \alpha_{pv} \) | Forecast error constant parameters | 0.02, 0.015 |
| \( \beta_w, \beta_{pv} \) | Forecast error scaling parameters | 0.16, 0.15 |
| \( \mu_k \) | Mean of load forecast error | 0 |
| \( \sigma_k \) | Std. dev. of load forecast error | 0.01 p.u. |
The economic parameters for the optimization are summarized below:
| Parameter | Description | Value / Range |
|---|---|---|
| \( k_1 \) | BESS power-related cost coefficient | 2.28 €/kW/day |
| \( k_2 \) | BESS energy-related cost coefficient | 0.14 €/kWh/day |
| \( a_0 \) | Daily fixed/amortized cost | 200 €/day |
| \( P_e \) | Electricity selling price | 0.45 €/kWh |
| \( P_f \) | Conventional generation fuel cost | 0.25 €/kWh |
| \( \eta_g \) | Grid generation efficiency | 0.75 |
| \( c_a \) | Renewable curtailment penalty | 0.18 €/kWh |
| \( P_{rated}^{bess} \) Search Space | BESS power capacity range | 300 kW to 900 kW |
| \( E_{rated}^{bess} \) Search Space | BESS energy capacity range | 2000 kWh to 8000 kWh |
4.2 Optimization Results and Analysis
A Genetic Algorithm was applied to solve the optimization problem under hundreds of scenarios generated from the uncertainty models. The algorithm converged to an optimal solution after 68 iterations.
Optimal BESS Configuration:
| Optimal Parameter | Value |
|---|---|
| Rated Power (\( P_{rated}^{bess} \)) | 616 kW |
| Rated Energy Capacity (\( E_{rated}^{bess} \)) | 5636 kWh |
| Minimum Net Daily Cost (\( f_{min} \)) | 696.8 €/day |
This configuration represents the best economic trade-off. A BESS of this size can capture a significant portion of the renewable surplus without incurring excessive capital costs that cannot be recovered through arbitrage and curtailment reduction.
4.3 Performance Evaluation
The impact of deploying the optimally sized battery energy storage system was analyzed over a typical 24-hour period with high renewable penetration.
Key Observations:
- Renewable Energy Curtailment Reduction: In the scenario without a BESS, significant curtailment occurred during midday hours (e.g., ~10:00 to 14:00) when PV generation peaked and load was moderate. With the optimal BESS in operation, a large fraction of this surplus was stored, drastically reducing the curtailed energy. The penalty cost \( P_c \) was reduced by approximately 65% compared to the no-BESS baseline.
- Load Shaping and Grid Support: During evening peak load hours (e.g., ~18:00 to 22:00) when renewable generation diminished, the BESS discharged its stored energy. This reduced the net power import \( P_{grid} \) from the main grid, lowering fuel costs and associated emissions. The discharge profile of the battery energy storage system effectively “shifted” the midday solar energy to the evening peak.
- Economic Balance: The optimization ensured that the system did not aim for 100% renewable curtailment elimination, as doing so would require a much larger, more expensive battery whose marginal cost would exceed the marginal benefit of saving the last few kWh of renewable energy. The solution finds the economically “sweet spot.”
The following table contrasts key performance metrics between the system without BESS and with the optimally configured BESS:
| Performance Metric | Without BESS | With Optimal BESS (616 kW / 5636 kWh) | Improvement |
|---|---|---|---|
| Daily Renewable Curtailment | ~1850 kWh | ~650 kWh | ~64.9% reduction |
| Daily Grid Import Energy | ~8250 kWh | ~7150 kWh | ~13.3% reduction |
| Renewable Self-Consumption Rate* | ~72% | ~89% | +17 percentage points |
*Percentage of total renewable generation consumed locally (by load or BESS).
5. Conclusion
This article has presented a comprehensive framework for optimizing the capacity of a battery energy storage system in an active distribution network with high penetrations of wind and solar power. The strategy explicitly addresses the dual uncertainties of renewable generation and load demand. By formulating an operational policy focused on renewable energy integration and an economic objective function that balances costs, revenues, and penalties for curtailment, the model determines the BESS size that offers the best economic value.
The case study on the IEEE 33-bus system demonstrates the effectiveness of the approach. The optimally sized battery energy storage system (616 kW / 5636 kWh) significantly enhances the utilization of renewable resources, reducing curtailment by approximately 65% and decreasing reliance on grid imports during peak periods. This leads to a higher rate of local renewable self-consumption. Crucially, the optimization proves that the most beneficial size is not necessarily the largest possible one, but the one that provides the optimal economic return on investment given the specific network conditions, renewable resource profiles, and market parameters. This methodology provides a valuable tool for system planners and utilities seeking to deploy battery energy storage systems cost-effectively to facilitate the clean energy transition.