With the global push toward carbon neutrality, the integration of distributed photovoltaic (PV) systems into distribution networks has surged. However, high PV penetration often leads to challenges such as equipment overloads and voltage violations due to insufficient hosting capacity. Distributed energy storage (DES) offers rapid and smooth voltage regulation, making it a key solution. This article proposes a collaborative optimization framework for DES and solar inverters, incorporating long-term battery degradation costs. A phased control strategy prioritizes economic efficiency, leveraging the synergistic potential of solar inverters and DES. A bi-level optimization model is formulated: the upper level minimizes DES investment and degradation costs to determine optimal locations and capacities, while the lower level minimizes operational costs through coordinated调度 of DES and solar inverters. Simulation results under various scenarios demonstrate enhanced voltage quality, reduced equipment overloads, and lower planning and operational expenses.
The proliferation of distributed PV systems introduces operational uncertainties, including voltage fluctuations and thermal overloading. Solar inverters, capable of reactive power control, provide a cost-effective means for voltage regulation. Meanwhile, DES supports active power management, enabling peak shaving and valley filling. However, DES deployment must account for battery degradation over time, influenced by charging/discharging patterns and seasonal variations. This study addresses these aspects through a holistic approach, optimizing both planning and调度 phases.
Synergistic Control Strategy for Solar Inverters and Distributed Energy Storage
The reactive power capability of solar inverters is constrained by their rated capacity and instantaneous active power output. The maximum reactive power adjustability, \( Q_{\text{max,PV}} \), is given by:
$$ Q_{\text{max,PV}} = \sqrt{S_{\text{INV}}^2 – P_{\text{PV}}^2} $$
where \( S_{\text{INV}} \) denotes the inverter’s rated capacity (typically 1.0–1.1 times the PV panel rating), and \( P_{\text{PV}} \) is the active power output. The voltage variation \( \Delta U_{\text{PV},i,j} \) at node \( i \) due to reactive power adjustment \( \Delta Q_{\text{PV},j} \) at node \( j \) is:
$$ \Delta U_{\text{PV},i,j} = \beta_{U-Q,i,j} \cdot \Delta Q_{\text{PV},j} $$
Here, \( \beta_{U-Q,i,j} \) is the reactive power-voltage sensitivity. The cost-effectiveness coefficient \( S_{U-C,\text{PV}} \) for solar inverter control is:
$$ S_{U-C,\text{PV}} = \frac{C_{\text{PV}}}{\beta_{U-Q,i,j}} $$
with \( C_{\text{PV}} = 0.067 \) USD/kvarh representing the unit reactive power control cost.
For DES, the state of charge (SOC) dynamics are described by:
$$ S_t = S_{t-1} + \frac{P_{\text{in,ESS}}(t) \Delta t}{\phi} – \frac{P_{\text{out,ESS}}(t) \Delta t}{\phi} $$
where \( S_t \) is the SOC at time \( t \), \( P_{\text{in,ESS}}(t) \) and \( P_{\text{out,ESS}}(t) \) are charging and discharging powers, and \( \phi \) is the storage capacity. The voltage change \( \Delta U_{\text{ESS},i,j} \) due to active power adjustment \( \Delta P_{\text{ESS},j} \) is:
$$ \Delta U_{\text{ESS},i,j} = \alpha_{U-P,i,j} \cdot \Delta P_{\text{ESS},j} $$
where \( \alpha_{U-P,i,j} \) is the active power-voltage sensitivity. The corresponding cost-effectiveness coefficient \( S_{U-C,\text{ESS}} \) is:
$$ S_{U-C,\text{ESS}} = \frac{C_{\text{ESS}}}{\alpha_{U-P,i,j}} $$
with \( C_{\text{ESS}} = 0.6 \) USD/kWh as the unit active power control cost. Comparing both coefficients:
$$ \frac{S_{U-C,\text{ESS}}}{S_{U-C,\text{PV}}} = \frac{C_{\text{ESS}}}{C_{\text{PV}}} \cdot \frac{\beta_{U-Q,i,j}}{\alpha_{U-P,i,j}} \leq \frac{0.6}{0.067} \cdot \frac{x_l}{r_l} $$
Given \( r_l/x_l \leq 6 \) for low-voltage networks, solar inverter control proves more economical. Thus, a phased strategy is adopted: solar inverters respond first to voltage deviations, followed by DES if violations persist.
Thresholds \( U_{1+,\text{cri}} \), \( U_{2+,\text{cri}} \), and \( U_{3+,\text{cri}} \) trigger incremental responses: reactive absorption by solar inverters, DES charging, and full-capacity DES utilization, respectively. Similarly, thresholds \( U_{1-,\text{cri}} \), \( U_{2-,\text{cri}} \), and \( U_{3-,\text{cri}} \) manage undervoltage conditions.
Bi-Level Optimization Model Incorporating Battery Degradation
Upper-Level Planning Model
The upper level minimizes total annual costs \( F \), encompassing DES investment \( C_{L,\text{inv}} \), dispatch costs \( C_0 \), and degradation costs \( C_{\text{ESS,cut}} \):
$$ \min F = C_{L,\text{inv}} + C_0 + C_{\text{ESS,cut}} $$
Dispatch costs aggregate seasonal expenses:
$$ C_0 = d_W C_W + d_S C_S + d_{S-A} C_{S-A} $$
where \( d_W \), \( d_S \), and \( d_{S-A} \) are winter, summer, and spring-autumn days, respectively. DES investment costs are:
$$ C_{L,\text{inv}} = \sum_{m=1}^{M} \left( \frac{c_{L,e,m} E_{L,\max,m} + c_{L,p,m} P_{L,\max,m} + c_{L,l,m} S_{L,l,m}}{T_L (1 + \tau)} \right) $$
for \( M \) DES units, with capacity \( E_{L,\max,m} \), power rating \( P_{L,\max,m} \), and footprint \( S_{L,l,m} \). Degradation cost \( C_{\text{ESS,cut}} \) derives from the Millner model:
$$ E_t = \alpha |P_{\text{ESS}}(t)| + \beta |\Delta S_t| + \mu \left( \frac{S_t + S_{t-1}}{2} \right)^\xi $$
where \( \alpha \), \( \beta \), \( \mu \), and \( \xi = 3.446 \) are battery-specific parameters. The degradation cost at time \( t \) is:
$$ C_{d,t} = \delta E_t $$
with \( \delta \) as the unit installation cost. The investment constraint is:
$$ C_{L,\text{inv}} \leq C_{L,\text{inv},\max} $$
Lower-Level Dispatch Model
The lower level minimizes daily operational costs \( C \):
$$ \min C = C_{Q,\text{PV}} + C_{\text{buy}} + C_{P,\text{PV}} + C_{P,\text{ESS}} $$
where \( C_{Q,\text{PV}} \) is solar inverter reactive control cost, \( C_{\text{buy}} \) is power purchase cost from the main grid, \( C_{P,\text{PV}} \) is PV generation payment, and \( C_{P,\text{ESS}} \) is DES active control cost. Specifically:
$$ C_{P,\text{PV}} = \sum_{t=1}^{T} \sum_{n=1}^{N_{\text{PV}}} C_D P_{\text{PV},n,t} \Delta t $$
$$ C_{\text{buy}} = \sum_{t=1}^{T} C_{\text{sub},t} P_{\text{sub},t} \Delta t $$
$$ C_{Q,\text{PV}} = \sum_{t=1}^{T} \sum_{n=1}^{N_{\text{PV}}} C_{\text{PV}} |Q_{\text{PV},n,t}| \Delta t $$
$$ C_{P,\text{ESS}} = \sum_{t=1}^{T} \sum_{n=1}^{N_{\text{ESS}}} C_{\text{ESS}} |P_{\text{ESS},n,t}| \Delta t $$
Constraints include power flow equations, device limits, and network security:
$$ P_i = \sum_{j=1}^{k} U_i U_j (G_{ij} \cos \delta_{ij} + B_{ij} \sin \delta_{ij}) $$
$$ Q_i = \sum_{j=1}^{k} U_i U_j (G_{ij} \sin \delta_{ij} – B_{ij} \cos \delta_{ij}) $$
$$ Q_{\text{PV},i,\min} \leq Q_{\text{PV},t,i} \leq Q_{\text{PV},i,\max} $$
$$ S_{\min} \leq S_t \leq S_{\max} $$
$$ P_{\text{ch}}(t) \leq B_{\text{ch}}(t) P_{\text{in}},\quad P_{\text{dis}}(t) \leq B_{\text{dis}}(t) P_{\text{out}} $$
$$ B_{\text{dis}}(t) + B_{\text{ch}}(t) \leq 1 $$
$$ U_{\min} \leq U_i \leq U_{\max} $$
$$ |P_{ij}| \leq P_{ij,\max} $$
Solution Methodology: Alternating Direction Method of Multipliers (ADMM)
The bi-level model is solved using ADMM, which decomposes the problem into subproblems coordinated via augmented Lagrangian. The general form is:
$$ \min f(x) + g(z) \quad \text{subject to} \quad Ax + Bz = C $$
The augmented Lagrangian is:
$$ L_\rho(x,z,\lambda) = f(x) + g(z) + \lambda^T (Ax + Bz – C) + \frac{\rho}{2} \|Ax + Bz – C\|^2 $$
Iterations proceed as:
- Solve upper-level DES planning subproblem.
- Solve lower-level dispatch subproblem.
- Update Lagrange multipliers \( \lambda \).
- Check convergence; repeat if necessary.
This approach ensures coordination between planning and调度 while handling nonlinear constraints efficiently.
Case Study and Results
A modified IEEE 33-node system with distributed PV at nodes 7, 13, 18, 28, 31, and 33 (500 kW each, power factor 0.9) is used. DES candidate locations are nodes 2-3 and 13-14. Seasonal load profiles and unit control costs are summarized below:
| Control Method | Unit Cost |
|---|---|
| Solar Inverter Reactive Control | 0.067 USD/kvarh |
| DES Active Control | 0.600 USD/kWh |
DES configuration results after optimization:
| Location | Capacity (kWh) | Max Power (kW) |
|---|---|---|
| 2-3 | 1,000 | 200 |
| 13-14 | 3,000 | 600 |
Two control schemes are compared:
- Scheme 1: DES-only control.
- Scheme 2: Coordinated solar inverter and DES control.
Control costs for Scheme 1:
| Control Method | Adjustment | Cost (USD) |
|---|---|---|
| Solar Inverter | 0 kvarh | 0.00 |
| DES | 4,154.52 kWh | 2,492.71 |
Control costs for Scheme 2:
| Control Method | Adjustment | Cost (USD) |
|---|---|---|
| Solar Inverter | 4,404.39 kvarh | 295.09 |
| DES | 3,235.11 kWh | 1,941.07 |
Scheme 2 reduces total cost by 256.55 USD compared to Scheme 1. Voltage quality improvements are evident:
| Season | Scheme 1 (kV) | Scheme 2 (kV) |
|---|---|---|
| Summer | 11.77 | 11.96 |
| Spring/Autumn | 11.74 | 11.97 |
| Winter | 11.70 | 11.78 |
No voltage violations occur post-DES deployment. The coordinated approach enhances economic efficiency and power quality.
Conclusion
This study presents a comprehensive framework for optimizing DES and solar inverters in distribution networks, accounting for long-term battery degradation. Key findings include:
- Solar inverters offer superior economic efficiency for reactive power control, but their capability is limited under high PV penetration.
- Phased coordination between solar inverters and DES maximizes flexibility and resource utilization.
- The bi-level model effectively balances investment and operational costs, incorporating battery degradation for realistic planning.
- Simulations confirm improved voltage profiles, reduced overloads, and cost savings.
Future work will explore real-time adaptability and integration with other distributed energy resources.