With the rapid growth of distributed solar power integration into distribution networks, high penetration levels of photovoltaic (PV) systems often lead to voltage violations, challenging grid stability and power quality. Traditional control strategies for solar inverters, such as droop control, may not fully address voltage issues and can result in significant power curtailment. This paper proposes a bi-level optimization framework for distributed energy storage system (ESS) placement and sizing, incorporating a multi-mode control strategy for solar inverters. The upper level minimizes ESS annual costs while maximizing revenue using an improved particle swarm optimization algorithm. The lower level employs a multi-mode control approach for solar inverters to mitigate voltage violations and coordinate with ESS operation, solved via quadratic programming in typical scenarios derived from K-means clustering with max-min distance criterion. Simulation results demonstrate the superiority of the multi-mode control in voltage regulation and reducing PV curtailment, alongside enhanced economic benefits for ESS deployment.
The integration of high-penetration distributed PV systems introduces bidirectional power flows, causing voltage rise during peak generation periods. Conventional solar inverter controls, like fixed power factor or droop methods, often fail to optimize voltage support and may lead to unnecessary power shedding. This work addresses these limitations by developing a coordinated approach where solar inverters dynamically adjust reactive and active power outputs based on grid conditions, while ESS provides additional flexibility for energy shifting and voltage support. The bi-level model ensures optimal ESS capacity allocation and operational strategies, improving overall system efficiency.
Solar inverter performance is critical in managing grid voltage. The proposed multi-mode control for solar inverters prioritizes reactive power adjustment, reducing reliance on active power curtailment. This strategy enhances the utilization of solar resources and minimizes losses. By integrating ESS, the system can store excess solar energy during high generation periods and discharge it during peak demand, further stabilizing voltage profiles. The following sections detail the scenario clustering, control strategy, optimization model, and case studies, supported by mathematical formulations and comparative analyses.
Solar Power Scenario Clustering
To handle the variability in solar generation, historical PV output data is clustered into representative scenarios using K-means algorithm with max-min distance criterion for initial center selection. This reduces computational complexity while capturing key patterns. The objective function minimizes the Euclidean distance between data points and cluster centers:
$$J = \sum_{b=1}^{E} \sum_{a \in G_b} \| \mathbf{x}_a – \mathbf{c}_b \|^2$$
where $\mathbf{x}_a$ represents PV output vectors, $\mathbf{c}_b$ are cluster centers, and $E$ is the number of clusters. The max-min distance criterion ensures diverse initial centers, avoiding local optima. For instance, with four clusters, typical scenarios include high, medium, low, and variable generation profiles, as shown in simulation results. This clustering enables efficient optimization across representative days, accounting for seasonal and diurnal variations in solar inverter outputs.
Multi-mode Control Strategy for Solar Inverters
The multi-mode control strategy for solar inverters dynamically switches between four operational modes based on grid voltage conditions and power changes. This approach enhances voltage regulation and minimizes active power curtailment compared to conventional droop control. The modes are:
Mode A: Normal Operation
Solar inverters operate at maximum power point tracking (MPPT), injecting active power $P_{j}^{\text{pv}} = P_{j}^{\text{mpp}}$ and reactive power $Q_{j}^{\text{pv}} = P_{j}^{\text{pv}} \tan(\phi_{j}^{\text{pv}})$, where $\phi_{j}^{\text{pv}}$ is the power factor angle. This mode applies when voltages are within limits.
Mode B: Reactive Power Reduction
If voltage at solar inverter terminal $V_{j}^{\text{tm}}$ exceeds threshold $V_2$ and active power increment $\Delta P_{j}^{\text{pv}} > 0$, inverters shift from capacitive to inductive reactive power output, gradually reaching $-Q_{j}^{\text{pv,max}}$ over time $t_{\text{DQ}}$:
$$Q_{j}^{\text{pv}}(t+1) = Q_{j}^{\text{pv}}(t) – \frac{[Q_{j}^{\text{pv}}(t-1) – Q_{j}^{\text{pv}}(t)] \Delta t}{t_{\text{DQ}} – n_{\text{DQ}} \Delta t}$$
This mitigates voltage rise without active power reduction.
Mode C: Active Power Coordination
When $V_{j}^{\text{tm}} \geq V_{\text{max}}$, solar inverters reduce active power $P_{j}^{\text{pv}}$ to prevent voltage violations. The adjustment follows:
$$P_{j}^{\text{pv}}(t+1) =
\begin{cases}
P_{j}^{\text{pv}}(t) \left(1 – \frac{\Delta t}{t_{\text{DP}} – n_{\text{DP}} \Delta t}\right) & \text{if } V_{j}^{\text{tm}} \geq V_{\text{max}} \\
P_{j}^{\text{pv}}(t) \left(1 + \frac{\Delta t}{t_{\text{RP}} – n_{\text{RP}} \Delta t}\right) & \text{if } V_{j}^{\text{tm}} < V_{\text{max}}
\end{cases}$$
Power is restored once voltages stabilize.
Mode D: Reactive Power Recovery
When solar generation decreases ($\Delta P_{j}^{\text{pv}} < 0$) or voltages drop below $V_{\text{max}} – \delta$, inverters increase capacitive reactive power to support voltage:
$$Q_{j}^{\text{pv}}(t+1) = Q_{j}^{\text{pv}}(t) + \frac{[Q_{j}^{\text{pv}}(t-1) – Q_{j}^{\text{pv}}(t)] \Delta t}{t_{\text{RQ}} – n_{\text{RQ}} \Delta t}$$
This mode ensures smooth transitions back to normal operation. The control logic prioritizes reactive power, reducing active power curtailment and enhancing solar inverter efficiency.
Bi-level Optimization Model
The bi-level framework optimizes ESS placement and sizing (upper level) and operational strategies (lower level). The upper level decisions include ESS locations, capacities, and power ratings, while the lower level determines daily charging/discharging schedules and solar inverter setpoints in clustered scenarios.
Upper Level Model
The objective is to minimize annualized ESS costs and maximize revenue, formulated as a weighted sum:
$$\min F_1 = \sum_{p=1}^{n_p} P_p \left( \alpha f_{1,p} – \beta f_{2,p} \right)$$
where $P_p$ is scenario probability, $\alpha$ and $\beta$ are weights ($\alpha + \beta = 1$). The cost and revenue components are:
$$f_{1,p} = \frac{C_{\text{es}}^{\text{inv}} + C_{\text{es}}^{\text{op}}}{365 \times T_s}, \quad f_{2,p} = C_{\text{es}}^{\text{in}} + C_{\text{es}}^{\text{ser}} + C_{\text{es}}^{\text{pvs}}$$
with investment cost $C_{\text{es}}^{\text{inv}} = \sum_{i=1}^{n_{\text{es}}} (c_e E_{\text{es},i} + c_p P_{\text{es},i}^{\text{max}})$, operation cost $C_{\text{es}}^{\text{op}} = \sum_{i=1}^{n_{\text{es}}} c_{\text{op}} P_{\text{es},i}^{\text{max}} \left( \frac{1+i_r}{1+d_r} \right)^\beta$, income from energy arbitrage $C_{\text{es}}^{\text{in}} = \sum_{t=1}^{n_t} \sum_{i=1}^{n_{\text{es}}} [\lambda_{\text{dis}} P_{\text{dis},i}(t) – \lambda_{\text{ch}} P_{\text{ch},i}(t)] \Delta t$, service fee $C_{\text{es}}^{\text{ser}} = \sum_{t=1}^{n_t} \sum_{i=1}^{n_{\text{es}}} \lambda_{\text{se}} [P_{\text{ch},i}(t) + P_{\text{dis},i}(t)] \Delta t$, and curtailment reduction revenue $C_{\text{es}}^{\text{pvs}} = \sum_{t=1}^{n_t} \sum_{j=1}^{n_{\text{pv}}} \lambda_{\text{pvs}} [\Delta P_{j}^{\text{pv0}}(t) – \Delta P_{j}^{\text{pv}}(t)] \Delta t$.
Constraints include ESS capacity and power limits:
$$E_{\text{es},i}^{\text{min}} \leq E_{\text{es},i} \leq E_{\text{es},i}^{\text{max}}, \quad N_{\text{es}}^{\text{min}} \leq N_{\text{es}} \leq N_{\text{es}}^{\text{max}}, \quad P_{\text{es},i}^{\text{max}} = \gamma E_{\text{es},i}$$
where $\gamma$ is the power-to-energy ratio.
Lower Level Model
The lower level maximizes PV revenue while minimizing curtailment in each scenario:
$$\max F_2 = \sum_{p=1}^{n_p} P_p \left( C_{\text{pv},p}^{\text{in}} – C_{\text{pv},p}^{\text{loss}} \right)$$
with PV income $C_{\text{pv},p}^{\text{in}} = \sum_{t=1}^{n_t} \sum_{j=1}^{n_{\text{pv}}} [\lambda_{\text{pv}} + \lambda_{\text{pv}}^{\text{bt}}] P_{j}^{\text{pv}}(t) \Delta t$ and curtailment penalty $C_{\text{pv},p}^{\text{loss}} = \sum_{t=1}^{n_t} \sum_{j=1}^{n_{\text{pv}}} \lambda_{\text{lo}} [P_{j}^{\text{mpp}}(t) – P_{j}^{\text{pv}}(t)] \Delta t$.
Operational constraints include power balance:
$$P_{\text{s}} = \sum_{n=1}^{n_{\text{bat}}} P_{\text{L},n} – \sum_{i=1}^{n_{\text{es}}} P_{\text{es},i} – \sum_{j=1}^{n_{\text{pv}}} P_{j}^{\text{pv}}$$
distribution power flow equations:
$$\begin{aligned}
P_I(t) &= \sum_{J} V_I(t) V_J(t) \left[ R_{IJ} \cos(\delta_{IJ}(t)) + X_{IJ} \sin(\delta_{IJ}(t)) \right] \\
Q_I(t) &= \sum_{J} V_I(t) V_J(t) \left[ R_{IJ} \sin(\delta_{IJ}(t)) – X_{IJ} \cos(\delta_{IJ}(t)) \right]
\end{aligned}$$
voltage prediction:
$$V_I(t+1) = V_I(t) + R_{II} \Delta P_I + X_{II} \Delta Q_I, \quad V^{\text{min}} \leq V_I(t) \leq V^{\text{max}}$$
ESS charging/discharging limits:
$$0 \leq P_{\text{ch},i} \leq P_{\text{es},i}^{\text{max}} B_{\text{ch},i}, \quad 0 \leq P_{\text{dis},i} \leq P_{\text{es},i}^{\text{max}} B_{\text{dis},i}, \quad B_{\text{ch},i} + B_{\text{dis},i} \leq 1$$
state of charge dynamics:
$$E_{\text{es},i}(t+1) = E_{\text{es},i}(t) + \left( \eta_{\text{ch},i} P_{\text{ch},i}(t) – \frac{P_{\text{dis},i}(t)}{\eta_{\text{dis},i}} \right) \Delta t, \quad E_{\text{es},i}^{\text{min}} \leq E_{\text{es},i}(t) \leq E_{\text{es},i}^{\text{max}}$$
and solar inverter constraints:
$$0 \leq P_{j}^{\text{pv}} \leq P_{j}^{\text{mpp}}, \quad Q_{j}^{\text{pv}} = P_{j}^{\text{pv}} \tan(\phi_{j}^{\text{pv}}), \quad (P_{j}^{\text{pv}})^2 + (Q_{j}^{\text{pv}})^2 \leq (S_{j}^{\text{pv}})^2$$
The lower level uses quadratic programming to solve for optimal ESS operation and solar inverter outputs, ensuring voltage stability and minimal curtailment.
Solution Methodology
The upper level employs an improved particle swarm optimization (PSO) with adaptive inertia weight and learning factors:
$$w = w_{\text{max}} – (w_{\text{max}} – w_{\text{min}}) \frac{k}{k_{\text{max}}}, \quad c_1 = c_{\text{max}} – (c_{\text{max}} – c_{\text{min}}) \frac{k}{k_{\text{max}}}, \quad c_2 = c_{\text{min}} + (c_{\text{max}} – c_{\text{min}}) \frac{k}{k_{\text{max}}}$$
where $k$ is iteration index. Velocity and position updates follow:
$$v_d^{k+1} = w v_d^k + c_1 r_1 (p_{\text{b}} – z_d^k) + c_2 r_2 (g_{\text{b}} – z_d^k), \quad z_d^{k+1} = z_d^k + v_d^{k+1}$$
The lower level applies quadratic programming to determine ESS schedules and solar inverter setpoints. The bi-level iteration continues until convergence, yielding optimal ESS configuration and operation.
Case Studies and Results
Simulations are conducted on IEEE 34-node and 123-node test systems with high PV penetration. Each node (except slack bus) hosts a 60 kW solar inverter with power factor 0.95. Typical scenarios are clustered into four profiles. Time-of-use tariffs for PV and ESS are summarized in Table 1.
| Period | Time | PV Sale ($\lambda_{\text{pv}}$) | ESS Discharge ($\lambda_{\text{dis}}$) | ESS Charge ($\lambda_{\text{ch}}$) |
|---|---|---|---|---|
| Valley | 00:00-08:00, 22:00-24:00 | 0.32 | 0.4 | 0.2 |
| Peak | 08:00-12:00, 17:00-21:00 | 0.65 | 0.75 | 0.55 |
| Flat | 12:00-17:00, 21:00-22:00 | 1.07 | 1.15 | 0.95 |
Key parameters for solar inverters and ESS are listed in Table 2.
| Parameter | Value | Parameter | Value |
|---|---|---|---|
| $c_e$ (per kWh) | 1800 | $[\alpha, \beta, \gamma]$ | [0.7, 0.3, 0.25] |
| $c_p$ (per kW) | 1000 | $[\eta_{\text{ch}}, \eta_{\text{dis}}]$ | [0.95, 1.1] |
| $c_{\text{op}}$ (per kW) | 25 | $[\lambda_{\text{se}}, \lambda_{\text{pvs}}, \lambda_{\text{lo}}]$ | [0.01, 0.01, 0.1] |
| $C_{\text{es}}^{\text{inv,max}}$ (10^4) | 500 | $[V^{\text{min}}, V_2, V^{\text{max}}]$ (p.u.) | [0.93, 1.06, 1.07] |
| $[i_r, d_r]$ | [0.02, 0.1] | $[E_{\text{es},i}^{\text{min}}, E_{\text{es},i}^{\text{max}}]$ (kWh) | [100, 1000] |
| $T_s$ (years) | 10 | $[N_{\text{es}}^{\text{min}}, N_{\text{es}}^{\text{max}}]$ | [1, 33] |
| $\Delta t$ (min) | 5 | $[t_{\text{DQ}}, t_{\text{RQ}}]$ (min) | [20, 20] |
| $[n_p, n_{\text{pv}}, n_t]$ | [4, 33, 24] | $[t_{\text{DP}}, t_{\text{RP}}]$ (min) | [5, 5] |
Four cases are analyzed: Case 1 (no control), Case 2 (comparison of droop and multi-mode control), Case 3 (ESS optimization with multi-mode control in IEEE 34-node), and Case 4 (extension to IEEE 123-node).
In Case 1, without control, voltage violations occur, exceeding 1.15 p.u. during peak solar generation. Case 2 shows that multi-mode control reduces annual PV curtailment to 969.64 MWh, compared to 1135.11 MWh with droop control. Additionally, multi-mode control maintains voltages within limits more effectively.
For Case 3, ESS optimization with multi-mode control on IEEE 34-node system yields optimal ESS placement at nodes 26 and 30 with capacities 756 kWh and 1068 kWh, and power ratings 189 kW and 267 kW. Results are summarized in Table 3.
| Scheme | Nodes | Capacity (kWh) | Power (kW) | ESS Cost (10^4) | ESS Revenue (10^4) | PV Revenue (10^4) |
|---|---|---|---|---|---|---|
| 1 | — | — | — | — | — | 349.17 |
| 2 | 26,30 | 756,1068 | 189,267 | 381.62 | 49.43 | 355.78 |
| 2 | 26,29 | 1056,798 | 264,199.5 | 387.96 | 48.29 | 356.18 |
| 2 | 24,29 | 1470,456 | 367.5,114 | 403.02 | 47.66 | 356.63 |
| 3 | 21,24,30 | 150,324,1410 | 37.5,81,352.5 | 394.29 | 48.42 | 356.32 |
| 3 | 19,24,29 | 894,516,528 | 223.5,129,132 | 405.53 | 47.26 | 356.24 |
| 3 | 21,23,33 | 660,540,762 | 165,135,190.5 | 410.56 | 46.94 | 356.18 |
| 4 | 18,23,27,29 | 216,168,1368,150 | 54,42,342,37.5 | 398.01 | 57.16 | 356.21 |
| 4 | 13,19,20,28 | 186,858,456,504 | 46.5,214.5,114,126 | 419.4 | 54.08 | 356.17 |
| 4 | 14,20,29,30 | 162,240,276,1356 | 40.5,60,69,339 | 425.62 | 56.06 | 356.66 |
The optimal scheme (Scheme 2, first row) reduces PV curtailment from 969.64 MWh to 108.94 MWh annually, a 88.76% improvement. Voltage profiles are maintained within 0.93–1.07 p.u., and ESS operation follows daily cycles, charging during low demand and discharging during peaks.
In Case 4, for IEEE 123-node system, ESS optimization with multi-mode control achieves similar benefits. The optimal configuration uses four ESS units, reducing curtailment from 14348.64 MWh to 1358.92 MWh (90.53% decrease). Voltage stability is enhanced across all scenarios, demonstrating scalability.
Conclusion
This paper presents a bi-level optimization approach for distributed ESS configuration, integrated with a multi-mode control strategy for solar inverters. The method effectively addresses voltage violations and PV curtailment in high-penetration distribution networks. The multi-mode control for solar inverters outperforms traditional droop control by prioritizing reactive power adjustment, minimizing active power reductions. The bi-level model ensures economically optimal ESS placement and operation, validated through IEEE test systems. Future work could explore real-time implementation and hybrid energy storage technologies.