Introduction
The integration of solar inverter into weak grids poses significant challenges due to voltage instability and limited active power transfer capabilities. Weak grids, characterized by low short-circuit ratios (SCRs), exacerbate interactions between grid-connected inverters and network dynamics, leading to oscillations or even instability. Traditional control strategies often fail to balance static power output requirements with dynamic stability constraints, particularly under varying grid conditions. This work proposes an adaptive Q-V droop gain tuning method for solar inverter to enhance active power stability while addressing voltage regulation and small-signal stability challenges.
Static Power Optimization via Q-V Droop Gain Adjustment
Problem Formulation
The static operating point of a solar inverter is constrained by:
- Voltage limits: Us,min≤Us≤Us,max (typically 0.95≤Us≤1.1p.u.).
- Current limits: Isd2+Isq2≤Ilim (e.g., Ilim=1.2p.u.).
- Active power requirement: Po=UsIsd, where Po is determined by maximum power point tracking (MPPT).
The Q-V droop relationship is defined as:Kv=Un−UsQ,
where Un is the nominal voltage reference (e.g., 1.2p.u.), and Q is the reactive power output.
Optimization Framework
The feasible Kv range is derived by solving:{min/maxKvs.t.UsIsd=Po,Isd2+Isq2≤Ilim,Us,min≤Us≤Us,max.
Key Results:
| SCR | Po(p.u.) | Kv,min | Kv,max |
|---|---|---|---|
| 1.5 | 0.8 | 0.12 | 0.25 |
| 2.0 | 0.8 | 0.18 | 0.32 |
| 3.0 | 0.8 | 0.24 | 0.41 |
The Kv range expands with higher SCRs and lower Po, enabling greater flexibility in voltage regulation.
Dynamic Stability Constraints and Impedance Modeling
Small-Signal Stability Analysis
The output impedance of a solar inverter in the dq-frame, accounting for frequency coupling and control dynamics, is modeled as:Zo=B−1A,
where A and B are matrices derived from linearized control equations (see Appendix for full derivation).
Stability Criterion
The system’s closed-loop poles determine stability. The critical pole λweakest (rightmost in the s-plane) must satisfy:Re(λweakest)<0.
A neural network (NN) approximates the relationship between Kv, grid inductance Lg, and λweakest:λweakest=fNN(Kv,Lg).
Stability Boundaries:{Lg<5mH⇒Kv<0.3for stabilityLg>5mH⇒Kv<0.2for stability
Adaptive Q-V Droop Gain Tuning Algorithm
Two-Stage Optimization
- Stage 1: Compute Kv,opt1 to minimize ∣Us−Un∣ under static constraints.
- Stage 2: Adjust Kv,opt1 to Kv,opt2 using a bisection method to ensure Re(λweakest)<0.
Algorithm Workflow:
- Estimate Lg using an extended Kalman filter (EKF).
- Solve Kv,opt1 via quadratic programming.
- Query λweakest using the pre-trained NN.
- If unstable, iteratively reduce Kv until Re(λweakest)<0.
Grid Impedance Estimation Using Extended Kalman Filter
The EKF estimates Lg from grid voltage and current measurements:xk+1yk=f(xk)+wk=Hxk+vk,
where x=[Δiα,Δiβ,Δusα,Δusβ,ωc,Lg−1]T, and f(⋅) encodes discretized grid dynamics.
Estimation Performance:
| True Lg(mH) | Estimated Lg(mH) | Error (%) |
|---|---|---|
| 3.0 | 3.1 | 3.3 |
| 4.5 | 4.4 | 2.2 |
| 5.0 | 4.9 | 2.0 |
Real-Time Simulation Validation
Case 1: Grid Impedance Variation
- Scenario: Lg transitions from 2.6mH to 5.0mH (SCR: 3.4 → 1.77) at Po=200kW.
- Results:
- Without adaptation: Us drops to 0.92p.u., violating constraints.
- With adaptation: Us stabilizes at 1.0p.u., Kv adjusts from 0.28 to 0.18.
Case 2: Power Demand Increase
- Scenario: Po increases from 200kW to 205kW under Lg=5.0mH.
- Results:
- Unstable Kv=0.22: Oscillations in Us and Is.
- Stable Kv=0.17: Us=1.02p.u., Is=1.15p.u..
Conclusion
The proposed adaptive Q-V droop gain tuning method ensures stable active power output for solar inverter in weak grids by harmonizing static voltage regulation with dynamic stability. Key innovations include:
- A two-stage optimization framework for Kv.
- Neural-network-based stability prediction for real-time adaptability.
- Robust grid impedance estimation via EKF.
Future work will extend this methodology to grid-forming inverters and validate it under ultra-weak grid conditions (SCR < 1.5).
Appendix: Impedance Model Derivation
The dq-frame impedance matrices A and B are derived from linearized control equations:A=BLf+BPI_I−Bdecpl−BPI_ICVPI_vdcRVdci−…B=BLfBCf−BplI_Vc−BFilterv+…
Full derivations are omitted for brevity but follow standard small-signal modeling techniques for voltage-source converters.