Introduction
The rapid integration of renewable energy sources into power grids has necessitated the widespread use of grid-tied inverters as critical interfaces for energy conversion. However, in weak grid environments characterized by high grid impedance, the operational stability and performance of grid-tied inverters face significant challenges. These challenges arise from voltage limit violations, power transfer limitations, and small-signal stability constraints, which collectively restrict the feasible operating range of such systems. This study investigates the critical short-circuit ratio (CSCR) that defines the boundary of stable operation for grid-tied inverters under multiple constraints. We propose a comprehensive analysis framework to quantify the CSCR and demonstrate how terminal voltage compensation strategies can expand the operational range by mitigating dominant constraints.
Key Concepts and Formulations
1. Short-Circuit Ratio (SCR) Definition
The SCR is defined as the ratio of the short-circuit capacity at the point of common coupling (PCC) to the rated capacity of the grid-tied inverter:γ=Us2RNRs2+Xs2=Us2RsRN1+η2,γ=RNRs2+Xs2Us2=RsRN1+η2Us2,
where UsUs is the grid voltage, RNRN is the inverter’s rated power, RsRs and XsXs are the grid resistance and reactance, and η=Xs/Rsη=Xs/Rs is the impedance ratio. The critical SCR (γminγmin) represents the minimum SCR required to maintain stable operation under specific constraints.
2. Operational Constraints
Three primary constraints govern the operational limits of grid-tied inverters:
- Voltage Limit Constraint: Ensures PCC voltage remains within permissible bounds (e.g., UPCC≥0.9 puUPCC≥0.9pu).
- Maximum Power Transfer Constraint: Determines the maximum active power the inverter can deliver to the grid.
- Small-Signal Stability Constraint: Guarantees system stability under dynamic perturbations.
2.1 Voltage Limit Constraint
The voltage drop at the PCC is derived as:ΔUPCC=1−12(1+2λ1+1+4(λ1−λ22)),ΔUPCC=1−21(1+2λ1+1+4(λ1−λ22)),
where λ1=1γP+ηQ1+η2λ1=γ11+η2P+ηQ and λ2=Qγ1+η2λ2=γ1+η2Q. Setting UPCC=0.9 puUPCC=0.9pu, the critical SCR for voltage stability is calculated as γV,min=2.29γV,min=2.29.
2.2 Maximum Power Transfer Constraint
Under unit power factor operation, the maximum active power PmaxPmax is:Pmax=Ug221+η2.Pmax=21+η2Ug2.
Equating Pmax=PNPmax=PN (rated power), the critical SCR for power transfer is:γP,min=2.γP,min=2.
2.3 Small-Signal Stability Constraint
Using impedance-based stability criteria, the system remains stable if the phase difference between grid and inverter impedances does not exceed 180∘180∘. For a purely inductive grid (η→∞η→∞), the critical SCR is derived as γS,min=1.85γS,min=1.85.
3. Comprehensive Critical SCR (CSCR)
The CSCR is determined by the most restrictive constraint:CSCR=max(γV,min,γP,min,γS,min)=2.29.CSCR=max(γV,min,γP,min,γS,min)=2.29.
This indicates that voltage limit violations dominate the operational boundaries in uncompensated systems.
Terminal Voltage Compensation Strategy
To address the voltage limit constraint, we propose a terminal voltage compensation method using thyristor-switched capacitors (TSCs). By injecting reactive power at the PCC, the voltage is stabilized near 1 pu1pu, effectively eliminating voltage violations. The compensated system’s impedance model is modified to:Zgrid,comp(s)=Zs(s)Zs(s)sC+1,Zgrid,comp(s)=Zs(s)sC+1Zs(s),
where CC is the compensation capacitance.
Impact on CSCR
After compensation, the dominant constraint shifts to small-signal stability. The revised critical SCR becomes:CSCRcomp=γS,min,comp=2.10,CSCRcomp=γS,min,comp=2.10,
expanding the operational range compared to the uncompensated case (CSCR=2.29CSCR=2.29).
Simulation and Experimental Validation
1. Simulation Setup
A Simulink model of a three-phase LCL-filtered grid-tied inverter was developed with parameters listed in Table 1.
Table 1: Inverter Parameters
| Parameter | Symbol | Value |
|---|---|---|
| DC-link voltage | UdcUdc | 800 V |
| Grid phase voltage | UgUg | 220 V |
| Current loop proportional gain | KpKp | 4 |
| Current loop integral gain | KiKi | 200 |
2. Voltage Limit Constraint Verification
Figure 1 shows the PCC voltage under varying SCR. At γ=2.29γ=2.29, UPCC=0.9 puUPCC=0.9pu, validating γV,minγV,min.
Table 2: Critical SCR Validation
| Constraint | Theoretical γminγmin | Simulated γminγmin |
|---|---|---|
| Voltage limit | 2.29 | 2.29 |
| Power transfer | 2.00 | 2.00 |
| Small-signal stability | 1.85 | 1.87 |
3. Compensated System Performance
After TSC compensation, the PCC voltage remains at 1 pu1pu, and the CSCR reduces to 2.102.10 (Table 3).
Table 3: Compensated System Critical SCR
| Constraint | Theoretical γminγmin | Simulated γminγmin |
|---|---|---|
| Small-signal stability | 2.10 | 2.06 |
Conclusion
- In uncompensated weak grids, the operational range of grid-tied inverters is primarily limited by voltage violations, with CSCR=2.29CSCR=2.29.
- Terminal voltage compensation using TSCs shifts the dominant constraint to small-signal stability, reducing CSCRCSCR to 2.102.10 and expanding the feasible operating range.
- The proposed framework provides a systematic approach to quantify and enhance the stability of grid-tied inverters in weak grid environments.
This study underscores the importance of multi-constraint analysis and targeted compensation strategies in optimizing the performance of grid-tied inverters, facilitating higher renewable energy penetration in modern power systems.