We investigate the critical challenges faced by grid-forming photovoltaic inverters operating without energy storage, particularly the difficulty of precise active power reservation under fluctuating irradiance and temperature, and the risk of DC-link voltage instability caused by power imbalance between the AC and DC sides during simultaneous source and grid disturbances. In this work, we construct and analyze a behavioral model of photovoltaic arrays, propose a sensorless maximum power point (MPP) estimation method that avoids additional irradiance or temperature sensors, and develop a dual-loop dual-update-cycle online correction algorithm for the power reserve coefficient of grid-forming inverters. Furthermore, we design a DC voltage stabilization switching strategy to handle extreme scenarios. The proposed control enables solar inverters to provide reliable active power support while maintaining DC voltage stability under both resource fluctuations and frequency variations. The effectiveness of our method is validated through RT-LAB hardware-in-the-loop experiments.
Energy security and the transition to clean energy are global priorities. The construction of large-scale renewable energy bases in desert, Gobi, and barren areas is a key initiative to achieve carbon neutrality and peak carbon emissions. However, these areas lack conventional thermal or hydropower support and often have weak grid strength, making it essential to enhance the voltage and frequency support capabilities of solar inverters. Traditional grid-following inverters rely on phase-locked loops for synchronization and suffer from instability in weak grids. In contrast, grid-forming inverters autonomously synchronize and provide inertia and frequency support. Nevertheless, for grid-forming photovoltaic inverters without storage, the randomness of solar irradiance and temperature variations, when combined with grid frequency disturbances, can cause severe DC overvoltage or undervoltage transients. To address these issues, we present a novel power reserve control strategy that ensures accurate active power reservation and robust DC voltage regulation.
1. Topology and Baseline Control of Grid-Forming Photovoltaic Inverters
The main circuit topology of a storage-free grid-forming photovoltaic inverter is shown in the figure above. The DC-link capacitor \(C_{dc}\) interfaces the photovoltaic array with the inverter bridge. The grid-forming control employs an active power loop to emulate the inertia and damping characteristics of synchronous generators. The swing equation is:
$$
J\omega_n \frac{d\omega}{dt} = P_{\text{ref}} – P_e – D_p \omega_n (\omega – \omega_g) – K_d(\omega – \omega_g)
$$
where \(J\) is the virtual inertia, \(D_p\) is the frequency-active power droop coefficient, \(\omega_n\) and \(\omega_g\) are the nominal and measured grid angular frequencies, \(P_{\text{ref}}\) is the active power reference, and \(P_e\) is the measured output power. In maximum power point tracking (MPPT) mode, the reference \(P_{\text{ref}}\) is generated by a DC voltage controller:
$$
P_{\text{ref}} = G_{dc}(s)(U_{dc,\text{ref}} – U_{dc})
$$
where \(U_{dc}\) is the measured DC voltage and \(U_{dc,\text{ref}}\) comes from the MPPT algorithm. The reactive power loop provides voltage support:
$$
E_m = D_q(U_n – U_m) + \frac{1}{K_{iq}s}(Q_{\text{ref}} – Q_e)
$$
where \(E_m\) is the internal voltage amplitude, \(U_n\) is the nominal voltage amplitude, \(U_m\) is the measured voltage amplitude, \(Q_{\text{ref}}\) and \(Q_e\) are the reactive power reference and measurement, \(D_q\) is the voltage droop coefficient, and \(K_{iq}\) is the integral gain.
2. Power Reserve Operation and the Need for Accurate Reserve Estimation
When operating in active power reserve mode, the inverter switches from MPPT to a fixed power reference \(P_{\text{ref}} = (1-m)P_{\max0}\), where \(m\) is the reserve coefficient and \(P_{\max0}\) is the maximum power at the moment of mode transition. The reserved capacity is \(P_{\text{reserve}} = m P_{\max0}\). However, during reserve operation, the MPPT algorithm is disabled, so \(P_{\max0}\) becomes outdated as irradiance and temperature change. This leads to inaccurate reserves and potential DC voltage instability when the grid frequency deviates. To solve this, we need a method to estimate the instantaneous maximum power of the photovoltaic array without additional sensors.
3. Photovoltaic Array Behavioral Model and Maximum Power Estimation
Using the single-diode equivalent circuit model, the output current of a photovoltaic array can be expressed as:
$$
I_{pv} = n_0 \left[ k_3 \left(1 – k_1 \exp\left(\frac{k_2 – U_{pv}/m_0}{k_5}\right)\right) + k_4 \left(1 – \exp\left(\frac{k_6 – U_{pv}/m_0}{k_5}\right)\right) \right]
$$
where the parameters \(k_1\) to \(k_6\) depend on irradiance \(S_{Lx}\) and temperature \(T_m\). The output power is:
$$
P_{pv} = U_{pv} I_{pv} = p_{pv}(U_{pv}, S_{Lx}, T_m)
$$
The maximum power point \(P_{\max}\) for a given \((S_{Lx}, T_m)\) satisfies:
$$
\frac{\partial p_{pv}(U_{pv}, S_{Lx}, T_m)}{\partial U_{pv}} = 0
$$
This equation is transcendental and difficult to solve in real time. We therefore construct two lookup tables offline (or periodically updated): one mapping \((U_{pv}, S_{Lx})\) to \(P_{pv}\) at each temperature, and another mapping \(S_{Lx}\) to \(P_{\max}\). During operation, we measure \(U_{pv}\) and \(P_{pv}\) locally, receive temperature \(T_m\) from the plant controller (communication is acceptable because temperature changes slowly), and then use the first table to look up the current irradiance \(S_{Lx}\). With \(S_{Lx}\) and \(T_m\), the second table gives the estimated maximum power \(P_{\max}\). This avoids any additional sensors.
| Parameter | Symbol | Value |
|---|---|---|
| Number of parallel cells | \(n_0\) | 10 |
| Number of series cells | \(m_0\) | 60 |
| Short-circuit current (STC) | \(I_{sc}\) | 8.5 A |
| Open-circuit voltage (STC) | \(U_{oc}\) | 37.5 V |
| Current at MPP (STC) | \(I_m\) | 7.8 A |
| Voltage at MPP (STC) | \(U_m\) | 30.0 V |
| Temperature coefficient | \(a\) | 0.0025 /°C |
| Irradiance coefficient | \(b\) | 0.0005 / (W/m²) |
| Series resistance | \(R_s\) | 0.1 Ω |
4. Dual-Loop Online Correction of the Reserve Coefficient
Given the different time scales of irradiance (seconds) and temperature (minutes to hours), we propose a dual-loop correction method. The first loop updates the lookup tables at a slow rate (e.g., every 5 minutes, or on temperature change). The second loop corrects the reserve coefficient \(m\) at a fast rate (e.g., every 0.1 s) based on the current estimated \(P_{\max}\).
Algorithm steps for the fast correction loop:
- Measure \(U_{dc}\) and \(P_{pv}\).
- From the first table, find the irradiance \(S_{Lx}\) that satisfies both voltage matching (with tolerance \(\varepsilon_{dc}\)) and power matching (with tolerance \(\varepsilon_{pv}\)).
- From the second table, obtain \(P_{\max}\) corresponding to \(S_{Lx}\) and current \(T_m\).
- Compute the updated reserve coefficient:
$$
m = 1 – \frac{P_{\text{ref}}}{P_{\max}}, \quad 0 \le m \le 1
$$
To meet the frequency regulation requirements (e.g., 6% to 10% of rated power per grid code), we set a target reserve, e.g., 20% of rated inverter power. The droop coefficient \(D_p\) is chosen so that the full reserve is released for a frequency deviation of 0.5 Hz:
$$
D_p = \frac{P_{\text{backup}}}{2\pi \cdot \Delta f_{\max}}
$$
With our online estimation, the maximum reserve error is reduced from 125% (conventional fixed-reserve) to 5% under simultaneous irradiance and temperature changes, as demonstrated in our simulation.
| Scenario | Conventional error (σ_err) | Proposed error (σ_err) |
|---|---|---|
| Irradiance step change (1000→800 W/m²) | 125% | 5% |
| Temperature ramp (17°C→45°C) | 80% | 4% |
| Combined variations | 110% | 5% |
5. DC Voltage Stabilization Switching Under Extreme Conditions
When the estimated maximum power \(P_{\max}\) falls below the sum of load demand and required frequency reserve, the solar inverter cannot sustain both functions. In such cases, we switch from power reserve mode to a DC voltage stabilization mode. The controller selects:
- If \(P_{\max} \ge P_{\text{backup}} + P_{\text{load}}\) → continue accurate power reserve mode (switch S2 to port 1).
- If \(P_{\max} < P_{\text{backup}} + P_{\text{load}}\) → enter DC voltage stabilization mode (switch S2 to port -1), where the DC voltage is regulated to a reference value \(U_{st}\) (e.g., the open-circuit voltage \(U_{oc}\)) and the active power output is temporarily reduced to zero.
A hysteresis of \(\Delta P_{pv}\) (e.g., 5% of rated power) prevents repeated switching near the threshold. This strategy avoids unwarranted tripping and ensures that the inverter remains operational, ready to resume frequency support when irradiance recovers.
6. Hardware-in-the-Loop Experimental Verification
We built an RT-LAB hardware-in-the-loop platform consisting of an OP5707 real-time simulator and a NST-VSG-500KTL grid-forming inverter controller. The main circuit parameters are listed below.
| Parameter | Value |
|---|---|
| Rated power | 500 kW |
| DC-link capacitance | 15 mF |
| DC reference voltage | 710 V |
| Switching frequency | 5 kHz |
| Filter inductance | 0.5 mH |
| Grid voltage | 400 V |
| Fundamental frequency | 50 Hz |
Test 1: Irradiance variation only. Initial conditions: \(S_{Lx}=1000\) W/m², \(T_m=25^\circ\)C. The irradiance dropped to 800 W/m² between t₃ and t₄. A 0.5 Hz frequency drop occurred at t₁ and t₅. The reserve coefficient automatically adjusted from 0.21 to 0.42, and the output power decreased accordingly. During each frequency event, the inverter successfully released 100 kW of active power. The DC voltage remained stable throughout, with no undervoltage tripping. This confirms that the proposed dual-loop algorithm accurately tracks the changing maximum power and adjusts the reserve in real time.
Test 2: Temperature variation only. Using a temperature profile based on an actual PV station in Baiyin, Gansu, the temperature increased from 17°C to about 45°C over 60 seconds (simulating a 15-hour period). The reserve coefficient increased from 0.15 to 0.38, then decreased as temperature fell. The solar inverter responded correctly to simulated frequency drops, always releasing the intended 100 kW. DC voltage remained within safe limits.
Test 3: Combined irradiance and temperature variation. A rapid irradiance drop from 1000 to 400 W/m² (due to cloud passage) was accompanied by a temperature decrease from 30°C to 20°C. The reserve coefficient changed from 0.25 to nearly 0.9, and the output power was reduced accordingly. Despite multiple frequency disturbances (each 0.5 Hz drop), the inverter maintained DC voltage stability and provided the expected frequency support. The maximum power estimation error remained below 5%.
Test 4: Extreme low irradiance without stabilization. When irradiance fell to 200 W/m², the inverter’s maximum power became insufficient to support both the load and the frequency reserve. Upon a subsequent frequency disturbance, the DC voltage collapsed, leading to inverter shutdown. Even after irradiance recovered, the inverter remained offline, losing all support capabilities.
Test 5: Extreme low irradiance with proposed stabilization. Under the same conditions, when \(P_{\max}\) dropped below the reserve requirement, the inverter switched to DC voltage stabilization mode. The DC voltage was held constant at \(U_{st}=710\) V, and the active output was limited to zero. When the frequency dropped, the DC voltage controller overrode the increased power command, maintaining zero net active power. After irradiance recovered (to 700 W/m²), the inverter seamlessly returned to normal reserve operation. No undervoltage tripping occurred, demonstrating the robustness of the proposed strategy.
7. Conclusion
We have presented a comprehensive control framework for grid-forming photovoltaic inverters that eliminates the need for additional sensors while achieving accurate active power reserve under varying irradiance and temperature. The dual-loop online correction method, combined with a DC voltage stabilization scheme, ensures that the solar inverter can provide reliable frequency support and maintain DC-link stability even under extreme source and grid disturbances. Experimental results confirm significant improvement in reserve accuracy (error reduced from 125% to 5%) and the ability to avoid unnecessary shutdowns. The proposed approach is readily applicable to existing photovoltaic installations without hardware modification, accelerating the adoption of grid-forming technology in weak-grid environments.