In recent years, the development of photovoltaic (PV) energy systems has shifted from focusing solely on high-cost PV modules to optimizing inverter topologies and control methods. Among various multilevel inverter structures, the cascaded H-bridge (CHB) converter has attracted considerable attention for grid-connected solar inverters due to its modularity, high reliability, and independent DC-link voltage capability. Each H-bridge unit can be fed by an individual PV panel, allowing independent maximum power point tracking (MPPT) and reduced sensitivity to partial shading. However, when the PV panels experience unequal irradiance or mismatched aging, the power generated from different units may be severely unbalanced. This imbalance leads to distorted grid currents and may force some H-bridge units into the over-modulation region, thereby violating the total harmonic distortion (THD) requirements and destabilizing the whole solar inverter system.
This paper presents a novel control strategy based on reactive power compensation that significantly expands the stable operating region of cascaded H-bridge solar inverters under power unbalance conditions. The proposed approach is derived from a rigorous mathematical analysis in the rotating reference frame (dq-frame), which yields explicit conditions for stable operation at unit power factor. Furthermore, a quantitative study reveals the impact of reactive power injection on the DC-link voltage stress and allows an optimized control structure to eliminate undesirable interactions with the voltage loop. Simulation and experimental results demonstrate that the proposed method maintains high-quality grid currents even when PV power is severely unbalanced.
The remainder of this paper is organized as follows. Section 1 establishes the mathematical model of a single-phase CHB solar inverter. Section 2 analyzes the over-modulation mechanism and derives the stability boundary under unit power factor. Section 3 presents the reactive power compensation strategy, power distribution principles, and the analysis of DC-link voltage stress. Section 4 describes the optimized control block diagram. Sections 5 and 6 provide simulation and experimental validations, respectively. Conclusions are drawn in Section 7.
1 Mathematical Model of the CHB Solar Inverter
The main circuit topology of a single-phase cascaded H-bridge solar inverter is shown in the figure below.
The topology consists of \(n\) H-bridge modules connected in series, followed by a filter inductor connected to the utility grid. Each H-bridge unit has a separate PV source on its DC side, enabling independent MPPT. In the figure, \(L_s\) and \(R_s\) represent the filter inductance and equivalent line resistance, respectively, while \(C_x\) is the DC-link capacitor. The switching devices are denoted as \(S_{x1}\)–\(S_{x4}\). The grid voltage and current are \(u_s\) and \(i_s\), respectively. For each unit, \(u_{dcx}\) and \(u_{Hx}\) denote the DC-side voltage and the AC-side voltage, while \(u_L\) is the voltage across the filter inductor. The PV output current, capacitor current, and H-bridge DC-side current are \(i_{PVx}\), \(i_{Cx}\), and \(i_{Hx}\), with \(x = 1, 2, \dots, n\).
Ignoring switching losses and line impedance, the mathematical model of the CHB inverter can be written from Kirchhoff’s laws as
$$
\begin{cases}
L_s \dfrac{di_s}{dt} = u_{AB} – u_s, \\[6pt]
C_x \dfrac{du_{dcx}}{dt} = i_{PVx} – i_{Hx}, \\[6pt]
u_{AB} = u_{H1} + u_{H2} + \dots + u_{Hn}.
\end{cases}
\tag{1}
$$
The switching function of the \(x\)-th H-bridge unit is defined as
$$
s_x = \frac{u_{Hx}}{u_{dcx}}.
\tag{2}
$$
Since sinusoidal pulse width modulation (SPWM) is used and the switching frequency is much higher than the grid frequency, harmonics in \(s_x\) can be neglected. Keeping the fundamental component, we define the modulation wave \(m_x\) as
$$
m_x = s_x \approx \frac{u_{Hx}}{u_{dcx}}.
\tag{3}
$$
Substituting (3) into (1) gives
$$
L_s \frac{di_s}{dt} = \sum_{x=1}^{n} m_x u_{dcx} – u_s.
\tag{4}
$$
Using phasor notation, (4) can be expressed in the frequency domain as
$$
j\omega L_s I_s = \sum_{x=1}^{n} \mathbf{M}_x U_{dcx} – \mathbf{U}_s,
\tag{5}
$$
where \(\mathbf{U}_s\) and \(\mathbf{I}_s\) are the phasors of the grid voltage and current, \(\mathbf{M}_x\) is the phasor of the modulation wave \(m_x\), and \(\omega\) is the angular frequency of the grid. Equation (5) is fundamental for the stability analysis presented in the next section.
2 Stability Analysis under Unit Power Factor
2.1 Over-modulation mechanism
A prerequisite for stable operation of a CHB solar inverter at unit power factor is that all H-bridge units remain within the linear modulation range. The modulation index \(M_x\) of each unit must satisfy
$$
0 \le M_x \le \frac{\sqrt{2}}{2}.
\tag{6}
$$
When the output powers of the PV panels differ significantly, over-modulation may occur. Consider a three-unit CHB system as an example. Let the output powers of the three units be \(P_1\), \(P_2\), and \(P_3\), and the total active power delivered to the grid be \(P_T = P_1 + P_2 + P_3\). If the irradiance on units 1 and 2 decreases, their power drops, thereby reducing \(P_T\) and consequently the grid current \(I_s\). To maintain unit 3 at its maximum power point, its AC-side voltage \(U_{H3}\) must increase. Since the maximum power point voltage \(U_{mpp\_3}\) of unit 3 is unchanged, its modulation index \(M_3\) must increase. If the power unbalance worsens, \(M_3\) eventually exceeds \(\sqrt{2}/2\), driving unit 3 into over-modulation and causing severe current distortion.
2.2 Derivation of the stable operating region
For a comprehensive analysis, we align the \(d\)-axis with the grid current and the \(q\)-axis leading the \(d\)-axis by \(\pi/2\). The AC-side voltage vector of each unit is decomposed into \(d\)-axis and \(q\)-axis components. Under steady-state operation with unit power factor, the vector diagram is shown in the original paper. The active power delivered by the \(x\)-th unit to the grid is
$$
P_x = U_{Hx\_d} I_s = M_{x\_d} u_{dcx} I_s,
\tag{8}
$$
where \(U_{Hx\_d}\) is the \(d\)-axis component of \(U_{Hx}\) and \(M_{x\_d}\) is the active component of the modulation wave. Since the total active power is \(P_T\) and the grid current is \(I_s\), using (8) we obtain
$$
M_{x\_d} = \frac{P_x}{P_T} \cdot \frac{U_s}{u_{dcx}}.
\tag{9}
$$
In a control scheme without reactive power compensation, all units share the same relative active and reactive voltage components. That is, the ratio of the reactive component \(M_{x\_q}\) to the active component \(M_{x\_d}\) is identical for all units and equal to \(U_L / U_s\), where \(U_L\) is the inductor voltage. Therefore,
$$
\frac{M_{1\_q}}{M_{1\_d}} = \frac{M_{2\_q}}{M_{2\_d}} = \dots = \frac{M_{n\_q}}{M_{n\_d}} = \frac{U_L}{U_s}.
\tag{10}
$$
The reactive component of each unit’s modulation index is thus
$$
M_{x\_q} = \frac{P_x}{P_T} \cdot \frac{\omega L_s I_s}{u_{dcx}}.
\tag{11}
$$
Combining (9) and (11), the total modulation index of the \(x\)-th unit is
$$
M_x = \sqrt{M_{x\_d}^2 + M_{x\_q}^2}
= \frac{P_x}{P_T} \cdot \frac{\sqrt{U_s^2 + (\omega L_s I_s)^2}}{u_{dcx}}
= \frac{P_x}{P_T} \cdot \frac{U_{AB}}{u_{dcx}}.
\tag{12}
$$
To ensure linear modulation, we require \(M_x \le \sqrt{2}/2\), which leads to
$$
\frac{P_x}{P_T} \cdot \frac{U_{AB}}{u_{dcx}} \le \frac{\sqrt{2}}{2}.
$$
Since \(U_{AB} \approx U_s\) when the inductor voltage is negligible, the condition becomes
$$
\frac{P_x}{P_T} \cdot \frac{U_s}{u_{dcx}} \le \frac{\sqrt{2}}{2}.
\tag{13}
$$
Rearranging (13) gives the necessary condition for stable operation at unit power factor:
$$
\frac{P_x}{P_T} \le \frac{\sqrt{2}\, u_{dcx}}{2 U_s}.
\tag{14}
$$
In practice, \(u_{dcx}\) is normally close to the maximum power point voltage \(U_{mpp\_x}\). If any unit violates (14), the system is prone to over-modulation, and the grid current THD rises to unacceptable levels.
3 Proposed Reactive Power Compensation Strategy
3.1 Required reactive current calculation
The grid current is again aligned with the \(d\)-axis. Let the active and reactive components of the grid current be \(I_{sp}\) and \(I_{sq}\), respectively, and let \(\phi\) be the phase angle between the grid voltage and the grid current. The active component is determined solely by the total PV power:
$$
I_{sp} = \frac{P_T}{U_s}.
\tag{15}
$$
This value is independent of any reactive power compensation because the DC-side power must be balanced by the active power delivered to the grid. The relation between the PV output current \(i_{PVx}\) and \(I_{sp}\) is
$$
i_{PVx} = m_x I_{sp},
\tag{16}
$$
which indicates that the unit with the largest modulation index delivers the largest PV current. Define \(i_{PV\_max}\) as the maximum PV current among all units.
The active component of the total AC-side voltage \(U_{AB\_d}\) is given by the sum of the active voltage components of each unit:
$$
U_{AB\_d} = M_{1\_d} u_{dc1} + M_{2\_d} u_{dc2} + \dots + M_{n\_d} u_{dcn}.
\tag{17}
$$
Using (16), each active modulation component can be expressed as \(M_{x\_d} = i_{PVx} / i_{PV\_max}\) (assuming all units operate at the same \(I_{sp}\)). Substituting into (17) yields
$$
U_{AB\_d} = \frac{i_{PV1} u_{dc1}}{i_{PV\_max}} + \frac{i_{PV2} u_{dc2}}{i_{PV\_max}} + \dots + \frac{i_{PVn} u_{dcn}}{i_{PV\_max}}.
\tag{18}
$$
From the voltage vector diagram under reactive compensation, we have
$$
\cos\phi = \frac{U_{AB\_d}}{U_s}, \qquad I_{sq} = I_{sp} \tan\phi.
\tag{19}
$$
The minimum reactive current needed is therefore
$$
U_{AB\_q} = U_s \sin\phi – \omega L_s I_s,
\tag{20}
$$
where \(U_{AB\_q}\) is the required reactive voltage component that must be allocated among the units that are not over-modulated.
3.2 Power distribution among units
For active power, the inherent series connection of H-bridge units implies that each unit’s AC-side voltage in the \(d\)-axis direction is proportional to its delivered power. Therefore,
$$
\frac{U_{H1\_d}}{P_1} = \frac{U_{H2\_d}}{P_2} = \dots = \frac{U_{Hn\_d}}{P_n} = \frac{U_{AB\_d}}{P_T}.
\tag{21}
$$
Equivalently, the active modulation index is
$$
M_{x\_d} = \frac{U_{Hx\_d}}{u_{dcx}} = \frac{P_x}{P_T} \cdot \frac{U_{AB\_d}}{u_{dcx}}.
\tag{22}
$$
For reactive power allocation, the principle is to ensure that after adding the reactive component, the total modulation index of every unit remains below \(\sqrt{2}/2\). For each unit, given its active component \(M_{x\_d}\), the maximum allowable reactive component is
$$
M_{x\_q(max)} = \sqrt{\left(\frac{\sqrt{2}}{2}\right)^2 – M_{x\_d}^2} = \sqrt{\frac{1}{2} – \left(\frac{U_{Hx\_d}}{u_{dcx}}\right)^2}.
\tag{23}
$$
This can also be expressed in terms of the AC-side voltage components:
$$
U_{Hx\_q(max)} = \sqrt{\frac{u_{dcx}^2}{2} – U_{Hx\_d}^2}.
\tag{24}
$$
In practice, the over-modulated unit (the one with the largest active power) should not participate in reactive power sharing; otherwise it will enter over-modulation again. The remaining units are assigned reactive power according to their maximum allowed reactive capability, i.e.,
$$
M_{x\_q} = \frac{ U_{Hx\_q(max)} }{ \sum_{k=1}^{n} U_{Hk\_q(max)} } \cdot \frac{U_{AB\_q}}{u_{dcx}}.
\tag{25}
$$
A vector diagram for a three-module system with \(P_1 > P_2 > P_3\) is given in the original paper. Unit 3 has the largest active power and is set to the maximum modulation index \(\sqrt{2}/2\); it receives no reactive component. Units 1 and 2 share the reactive voltage in proportion to their available margins.
3.3 Impact of reactive compensation on DC-link voltage stress
The DC-link voltage stress is characterized by the maximum instantaneous value of \(u_{dcx}\). The DC-link voltage consists of a DC component and a superimposed AC ripple:
$$
u_{dcx} = \bar{u}_{dcx} + \tilde{u}_{dcx},
\tag{26}
$$
where \(\bar{u}_{dcx}\) is the average value (equal to \(U_{mpp\_x}\) under normal MPPT operation) and \(\tilde{u}_{dcx}\) is the ripple. The peak value is
$$
\max(u_{dcx}) = \bar{u}_{dcx} + \frac{U_{p-p\_x}}{2},
\tag{27}
$$
where \(U_{p-p\_x}\) is the peak-to-peak ripple. The ripple is caused by the instantaneous power oscillation at twice the grid frequency. Using the capacitor impedance relation, we have
$$
U_{p-p\_x} = \frac{S_x}{\omega_1 C_x \bar{u}_{dcx}},
\tag{28}
$$
where \(\omega_1 = 2\pi (2f_g) = 4\pi f_g\) is the second-harmonic angular frequency, \(f_g\) is the grid frequency, and \(S_x\) is the apparent power of the \(x\)-th unit given by
$$
S_x = \sqrt{P_x^2 + Q_x^2}.
\tag{29}
$$
Combining (27)–(29), the DC-link voltage stress is
$$
\max(u_{dcx}) = u_{mpp\_x} + \frac{\sqrt{P_x^2 + Q_x^2}}{4\pi f_g C_x u_{mpp\_x}}.
\tag{30}
$$
To quantify the effect, we consider a two-unit example with parameters \(u_{dc1} = u_{dc2} = 200 \, \text{V}\), \(P_1 = 300 \, \text{W}\), \(C_1 = C_2 = 470 \, \mu\text{F}\), \(U_s = 220 \, \text{V}\). As the power ratio \(P_2/P_1\) increases, the system first operates at unit power factor until the stability limit is reached. Beyond that, reactive power compensation is applied, increasing the voltage stress of unit 1. The analysis shows that even under severe unbalance (e.g., \(P_2/P_1 = 4.5\)), the voltage stress increase is only about 18.5 V (12.04%), which is acceptable for practical design.
4 Optimized Control Architecture
The overall control block diagram of the proposed reactive power compensation strategy for cascaded H-bridge solar inverters is depicted in the original paper. It consists of three main parts:
(1) Independent MPPT control: Each PV panel is forced to operate at its own maximum power point. The MPPT algorithm provides the reference voltage \(U_{mpp\_x}\) for the corresponding DC-link voltage loop.
(2) Grid current control: A quasi-proportional-resonant (Q-PR) controller is used to achieve zero steady-state error for the sinusoidal current reference. The grid voltage feed-forward is included to decouple the current controller from the grid voltage, enhancing disturbance rejection.
(3) Over-modulation detection and reactive power allocation: The controller monitors the PV voltages and powers continuously. If the condition (14) is satisfied, the system runs at unit power factor. Otherwise, reactive power compensation is triggered. First, \(U_{AB\_d}\) is calculated via (18). Then, the required reactive current reference \(I_{sq}^*\) and reactive voltage component \(U_{AB\_q}^*\) are obtained from (19) and (20). Finally, the active and reactive modulation indexes are generated according to (22) and (25).
It is important to note that in some existing implementations (e.g., reference [18] in the original paper), the reactive power compensation is realized by shifting the phase angle of the total current reference. This approach implicitly reduces the active current component, which interferes with the voltage loop. In our proposed method, the current reference is composed of two independent parts: \(i_s^* = i_{sd}^* + i_{sq}^*\), where \(i_{sd}^*\) depends solely on the PV power and \(i_{sq}^*\) is the calculated reactive current. Consequently, the reactive power injection does not alter the output of the voltage loop PI controllers, preserving their physical meaning and simplifying the controller design.
The key equations for the control implementation are summarized in the table below.
| Item | Formula | Description |
|---|---|---|
| Active current reference | \(I_{sp}^* = P_T^* / U_s\) | From total PV power reference |
| Reactive current reference | \(I_{sq}^* = I_{sp}^* \tan\phi\) | Minimum required reactive current |
| Active modulation index | \(M_{x\_d} = \frac{P_x}{P_T} \frac{U_{AB\_d}}{u_{dcx}}\) | Active power distribution |
| Reactive modulation index | \(M_{x\_q} = \frac{U_{Hx\_q(max)}}{\sum U_{Hx\_q(max)}} \frac{U_{AB\_q}}{u_{dcx}}\) | Reactive power distribution |
| Stability condition | \(\frac{P_x}{P_T} \le \frac{\sqrt{2} u_{dcx}}{2 U_s}\) | Unit power factor stable region |
| Voltage stress | \(\max(u_{dcx}) = u_{mpp\_x} + \frac{\sqrt{P_x^2+Q_x^2}}{4\pi f_g C_x u_{mpp\_x}}\) | Effect of reactive power |
This optimized architecture ensures that the voltage loop is completely decoupled from the reactive power compensation. Thus, the voltage controller can be designed based only on the active power dynamics, which greatly enhances the reliability and robustness of the entire solar inverter system.
5 Simulation Results
To verify the effectiveness of the proposed control strategy, a 9-level single-phase CHB solar inverter was simulated in MATLAB/Simulink. The PV panels were modeled as voltage sources with series resistors. The simulation parameters are given in the following table.
| Parameter | Symbol | Value |
|---|---|---|
| Number of cascaded H-bridge units | n | 9 |
| Grid voltage (RMS) | \(U_s\) | 660 V |
| Switching frequency | \(f_s\) | 20 kHz |
| Filter inductance | \(L_s\) | 4.5 mH |
| DC-link capacitance | \(C_x\) | 940 μF |
| Open-circuit PV voltage | \(U_{oc}\) | 160 V |
| Maximum power point voltage | \(U_{mpp\_x}\) | 135 V |
| Rated active power per unit | \(P_x\) | 750 W |
5.1 Balanced power operation
The system was first tested under balanced power conditions. During the initial 0.2 s, the DC-link voltages gradually decreased from their open-circuit values to the MPPT voltages, and the PV power reached the maximum output. The grid current THD was measured as 0.69%, demonstrating excellent steady-state performance and fast dynamic response of the proposed control scheme.
5.2 Unbalanced power operation
In the unbalanced scenario, the system was operating with balanced power until \(t = 0.3 \, \text{s}\). At that moment, the equivalent resistors of units 2–9 were changed to reduce their powers from 750 W to values between 650 W and 300 W, while unit 1 remained at 750 W. This severe unbalance caused units 1 and 2 to violate the stability condition (14). Figure 11 in the original paper compares the responses without and with reactive power compensation. Without compensation, the DC-link voltages of the low-power units undergo drifting, and after about 0.1 s all units reach the MPPT points, but units 1 and 2 enter over-modulation. The grid current THD is 16.35%, far exceeding the acceptable limit. With the proposed reactive power compensation, the THD is reduced to 1.82%, satisfying grid-connected inverter standards.
5.3 Effect of reactive compensation on the voltage loop
To isolate the influence of reactive power injection on the voltage loop, a separate test was performed in which the inverter was running normally and a step reactive power command was applied at \(t = 0.3 \, \text{s}\) without changing PV powers. The optimized method produced no visible disturbance on the DC-link voltages, while the conventional method caused a temporary power drop of about 6.67% (50 W per unit) before recovering after 150 ms. These results confirm that the proposed control scheme successfully eliminates the coupling between the reactive power path and the voltage regulation loop.
6 Experimental Validation
A laboratory prototype of a 3-unit CHB solar inverter was built to further validate the proposed strategy. The experimental parameters are listed below.
| Parameter | Symbol | Value |
|---|---|---|
| Number of cascaded H-bridge units | n | 3 |
| Grid voltage (RMS) | \(U_s\) | 220 V (mains) |
| Filter inductance | \(L_s\) | 4.5 mH |
| DC-link capacitance | \(C_x\) | 470 μF × 2 |
| Switching frequency | \(f_s\) | 20 kHz |
The PV panels were emulated by voltage sources with series resistors. The parameters for the three units are given in the following table.
| Parameter | Unit 1 | Unit 2 | Unit 3 |
|---|---|---|---|
| Open-circuit voltage \(U_{oc}\) / V | 160 | 165 | 165 |
| Maximum power point voltage \(U_{mpp\_x}\) / V | 135 | 142 | 142 |
| Rated active power \(P_x\) / W | 750 | 750 | 800 |
6.1 Balanced power experimental results
Under balanced operation, the DC-link voltage of unit 1 showed a 100 Hz ripple around its reference of 135 V, which is inherent to single-phase systems. The grid current THD was measured as 1.71%, while the grid voltage THD was 4.19%. The current was in phase with the voltage, confirming the unit power factor operation.
6.2 Unbalanced power experimental results
In the unbalanced test, the internal resistors of units 1 and 2 were changed to reduce their powers from 750 W to 350 W and 380 W, respectively, while unit 3 remained at 800 W. Without reactive compensation, unit 3 violated the stability condition and entered over-modulation, causing severe current distortion with THD as high as 36.23%. With the proposed reactive compensation, the grid current THD was reduced to 3.64%, and the DC-link voltage of unit 3 remained stable at around 135 V. These results clearly demonstrate the effectiveness of the proposed method in extending the stable operating region of cascaded H-bridge solar inverters.
6.3 Voltage loop decoupling verification
The decoupling effect was verified experimentally by applying a reactive power step without changing any PV power. In the pre-optimization scheme, the DC-link voltage experienced an upward transient of 6.5 V and required about 240 ms to settle back to the MPPT point. In contrast, the optimized control scheme produced no noticeable transient on the DC-link voltage, confirming that the reactive compensation is transparent to the voltage loop.
7 Conclusion
This paper has presented a comprehensive reactive power compensation strategy for cascaded H-bridge solar inverters to overcome the detrimental effects of PV power imbalance. The main conclusions are as follows:
1) The stable operating region of a CHB solar inverter under unit power factor is analytically derived. The condition given by (14) clearly indicates when reactive power compensation becomes necessary.
2) The proposed reactive power distribution algorithm assigns active power proportionally and reactive power according to the maximum allowable modulation index of each unit. Consequently, the system remains within the linear modulation range even under severe power mismatch.
3) The optimized control architecture decouples the reactive power injection from the voltage loop, preserving the physical meaning of the voltage controller output and simplifying the controller design. Both simulation and experimental results confirm that the voltage loop is no longer affected by reactive power compensation.
4) The effect of reactive power on DC-link voltage stress is quantified. Under extreme unbalance, the voltage stress increase remains within acceptable limits for practical design.
The proposed method is simple, effective, and easily implementable in existing digital controllers. It offers a practical solution for expanding the stability boundary of cascaded H-bridge solar inverters, thereby improving their reliability and power quality in real-world photovoltaic applications.