Design and Control of an Enhanced Grid-Tied PV CHB Inverter

Abstract

This paper deals with the design and control of an enhanced grid-tied photovoltaic (PV) cascaded H-Bridge (CHB) inverter, which suffers from issues related to operation in the overmodulation region in the case of a deep mismatch configuration of PV generators (PVGs). This can lead to reduced system performance in terms of maximum power point tracking (MPPT) efficiency, or even instability (i.e., a lack of control action). The proposed solution is to insert into the cascade a power cell fed by a battery energy storage system (BESS) with the aim of providing an additional power contribution. The latter is useful to reduce the modulation index of the cell, delivering more power than the others when a preset threshold is crossed. Moreover, a suitable hybrid modulation method is used to achieve the desired result. A simulated performance in a PLECS environment proves the viability of the proposed solution and the effectiveness of the adopted control strategy.

1. Introduction

The achievement of the goals set for energy transition requires an ever-increasing share of renewable sources in the energy mix [1]. This has directed research interest towards energy conversion circuits enabling the integration of renewable energy into the grid with high efficiency, both for low-power and medium-to-high-power systems.

In particular, solar photovoltaic (PV) technology is the most suitable for distributed generation, even on a small or medium scale. This makes it the main source of renewable energy for enhancing the local self-consumption of electric energy by turning consumers into prosumers.

In fact, grid-connected PV systems account for a major part of the current installed capacity, driving the continuous evolution of power converters to achieve high efficiency and output current quality, transformerless operation, compactness, reliability, and maximum power harvesting.

In this scenario, multilevel inverters represent a viable solution to meet the idea of the Distributed Power Generation System (DPGS), which is related to the concept of a “smartgrid” where the production of electricity is no longer centralized, but also carried out by small-scale energy systems installed near the energy consumer. If properly adopted, this approach could help strengthen grid resiliency, decreasing the environmental impact of electricity generation while increasing energy efficiency.

Among the different multilevel architectures, the cascaded H-Bridge (CHB) inverter has become the most used circuit topology in PV application thanks to its modular nature based on a multi-input single-output (MISO) configuration [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21]. The multiple inputs permit inserting a single dc source (e.g., a panel or a string) into each inverter cell, thus increasing the granularity level of the PV system. This means that it is possible to perform a dedicated maximum power point tracking (MPPT), thus leading to a distributed MPPT (DMPPT), which has a remarkable impact on the harvesting of the maximum available power in shaded conditions. Further significant features of the CHB are related to the multilevel architecture and are well known: the sharing of the voltage boost among the cells in the cascade (i.e., lower voltage and power ratings for the power semiconductor devices); a multilevel output voltage waveform with a number of voltage steps depending on the number N of cells; and a lower switching frequency for the power semiconductor devices, but with a higher equivalent output switching frequency, increasing with N. These characteristics also provide a better output quality, or rather, reduced total harmonic distortion (THD), weight, and volume of the line filter inductor.

Generally, a PV CHB inverter consists of N HBs in a series connection, thus obtaining a single-stage configuration, even in the case of transformerless grid-connected application. In the case of single-panel application, the dc source of each HB is represented by a single PV module. As aforementioned, all of the PV modules can individually operate at their MPP, thus improving the overall MPPT efficiency. Nevertheless, some limitations must be considered. The first is related to the need for an overall dc-link voltage (i.e., the sum of the single dc-link voltages) that is always greater than the grid peak voltage to ensure the PV active power is transferred to the grid. This translates to a minimum PV voltage reference, thus leading to a reduced MPPT range. This lower limit is also required to prevent the performed voltage range from including the flat region of the I-V curve, which is where the PV generator (PVG) is a constant current source. The second drawback is the inherent variability and the energy production uncertainty of PVGs, which is a challenge to be addressed to guarantee the system’s stable operation. Indeed, the different irradiance and/or temperature of the PVGs determine an uneven power distribution among the cells in the cascade. This power mismatch must be properly considered to ensure that each cell can separately operate at its own MPP. In fact, the cells in the series share the same output current, and the power that each cell delivers is directly correlated with the amplitude of its voltage modulation index [5,10,11,14,15]. Practically, the cells managing higher PV power have higher modulation indices than the others. This behavior is exacerbated by a condition of deep mismatch between the PVGs so that the most powerful cell can reach the overmodulation region (i.e., a modulation index greater than one), also with the possibility of achieving square-wave operation (i.e., a modulation index of 4/π). The latter represents the maximum allowable cell power capability; thus, any further increase in the generated power cannot be managed by the cell. In such a situation, the excess power can only charge the dc-link capacitor, whose voltage increases moving away from the desired reference (i.e., MPPT failure). This means that the control is no longer able to work correctly, thus determining possible system instability. This condition is due to the difference between the input PV power (i.e., MPP power) and the maximum allowable power transmission of a cell corresponding to a modulation index equal to 4/π (i.e., square-wave operation, namely, a modulation waveform no longer expandable) [8]. The maximum reachable modulation index with no detrimental effect on the circuit performance (e.g., MPPT efficiency, output current distortion, dc-link voltage control, etc…) depends on the used modulation strategy [22]. Conventional modulation techniques based on multicarrier PWM (e.g., Phase-Shifted PWM, Level-Shifted PWM), if also modified to address a power imbalance, cannot exploit a modulation index greater than one without this leading to worsened inverter output harmonic content, which is not acceptable [14,15].

Different modulation strategies have been proposed to overcome the power mismatch issue by extending the operating range of the PV CHB inverter [22]. In [11], a reactive power control is adopted, while [23] proposes a discontinuous modulation. Instead, in [8,24,25,26], a harmonic compensation strategy is proposed, while in [9,14,27], a hybrid modulation method is implemented. Among these possible solutions, the hybrid modulation strategies seem to be the ones which better fit the scope of obtaining higher modulation indices up to 4/π, as well as in the case of deep power mismatch with no negative impact on the harmonic content of the grid current and MPPT efficiency (and also if the PVG’s operating voltage moves away from its MPP). In fact, as clearly stated in [27], a higher modulation index allows the overmodulated cell to work closer to the corresponding MPP, thus improving the harvesting of the available power.

From the above discussion, it appears that the first issue to be addressed is the reduced MPPT range due to the single-stage circuit configuration. This limitation can be easily overcome by considering a double-stage configuration [28]. This comprises a boost dc-dc power stage, which ensures MPP tracking with a wider voltage range, avoiding a minimum PV voltage reference. The main drawback of the double-stage configuration is an increased number of components with additional power losses. Nevertheless, it should be noted that the obtained voltage boost allows the reduction in the total cell number N needed for the grid connection.

The second issue to be addressed is related to the overmodulation arising from the power mismatch among the PV modules. In fact, the double-stage configuration does not solve this trouble, which is only transferred to the inverter dc-link.

To solve the problem, this work focuses on the hybrid modulation strategy proposed in [29]. The method relies on a sorting algorithm accounting for the dc-link voltage error (i.e., the difference between the voltage reference and the actual value) to decide which cell must be charged/discharged more than the others. This result is reached at each sorting cycle by considering that only one cell will be in PWM, while the others can be in inserted, ±1 states and/or bypassed, 0 states. This technique permits the most powerful cells in the overmodulation region up to square-wave operation (i.e., a modulation index of 4/π) thanks to the simultaneous use of the PWM, ±1, and 0 modes. However, as previously discussed, the input power can increase more than the maximum allowable cell capability corresponding to a modulation index of 4/π, thus causing a further increase in the dc-link voltage that no longer follows the desired reference. This condition is not acceptable, and corresponds to a loss of dc-link voltage control. To prevent this undesired scenario, a modulation index control is typically embedded in the MPPT algorithm [10]. It provides an upper limit for the possible modulation index whose value depends on the chosen modulation strategy (i.e., in our case, it can theoretically be 4/π). When the modulation index of a cell exceeds the fixed upper limit, the MPPT execution is interrupted while the PV voltage reference is incremented at a fixed amount, thus reducing the cell current/power for the purpose of bringing the modulation index back within the allowed range. Unfortunately, the outcome of this control action is a shift from the MPP voltage, which translates to a MPPT efficiency depletion. To avoid this, one possible way is to reduce the power share of the overmodulated cell by injecting additional power by means of an additional cell fed by a battery. This solution has already been investigated by the authors in [30], where the additional battery cell can provide suitable power as the output of a PI regulator that uses the modulation index error as its input, which is the difference between the actual modulation index and the set limit. This approach increases the total power output from the CHB, helping to lower the modulation index and return it more rapidly to a safe range. Moreover, an appropriate selection of the limit for the battery intervention can prevent the modulation index control embedded in the MPPT algorithm from activating, ensuring that the PV operating point does not stray from the MPP, also in the case of a heavy mismatch configuration. Obviously, the additional power cell increases the circuit’s total cost. Nevertheless, it represents only a small percentage of the entire system, thus not having a significant impact. On the other hand, this disadvantage is offset by the undeniable advantage of recovering the energy produced by the PV panels that would otherwise have been largely wasted. In fact, a mismatch condition between the PVGs is very likely due to different irradiation, aging, and temperatures. Furthermore, the proposed solution is better than the one proposed in [28], where the batteries are integrated into the CHB structure in a split manner at the dc-link of each cell. In fact, although this configuration may solve the problem of overmodulation, it is more burdensome in terms of weight and volume due to the needs of the N batteries and N corresponding bidirectional dc-dc converters. This is reflected in the higher cost and lower efficiency. Moreover, the integration of a single battery into the dc-link of a single cell to avoid an additional inverter stage is not useful for avoiding overmodulation issues. In fact, if the modulation index of a cell becomes greater than the preset limit, the battery shall provide the necessary power to compensate for the overmodulation condition. Therefore, the cell where the battery is embedded shall output the PV power plus the additional battery power. The consequence is that the modulation index of the overmodulated cell reduces, but the cell equipped with the battery, which must manage the additional battery power, could easily reach overmodulation. Consequently, the overmodulation problem is not solved, but transferred to the cell with battery.

In this paper, the system operation principle is highlighted by also discussing the charging/discharging process of the battery. The battery charging can occur during CHB normal operation (i.e., PV uniform condition) when no overmodulation control is activated. This means that all of the power needed to charge the battery is provided by the CHB itself with no grid support. Furthermore, the battery discharges only when its power needs to compensate for the excess power produced by a cell in overmodulation, or rather, the battery cell is not intended to provide power to a local load or to the grid.

Therefore, the main proposal of this paper is the possibility of increasing the total power delivered by the PV cells in the case of a large power imbalance thanks to the presence of an additional battery cell. A simulated system performance obtained in a PLECS environment demonstrates that the well-known limit on power harvesting from the PV cells can be exceeded with the proposed circuit topology and control strategy.

The paper is organized as follows: Section 2 reports the system configuration and its principle of operation, while in Section 3, the modulation technique and control strategy are described. Section 4 shows the numerical results obtained. Finally, conclusions are drawn in Section 5.

2. System Configuration and Principle of Operation

The electric architecture of the system under study is shown in Figure 1. It represents a CHB inverter where the first N cells are supplied by PVGs, while the N + 1-th cell is fed by a battery energy storage system (BESS). Each cell is in a double-stage configuration. The first stage is a dc-dc converter, which represents a boost converter, in the case of the N PV cells, and a bidirectional (i.e., buck–boost) converter in the case of the BESS cell. The boost dc-dc converter in the PV cells performs the tracking of the MPP, while the buck–boost converter in the BESS cell copes with the charging/discharging process of the battery. The second stage is a dc-ac converter in the H-Bridge configuration, which addresses the connection to the grid by controlling the output current.

A more detailed view of the generic power cell is presented in Figure 2. A power decoupling capacitor C_dci is located at each inverter dc-link: v_dci is the dc-link voltage, and i_dci and i_Hi are the dc-dc output current and the inverter input current, respectively. The PV decoupling capacitor is C_pvi, while C_bat is the one for the BESS cell. The PV module output voltage and current are v_pvi and i_pvi. The BESS output voltage and current are v_bat and i_bat.

The inverter output voltage is v_Hi, while v_L and v_grid are the voltage across the output filter inductor and the grid voltage, respectively. The total CHB output voltage (see Figure 1) is

$$v_{inv,tot}=v_{inv}+v_{inv,bat}={\displaystyle \sum _{i=1}^{N+1}{v}_{Hi}}$$

(1)

or rather, the sum of the output voltage of the PV cells and the one of the BESS cell. This distinction will be clearer in the following.

The cells’ output voltage v_Hi can be expressed as

$$\begin{array}{cc}{v}_{Hi}=h_i{v}_{dci}& i=1,\dots ,N\end{array}+1$$

(2)

where the control signal h_i can assume three discrete values based on the HB switching state: +1 (i.e., v_hi = v_dci), −1 (i.e., v_hi = −v_dci), and 0 (i.e., v_hi = 0).

The dynamic behavior of the lossless system can be described as follows:

$${\displaystyle \frac{d{i}_{grid}}{dt}}={\displaystyle \frac{v_L}{L}}={\displaystyle \frac{v_{inv,tot}-v_{grid}}{L}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{N+1}{v}_{Hi}}-v_{grid}}{L}}={\displaystyle \frac{{\displaystyle \sum _{i=1}^{N+1}{s}_i{v}_{dci}}-v_{grid}}{L}}$$

(3)

$$\begin{array}{cc}{i}_{outi}=i_{Hi}+C_{dci}{\displaystyle \frac{d{v}_{dci}}{dt}}& i=1,\dots ,N+1\end{array}$$

(4)

where instead of h_i, it is considered a continuous switching function, s_i; Equation (3) is the voltage–current relationship for the line inductor, while Equation (4) represents the Kirchhoff’s current law at each dc-link node, and i_outi is the output current of the i-th dc-dc converter.

To illustrate the system’s operation, the voltage modulation index of the i-th cell must be introduced:

$$\begin{array}{cc}{m}_i={\displaystyle \frac{{\overline{i}}_{Hi}{V}_{Hi,1}}{{\displaystyle \sum _{j=1}^{N+1}{\overline{v}}_{dcj}{\overline{i}}_{Hj}}}}& i=1,\dots ,N+1\end{array}$$

(5)

where V_Hi,1 is the amplitude of the fundamental harmonic of v_hi, while the average values have been considered for the i-th dc-link voltage and HB output current. Equation (5) derives from an average power balance with a unit power factor in a lossless system without considering the higher-order harmonics. The value of the modulation index is useful to discriminate between a cell operating in the linear modulation range (i.e., m_i < 1) and a cell in the overmodulation zone (i.e., m_i > 1). In this latter case, as mentioned in the introduction, the theoretical maximum upper limit is m_i = 4/π (i.e., square-wave operation) corresponding to the maximum capacity of the cell to transfer the produced PV power, whose further increase causes an uncontrolled rise in the dc-link voltage.

To better understand the system’s behavior, it is useful to distinguish the two main operating conditions: the uniform condition, corresponding to the PV modules subject to the same irradiance level, and the mismatch condition corresponding, to the PV modules under different irradiance levels.

In the former case, no overmodulation occurs. Each PV cell handles the same power, and the battery can be charged. As can be inferred from Equation (5), the power requested by the battery leads to an increase in the modulation indices of the PV cells which, however, remain lower than one. In the latter case, a cell can experience overmodulation due to the uneven power distribution among the PVGs. In this scenario, the battery can be discharged so that the provided power is added to the denominator in Equation (5), lowering the modulation index. A detailed description of the control action and its constraints will be provided in Section 3.

3. Control Section

The block diagram of the adopted control strategy is shown in Figure 3. It can be divided into five sub-groups: the distributed MPPT (DMPPT) control, along with the embedded modulation index control (Figure 3a,b); a centralized grid current control outcoming the voltage references for the modulator stages, including the hybrid modulation scheme, used for the N PV cells, and the sinusoidal pulse width modulation (SPWM) scheme, used for the BESS cell (Figure 3c,d); and a battery charging/discharging control (Figure 3e).

3.1. DMPPT Control

The DMPPT control allows for the individual tracking of each PVG’s MPP by means of a typical perturb and observe (P&O) algorithm. The execution time is T_MPPT = 50 ms, while the perturbation voltage step is ΔV_pv = 0.3 V. The embedded modulation index control (see Figure 4) turns on when a modulation index m_i (i = 1,…, N) result is higher than a preset upper limit m_UL, or if the stability flag is already high and the modulation index is within the range [m_LL, m_UL). In such a situation, the stability flag (flag_stab_i) goes high, thus determining the interruption of the MPPT execution and the flag signal in Figure 4, which represents that the sign of the perturbation is fixed to one, and thus the PV voltage reference is incremented by ΔV_pv. This happens at every iteration of the MPPT algorithm until the stability flag remains high. The latter returns low only when the modulation index becomes less than a preset lower limit m_LL, which is set lower than m_UL by a small amount. In practice, the modulation index control enters a hysteresis cycle. The undesired consequence is that the PV operating point moves away from the MPP, thus worsening the MPPT efficiency. Nevertheless, it is worth noting that the maximum reachable value of the upper limit m_UL depends on the adopted modulation technique, thus influencing the extension of the CHB stable operating range even under deep mismatch conditions. However, one of the objectives of this work is to avoid the activation of this control by compensating for the overmodulation with the power provided by the battery (this concept will be detailed in Section 3.4). Finally, from the PV voltage error, through a PI controller, the switching signal of the boost converter is obtained to properly and individually track the MPP of the corresponding PVG (see Figure 3a).

3.2. Grid Control

The main task of this block is to control the grid current, with the aim of transferring the produced active power from the PVGs to the network with the unit power factor and a low total harmonic distortion (THD) to agree with the grid code. In the cascade configuration, the output current is the same for each cell, so the control loop must be unique for all (see Figure 3c). Firstly, the dc-link voltage error is calculated and then processed by a PI, whose output is compensated by the PVGs and battery contributions to obtain the amplitude I_grid,ref of the current reference. Grid synchronization is achieved by means of a phase-locked loop (PLL) circuit, thus resulting in the sinusoidal current reference i_grid,ref. In particular, the used grid synchronization technique is based on a Second-Order Generalized Integrator (SOGI) PLL [31]. The grid current error (i.e., i_grid,ref − i_grid) is the input of a proportional resonant (PR) controller whose resonance frequency corresponds to the grid frequency, thus removing the alternating steady-state error. The PR controller provides the filter inductor voltage reference v_L,ref, and then the Kirchhoff’s voltage law gives the CHB voltage reference v_inv,ref,tot. Once this total voltage reference is obtained, it is necessary to subtract the portion related to the BESS cell (v_inv,ref,bat) to have the voltage reference v_inv,ref only related to the cascade of the PV cells. In fact, while the HBs of the PV cells are controlled through hybrid modulation, the HB of the BESS cell is controlled thanks to SPWM. The share is calculated based on the battery charging/discharging power, as will be discussed below.

3.3. Modulation Strategy

The adopted modulation strategy provides two different control methods: hybrid modulation for the PV cells and SPWM for the BESS one (see Figure 3d). The used hybrid modulation technique is the one proposed in [29]. The strategy adopts an algorithm for the cells’ sorting based on the dc-link voltage error to determine which cells should be inserted (±1 mode), which should be bypassed (0 mode), and which single cell should be in PWM mode at each iteration. This technique allows the most powerful cells to achieve an overmodulation region up to square-wave operation (i.e., m_i = 4/π) without negatively impacting the output power quality. On the other hand, the BESS cell is not involved in the previous sorting, where it is individually controlled by SPWM, thus not introducing distortion to the CHB output. The next sub-section will explain as the BESS cell inverter voltage reference v_inv,ref,bat is obtained.

3.4. BESS Converter Control

The control of the bidirectional dc-dc converter starts with the battery power ΔP. The value of ΔP depends on the battery operating state, namely, if the battery is in charging or discharging mode. This condition is decided by the flag_bat status, which is established by a comparison between the current modulation index of the PV cells and the preset lower limit m_LL. Consequently, the battery is charged until m_i is less than m_LL (i.e., the PV cells can also be in the overmodulation region (m_i > 1), but in a condition manageable by the hybrid modulation strategy). In such a case, the battery charging power is chosen for a C-rate equal to one. Instead, in cases where the modulation index of a PV cell is to become greater than m_LL, the battery reverses its operation to provide the necessary power to compensate for the overmodulation condition. The choice of the battery action when m_i becomes higher than m_LL is to avoid the modulation index control embedded in the MPPT algorithm from intervening. In fact, as anticipated, this would cause the PV cell to move away from the MPP. Obviously, the power provided by the battery must be carefully chosen. ΔP is calculated through a PI regulator with the modulation index error as the input (i.e., the difference between the highest modulation index m_i,om and m_LL). Once ΔP is identified, the battery current reference i_bat,ref can be calculated so that from the current error, the proper duty cycle can be derived. In addition, it is worth noting that ΔP is used to obtain the BESS cell inverter voltage reference as follows:

$$v_{inv,ref,bat}=v_{inv,ref,tot}{\displaystyle \frac{\mathsf{\Delta }P}{P_{tot}}}$$

(6)

where $P_{tot}={\displaystyle \sum _{i=1}^{N+1}{\overline{v}}_{dci}{\overline{i}}_{Hi}}={\displaystyle \sum _{i=1}^N{\overline{v}}_{dci}{\overline{i}}_{Hi}}+{\overline{v}}_{dc(N+1)}{\overline{i}}_{H(N+1)}=P_{PV,tot}+P_{bat}$.

It is worth noting that the BESS cell inverter voltage reference represents a portion of the total CHB voltage reference. As aforementioned, it is obtained from ΔP, which represents a fixed quantity (i.e., P_bat,charge, see Figure 3e) during the charging process at 1C, whereas it is the output of a PI regulator whose input is the modulation index error (see Figure 3e) during the discharging process. The latter permits it to output the additional power used to compensate for the overmodulation condition. The result is an increase in the total power delivered by the PV cells in the case of a large power imbalance (i.e., the exceeding of the well-known limit on power harvesting from the PV cells in the case of deep mismatch). Finally, it is worth noting that if the battery is not able to provide all of the additional power used to compensate for the overmodulation condition, the effect will be incomplete, thus reducing the obtained MPPT efficiency. On the other hand, if a battery failure occurs, the system will return to the scenario with only the PVG cells, thus leading to a further reduction in the MPPT efficiency.

4. Simulated Performance

A complete set of numerical simulations in a PLECS environment has been performed to prove the viability of the proposed solution to the overmodulation problem, with the aim of enhancing the PV power harvesting. The system architecture comprises N = 9 PVG cells plus one additional BESS cell, thus leading to a (2N + 1) + 1-level CHB circuit. The worst case for a power mismatch scenario determining an overmodulation condition corresponds to a single cell being more powerful than all others which are subjected to the same lower irradiance level [27]. In fact, the modulation index is a monotonic increasing function of the irradiance mismatch. More specifically, positive mismatches (i.e., the difference between the constant irradiance of the more powerful cells and the irradiance of the remaining cells) bring the more powerful cells into the overmodulation region. The growth rate of m_i rises with the increase in the uneven irradiated PV cells. The simulation has a total duration of 2.5 s. Initially, all of the PVGs are under 1000 W/m² (i.e., uniform condition) until 1 s, when the first cell remains subjected to 1000 W/m² while the irradiance of the other N − 1 cells decreases linearly to 542 W/m² over a period of 1 s, thus leading to a positive mismatch, as previously described. This uneven power distribution is then preserved until the simulation ends (i.e., a time of 2.5 s). It represents the deep mismatch condition, which can bring the first cell in the overmodulation region up to square-wave operation (i.e., m₁ = 4/π). The preset upper and lower limits for the modulation index are m_UL = 4/π and m_LL = (4/π − 0.01), respectively. The used values of the main circuit parameters are listed in

Table 1. Circuit parameters.

Parameter	Value
Cdci	6 mF
Cpvi	50 μF
Cbat	100 μF
L	10 mH
RL	0.8 Ω

.

Indeed,

Table 2. Control parameters.

Control Section	Parameter	Value
PV converter	kp	1 × 10−5
(PI)	ki	2
Battery Converter	kp	10
(inner loop PI)	ki	100
Battery Converter	kp	Ptot/mLL
(outer loop PI)	ki	100 × (Ptot/mLL)
CHB Converter	kp	1
(PI)	ki	100
CHB Converter	kp	35
(PR)	kr	2 × 105

reports the numerical values of the control parameters.

The PV panel presents an MPP point at standard test conditions (STCs), with P_MPP = 331.55 W corresponding to V_MPP = 37.6 V and I_MPP = 8.82 A, while the battery has a capacity of 5 Ah, with a rated voltage of 36 V and a total internal resistance of 30 mΩ. The time behaviors of the modulation index, stability flag, battery flag, and MPPT voltage reference of the first cell are shown in Figure 5.

As can be inferred from Figure 5, the battery flag goes high when m₁ > m_LL, thus inverting its operation from charging to discharging mode to provide the suitable power (see Figure 3e) needed to reduce m₁.

Nevertheless, the modulation index m₁ increases again, overcoming the maximum upper limit m_UL = 4/π and thus activating the modulation index control embedded in the MPPT algorithm. The stability flag of cell#1, flag_stab₁, goes high, the MPPT algorithm is stopped for cell#1, and the quantity ΔV_pv = 0.3 V is added to the MPPT reference of cell#1 with the aim of reducing the corresponding current. This behavior determines a drift from the MPP. However, this happens only once thanks to the presence of the battery, which ensures the modulation index is in a safe region, thus avoiding a forced increase in the MPPT reference which would result in a further shift from the MPP (i.e., a reduced MPPT efficiency).

To prove this claim, Figure 6 shows the time behaviors of the modulation index, stability flag, and MPPT voltage reference of the first cell in the absence of the battery (i.e., a CHB with only nine PVGs as dc sources). It can be noted that the stability flag of cell#1 goes high more than once, thus determining a substantial increase in the MPPT reference, which results in a relevant shift from the MPP with a consequent reduced MPPT efficiency. This can be easily deduced from Figure 7, where the average PV power produced by cell#1 is reported in the case of the presence of the additional battery cell and with only PVGs.

It can be noted that when the full mismatch condition is reached at a time of 2 s, the PV power of cell#1 in the case of only PVGs deviates from the MPP, with a lack of possible production with respect to the case of the presence of the BESS cell.

Moreover, Figure 8 reports the dc-link voltages in both cases. It is worth noting that without the battery cell, the dc-link voltage of cell#1 moves away from the desired reference of 48 V. Practically, the dc-link voltage control of cell#1 is lost, such as the system is no longer stably controlled. To better understand this behavior, one can refer to Figure 9, which shows the output voltage v_H1 of the first cell under the deep mismatch condition in the presence of the battery cell (Figure 9a) and in its absence (Figure 9b).

It can be observed that with only the PVGs (Figure 9b), the first cell reaches practically full square-wave operation (i.e., m₁ = 4/π) corresponding to the maximum power transfer capability of the HB. This means that the excess power produced by the PVGs cannot be transferred to the output, and is therefore transferred to the dc-link capacitor, thus resulting in an increase in the dc-link voltage (see Figure 8b). Simultaneously, the intervention of the modulation index control determines a shift in the PV voltage from the MPP (see Figure 6), which translates into a PV power reduction (see Figure 7) until reaching an operating point at a lower current/power that can be managed by the cell. As aforementioned, this can reflect on the dc-link voltage, but in a safe manner. In fact, as can be inferred from Figure 5, when the modulation index of cell#1 is increasing in trend, the corresponding dc-link voltage starts to grow slightly (see Figure 8.b before t = 2 s). Then, the modulation index exceeds the maximum upper limit m_UL, and thus the flag_stab goes high and the MPPT execution is stopped, while the MPPT voltage reference is increased by the fixed quantity ΔV_pv, followed by the PV voltage (see Figure 6). This effect also slightly involves the dc-link voltage, but for the time interval in which the flag_stab remains high, the modulation index returns to the safe region (i.e., lower than the lower limit m_LL) and the effect is canceled.

On the other hand, in the presence of the battery cell, the output of cell#1 does not fully operate in square wave (see Figure 9a), ensuring stable system operation in the case of a deep mismatch as well, thus avoiding a detrimental effect on the MPPT efficiency. All of the above evidences that the considered deep PV power mismatch configuration cannot be handled by the CHB with only PVGs, whereas this is possible thanks to the presence of the battery cell, which ensures enhanced circuit performance.

Finally, Figure 10 shows the grid behavior (voltage and current) along with the inverter output voltage (modulated voltage) during the discharging phase of the battery. It can be noted that the overall system performance is not affected by the modulation index control with the additional BESS.

In fact, the power factor is practically unity, and the grid current total harmonic distortion (THD) is about 1.62%, which is well below the limit of 5% imposed by the grid code.

5. Conclusions

This paper focuses on the design and control of an enhanced grid-tied PV CHB inverter. The main goal is to extend the operating range of the inverter, also in the case of the heavy power mismatch configuration of the PVGs. The circuit enhancement is due to the introduction of an additional power cell fed by a BESS. The latter allows for additional power, which is useful to compensate for the overmodulation condition. In fact, the additional power helps to reduce the modulation index of the most powerful cell, thus ensuring suitable and manageable system operation in the case of deep mismatch as well.

In other words, an increase in the total power delivered by the PV cells is obtained in the case of a large power imbalance, thus exceeding the well-known limit on power harvesting from the PV cells in the case of deep mismatch. In addition, dedicated power management of the CHB with the additional battery cell is proposed.

In addition, a hybrid modulation technique is used to achieve an overmodulation region up to square-wave operation (i.e., m_i = 4/π) without negatively impacting the output power quality, while the proposed control of the battery cell allows it to choose the suitable power to be provided to obtain the desired result. Furthermore, it must be highlighted that the battery cell is not involved in the hybrid modulation, whereas it is individually controlled by sinusoidal PWM, whose voltage reference is properly carried out. In such a way, the additional battery cell participates in the cascade without introducing distortion into the CHB output.

The obtained results, both with and without the additional BESS cell, provide compelling evidence of the superior performance achieved with the proposed circuit and power management method. The integration of the BESS cell not only enhances the inverter’s overall efficiency, but also demonstrates significant improvements in power management, modulation capability, and system reliability. These results clearly prove the effectiveness of the proposed approach, making it a promising solution for addressing the challenges of modern energy conversion systems.