Optimized Reactive Power Control of Module Power Imbalance of Cascaded Converter

In the medium voltage cascaded converter, the power imbalance problem among the converter modules will result in certain converter modules over-modulated. Reactive power control (RPC) scheme is an effective way to balance the cascaded converters. Many efforts, such as reactive power sharing control and apparent power sharing control have been used to solve the overmodulation issue. This paper investigates an optimization RPC scheme to solve the overmodulation problem by minimizing the total required reactive power. An improved minimum reactive power (IMRP) RPC scheme is presented, the feasible area for the active power imbalance is evaluated and compared with the reactive power sharing control scheme and the apparent power sharing control scheme. Finally, the control scheme is veriﬁed on a three-module scaled-down cascaded inverter.


I. INTRODUCTION
Continuously declining the cost of photovoltaic generation has promoted the rapid growth of PV installations [1], [2]. Solar energy is expected to be one of the dominant energies in the future. To reduce the per-wat cost of PV, the scale of centralized PV inverters is continuously increasing. Nowadays, the DC bus voltage is increased up to 1.5kV, and the power rating of the centralized inverter is increased up to MW. To further increase the inverter power, the MV inverter could be directly connected to the MV utility line. In this way, the bulky 50Hz transformer is needless, which not only saves space but also reduces the standby loss of the transformer.
The PV medium-voltage cascaded converters, which eliminate the bulky low-frequency transformer in conventional centralized PV plants, are potential candidates for nextgeneration utility-scale PV plants [3]- [8]. In the construction of high-efficiency cascaded photovoltaic systems, two conventional structures are the three-stage structure and the two-stage structure, which are depicted in Fig. 1 [9], [10]. In [10], the three-stage structure is used, the active power imbalance problem can be solved by the parallel connection of the first-stage power converter. Compared with the threestage structure, the two-stage structure of Fig. 1(b) is more advantageous on efficiency and cost due to the reduction of power stages.
An important issue in the control system design of twostage converters is active power imbalance. The output power of the PV panels varies with the change of irradiance, temperature, etc. In three-stage converters, the active power imbalance problem is solved by constructing a DC bus to distribute power through power modules. For two-stage converters, the DC bus is unavailable due to the absence of the front-end DC/DC converters. Thus, the active power imbalance is a critical problem for the PV medium-voltage two-stage cascaded converters.
Many sophisticated methods have been proposed to solve the active power imbalance problem. The simplest way is controlling the module voltage which is proportional to its active power [11]. The module with the largest active power will over-modulate easily under the power imbalance condition, as shown in Fig. 2(b). In [12], the voltage balance circuit is inserted between the power module to balance the active power. Some researchers found that the voltage balance circuit will increase the system cost and lower reliability and efficiency. And in [8], the three-port LLC converter is adopted as the isolated stage. A DC bus is built up using the third port  to maintain power balance across modules. Some researchers found that this method needs some additional circuit elements.
Reactive power control (RPC) is an effective way to solve the overmodulation issue, which has an advantage in cost. Literature [13] proposed a control approach that autonomously compensates reactive power contents of inverters caused by the active and reactive power changes by shifting their phase angle with local AC current measurement. In [14], a reactive power control strategy is proposed for improving the converter's adaptability to the power imbalance. In this RPC scheme, the reactive power is evenly distributed to each power module, which is called as reactive power sharing (RPS) scheme. This control strategy is easy to be used, and some researchers found that the maximum power imbalance control adaptability is limited by the same reactive power control for each module. In [15], another RPC scheme is proposed where the apparent power of each power module is equal, which is called as apparent power sharing (APS) scheme. This control scheme efficiently improves the adaptability to the power imbalance, and the required reactive power is higher during some imbalance conditions, which will increase the power losses. In [16], the reactive power for each power module is distributed inversely to its corresponding active power. This control scheme improves adaptability and it also required more reactive power.
Apart from the RPC scheme, other methods, such as common mode injection [17], [18], third harmonic injection [19], [20], space vector modulation [21], or square-wave modulation [22], are also proposed to improve the adaptability of the cascaded photovoltaic converters. Due to the limitation of the modulation ratio, the RPC scheme is a more efficient method for the module active power imbalance problem. The RPC scheme can be easily combined with these modulation schemes [15], [22], [23], which will further improve the adaptability of the cascaded photovoltaic converters.
Two key indexes of the RPC scheme are the tolerance range of active power imbalance and the minimum required reactive power. Less reactive power means lower power losses of the converter, i.e., higher efficiency of the converter. This paper proposes an improved reactive power control scheme of enhancing adaptability to the power imbalance. It is based on the idea of minimum reactive power distributing and adding reactive power calculation control loops for each power module. The proposed scheme is designed to achieve a wider tolerance range and less reactive power. This paper is organized as follows. Section I introduces the background of the active power imbalance control method for the cascade inverter. Section II analyzes the inverter operating feasible area under active power imbalance conditions for the traditional RPC scheme. Section III describes the improved RPC scheme with the minimum required reactive power. Section IV analyzes the inverter operating feasible area for the proposed control scheme. A scaled-down prototype is built up and the experimental results verify the proposed RPC scheme in Section V.

A. THE POWER IMBALANCE PROBLEM
For simplicity, a cascaded PV converter with 3 modules is considered in this paper, as shown in Fig. 1. Assuming the active power, reactive power, and apparent power of each module is P i , Q i , and S i , respectively. The fundamental RMS value element of an output voltage for each module is V si . The fundamental RMS value of the total output voltage is V s . The fundamental RMS value of the grid voltage and current are V g and I g , respectively. P g is the total output active power, Q g is the total output reactive power.
The phasor diagram with P g > 0 and Q g = 0 is depicted in Fig. 2. To distinguish the active and reactive elements of the grid current I g , the d axis is aligned with the grid voltage phasor and the q axis is lagged d axis by 90 degrees. Also, to distinguish the active and reactive components of the output voltage V s , the d axis is aligned with the grid currentİ g and the q axis is lagged d axis by 90 degrees.
The relationship between the total output voltageV s , the grid voltageV g , and the grid currentİ g iṡ where X L = ω g L g . ω g is the angular frequency of the grid voltage, L g is the filter inductor. When power modules don't output reactive power, the output voltage of each power moduleV si is proportional to the corresponding module active power P i , which iṡ When the output power P i is balanced, i.e., P 1 =P 2 =P 3 , the output voltage magnitude of each power module will be the same, i.e., V s1 =V s2 =V s3 . The phasor diagram is depicted in Fig. 2(a). When the output power P i is unbalanced, such as P 1 <P 2 <P 3 , the output voltage V s3 will be the largest magnitude among the three power converter modules. In some cases, when the power imbalance is serious, module 3 will tend to be over-modulated, as shown in Fig. 2(b). The over-modulation might result in instability and higher current THD. Reactive power control [13]- [15] is a good solution to adapt to the power imbalance.

B. ANALYSIS OF THE CONVENTIONAL RPC SCHEMES
In this section, the aforementioned RPS and APS schemes will be analyzed. The feasible area of these two schemes and the required reactive power will be calculated.
The output voltage d components of each module V sid can be calculated as: where V sd is the d components of total output voltage, N is the Module number. By ignoring the impact of the filter inductor in (1), the output voltage q components of each module V siq can be calculated as where V sq is the q components of total output voltage. With (3), (4), the output voltage of the inverter side is For the RPS scheme, the reactive power of each module will be the same, that is [14] N is the number of the power modules. The output voltage of each power module is limited by V max , which is By limiting the voltage of the power module at V max , the required minimum reactive power for the RPS scheme can be calculated from(5), (6), which is where P m is the output active power of the power module which outputs the maximum active power. Besides the limitation of the power module output voltage, the power module apparent power should not exceed the maximum allowable value S max , which is The boundaries of the feasible area for the RPS scheme in (9) are shown in Fig. 3(a). The red line is the output voltage limitation of module 1, the green line is the output voltage limitation of module 2, the blue line is the output voltage limitation of module 3. The feasible area of the RPS scheme is the intersection of the feasible area of three modules. The parameters used in the calculation are listed in Table 1. The active power of P 3 is 70 kW.
For the APS scheme, the module apparent powers are controlled to be the same. The reactive power of each module can be calculated by [15]:   [15], The boundaries of the feasible area for the APS scheme in (12) are shown in Fig. 3(b).
It can be seen from Fig. 3(b) that the feasible area of the APS scheme is different from the RPS scheme.
S re f ≤ S max (12) The minimum required reactive power of the RPS scheme and APS scheme in (8) and (10) are shown in Fig. 4.

III. THE PROPOSED REACTIVE POWER CONTROL SCHEME
Since the basic idea of the proposed scheme is minimizing the reactive power, the proposed scheme is called an improved minimum reactive power (IMRP) control scheme.

A. THE CONTROL SYSTEM OF THE IMRP SCHEME
In order to reduce the minimum required reactive power and extend the unbalanced active power feasible area, this paper proposed an optimization RPC scheme to solve the overmodulation problem by minimizing the total required reactive power and adding a module reactive power calculation control loop. The control system diagram proposed in this paper is depicted in Fig. 5. The proposed control system contains 7 units. The DC voltage control module samples the input DC voltage of each power module V dci and calculates the module active power P i . The module active powers P 1 to P N are then sorted in the active power sorting unit as: The sorted active powers P ki are input to the reactive power calculation unit, where the total reactive power Q g and the reactive power of each module Q ki will be calculated. The Q g and the P g will then be sent to the current control unit. The reference current i dref and i qref will be calculated by P g and Q g . The output of the current control unit is the d -q components of the system output voltage, i.e., v sd and v sq . The module reactive power Q ki is re-sorted to Q i in the re-sorting unit. The P i , Q i , P g , and Q g are used to distribute the v sd and v sq to each of the power modules and the output voltage v si of each power module are derived in the voltage distributor unit. Finally, the v si are passed through the modulation unit and the switching signals are generated to control the power switches. The phase angle of the grid current θ ig is, Where Q g is the total reactive power, P g is the total active power, the grid voltage phase θ vg is obtained through a phase locked loop (PLL).
The key component of the proposed control system is the reactive power calculation module, in which the aggregate reactive power Q g and module reactive power Q i are determined. Q i is the re-sort result of Q ki .

B. THE REACTIVE POWER CALCULATION OF THE IMRP SCHEME
For simplicity, this section considers a cascaded PV converter with three modules to derive the equation of Q g and Q ki .
To avoid over-modulation, the module voltage V i shouldn't exceed the maximum voltage V max .
According to (3) (15): To solve Q g , take the equivalent transformation of (16): When the output power P i is unbalanced, such as P 1 <P 2 <P 3 , the output voltage V s3 will be the largest among the three power converter modules. In the actual operation of the three modules converter, V s3 is most likely to overmodulation. So, when Q g is low, the V s3q is set to zero. Based on the above analysis: The cascaded PV converter has four operating conditions: without over-modulation, slightly over-modulation, moderate over-modulation, and serious over-modulation.
The first operating condition is without over-modulation. When (18) has no real solution, the reactive power injected is needless. In this condition, the active power imbalance is absence. The calculation of Q g and Q ki are shown in (19). The without over-modulation vector diagram is shown in Fig. 6(a).
The second operating condition is slightly over-modulation. In this condition, only the active power smallest module injects the reactive power. The calculation of Q g and Q ki are shown in (20). The slightly over-modulation vector diagram is shown in Fig. 6(b). The module phasor diagram of slightly  over-modulation is shown in Fig. 7(a).
The third operating condition is moderate over-modulation. In this condition, both module 1 and module 2 inject the reactive power. The calculation of Q g and Q ki are shown in (21). The moderate over-modulation vector diagram is shown in Fig. 6(c). The module phasor diagram of moderate overmodulation is shown in Fig. 7(b).
The last operating condition is serious over-modulation. In this condition, all three modules inject the reactive power.
The calculation of Q g and Q ki are shown in (22). The serious over-modulation vector diagram is shown in Fig. 6(d). The module phasor diagram of serious over-modulation is shown in Fig. 7(c).  (23), which is a transcendental equation. This paper uses binary search to solve this equation. The precision of binary search is set as 5%. (23) Extend the formulas of Q g and Q ki to the general case. The cascaded PV converter has N modules and the active power numerical relationship is random. The equations calculating the Q g and the Q ki are: The basic idea of the proposed scheme is to determine the minimum required reactive power Q g and distribute the Q g to each of the power modules in a reasonable way. Specifically, when the power imbalance is relatively slight, the required reactive power Q g is relatively small, thus only part of the power modules are controlled to output reactive power. On the other hand, when the power imbalance is relatively serious, more power modules will be driven to support the increasing reactive power requirement.
The minimum required reactive power is listed in (24). The first line of (24) means that there is no need to output the reactive power. The second line of (24) means that the module which outputs maximum active power doesn't need to output the reactive power, the output voltage of the maximum active power module reaches the maximum value V max . The third line of (24) means that all modules need to output the reactive power with the same apparent power.
The output reactive power for each module is listed in (25). The first line of (25) means that this module does not need to output the reactive power. The second line of (25) means that this module can output the remaining reactive power. The third line of (25) means that this module outputs the maximum reactive power. The fourth line of (25) means that all the module outputs the same apparent power. The flow chart of the reactive power calculation is shown in Fig. 8.

C. THE MINIMUM REQUIRED REACTIVE POWER AND THE FEASIBLE AREA OF THE IMRP SCHEME
The minimum required reactive power of the IMRP scheme is depicted in Fig. 9. It can be seen from Fig. 9 that the reactive power of the proposed IMRP scheme is not larger than both the APS and RPS scheme.
The feasible area of the proposed IMRP scheme is Where the S ref is calculated by (23). The feasible area result for P 1 = 0∼100 kW, P 2 = 0∼100 kW, and P 3 = 70 kW is depicted in Fig. 10 . It can be seen from Fig. 10 and Fig. 3, the feasible area of the proposed IMRP scheme is not smaller than both the APS and RPS scheme.
Compared with the RPS scheme in reference [14] and the APS scheme in reference [15], the advantage of the proposed IMRP scheme is that it minimizes the reactive power required by the system. The reactive power of the proposed IMRP scheme is not larger than both the APS and RPS scheme under all working conditions and the feasible area of the proposed IMRP scheme is not smaller than both the APS and RPS scheme.

IV. EXPERIMENTAL RESULTS
To verify the proposed scheme, a 3kVA three-module cascaded PV converter prototype is built up, the main circuit is shown in Fig. 11. The parameters of the prototype are listed in Table 2. Based on the experimental parameters, the feasible    area of the IMRP scheme is depicted in Fig. 12. The feasible area of the RPS and APS scheme is depicted in Fig. 13. Fig. 14 shows the required reactive power of three PRC schemes with P 2 = 400 W, P 3 = 500 W. It can be seen from Figs. 12-14 that the IMRP scheme has a larger feasible area  and smaller required reactive power than the RPS scheme. It can be seen from Figs. 12 and 13 that the IMRP scheme has the same feasible area as the APS scheme and it can be seen from Fig. 14 that the IMRP scheme has smaller required reactive power than the APS scheme. Three operation conditions, O 1 (P 1 = P 2 = P 3 = 500 W), O 2 (P 1 = P 2 = 250 W, P 3 = 500 W), O 3 (P 1 = 100 W, P 2 = 100 W, P 3 = 500 W) are chosen. The operation curves without reactive power injection (before t 0 ) are shown in Fig. 15(a) and (c). Before t 0 , the system works at balance condition O 1 . At time t 0 , the system is switched to imbalance conditions. The system operation curves under O 2 and O 3 (after t 0 ) are shown in Fig. 15(a) and (c), respectively. It can be seen from the figures that the output current will be distorted under imbalance conditions with Q g = 0. The distortion will become more serious with the imbalance getting worse. The operation curves using the proposed IMRP control scheme are shown in Fig. 15(b) and (d). It can be seen from these figures that the system works normally under imbalanced conditions. The transient performance of the proposed IMRP control scheme is also verified. As can be seen from these figures, the switching interval between different imbalance conditions is shorter than one single fundamental period. In order to verify that the IMRP scheme has a larger feasible area than the RPS and APS, O 4 (P 1 = 800 W, P 2 = P 3 = 500 W) is chosen. As shown in Figs. 12 and 13, only the IMRP scheme can work in this operation. The operation curves are shown in Fig. 16. It can be seen that the system works normally.

V. CONCLUSION
This paper discusses the RPC scheme for controlling the power imbalance of cascaded photovoltaic converter. Based on the detailed mathematical analysis, the proposed IMRP scheme reasonably distributes the reactive power to every power module. The feasible area and required reactive power of the IMRP scheme are also analyzed. The proposed IMRP scheme not only improves the adaptability of the cascaded photovoltaic converter but also reduces the required reactive power. The theoretical and experimental results show that the