A Decentralized Control Strategy for Series-Connected Single-Phase Two-Stage Photovoltaic Grid-Connected Inverters

: Currently, most of the series inverter control methods rely on communication, which greatly reduces the reliability of the system and increases the cost. To address the above problems, this paper proposes a decentralized control strategy for series-connected single-phase two-stage grid-connected photovoltaic (PV) inverters. By improving the traditional droop control, the frequency and phase of the output voltage of each inverter unit are consistent and self-synchronized with the power grid while relying only on local independent control, and each unit realizes maximum power point tracking (MPPT). At di ﬀ erent light intensities, each series-connected unit can quickly track the new maximum power point while still maintaining self-synchronization between the units. The system stability is analyzed via the small signal method, and su ﬃ cient conditions for system stability are obtained. Finally, the simulation results verify the correctness and e ﬀ ectiveness of the proposed control strategy.


Introduction
Fossil energy such as oil, coal, and natural gas is irrecoverable and will eventually be consumed, while also causing many environmental pollution problems.Therefore, the development of renewable energy such as marine energy, wind energy, and solar energy has received extensive attention from the international community, and its development has been highly valued by all countries.Solar energy has developed more rapidly than other renewable energy sources due to its advantages of large supply, universal distribution, and economic feasibility, thereby promoting the development of the photovoltaic power generation industry, so the structure and control method of the PV grid-connected power generation system has become a research hotspot [1][2][3].
At present, the inverter clusters in most photovoltaic power plants are connected to the AC bus in parallel, and then accessed by medium-or high-voltage transmission lines through the power-frequency step-up transformer.This system structure has the advantages of high reliability, plug-and-play, and high flexibility.However, step-up transformers increase the cost and complexity of the system.Moreover, it is difficult for a single inverter to further increase power and voltage levels.Compared to the parallel-connected inverter structure, the output voltage of each inverter in the series-connected structure is superimposed, which enables the cluster of low-voltage PV inverters to be directly connected into the medium-high voltage power network without the need for a step-up transformer.Therefore, researchers have proposed a lot of series-connected inverter structures, such as the Cascaded H-Bridge (CHB), quasi-Z-source cascade inverter, hybrid cascade inverter, and other topologies [4][5][6][7][8][9].Due to their modular, easily extensible structural layouts, they have been widely used in PV power generation systems.
Currently, most series-connected inverters use centralized control methods [10][11][12].However, centralized control relies on a real-time communication network and a powerful central controller.If a communication failure occurs, it can cause the system to be paralyzed and unable to work properly.In addition, when the amount of series-connected inverters is large and the distance between the units is far, its implementation will become more difficult.
Therefore, researchers developed some interest in the distributed control strategies based on weak communications [13][14][15][16][17]. Study [13] proposes a distributed autonomous control method for series photovoltaic inverters.It designs a fast output active power control loop and a slow voltage control loop, which can enable all PV modules to achieve MPPT.Study [16] proposes an improved distributed power control strategy in which all series-connected inverter units are controlled as voltage sources to regulate the output active and reactive power.In addition, study [17] proposes a current-voltage hybrid control scheme for a series-connected inverter, in which the main inverter is controlled as a current source to regulate the common current of the line, and the other inverters are controlled as voltage sources to establish voltage at the common coupling point (PCC).In this way, the MPPT operation of all series-connected inverter units can be well ensured.The common feature of studies [14][15][16][17][18] is the need for low-bandwidth communication links to transmit grid synchronization signals to each inverter.To save costs and avoid the adverse effects of communication failures, studies [18][19][20][21][22][23] propose decentralized control methods without communication for series-connected inverters.Study [18] uses a P-ω droop control strategy to self-synchronize the output voltage of all series-connected inverter units, but it can only be applied when the line impedance is inductive.Furthermore, the previous work [19] proposes a φ-ω power factor angle droop control strategy, which can be applied to any line impedance characteristics.However, [18,19] are implemented on the premise that all inverter units are the same, and the output voltage amplitude of all units is fixed, so it is not suitable for inverters with different capacities.In study [22], the improved ω-p droop control requires simultaneous active power and inverter DC side voltage, and is influenced by the droop coefficient and PI controller parameters, making the entire system relatively complex.In the same way, the output voltage amplitude of all the units are equal, meaning that the output apparent power of all units is the same.Furthermore, study [23] proposes a decentralized control strategy for cascaded single-stage PV inverters with a MPPT capability, which can realize the series connection of inverter units with different power generation capabilities.It can only be used in single-stage PV inverter series systems.
According to the above analysis, to get rid of the dependence on communication, reduce the complexity and cost of the system, and realize the series connection of inverters with different capacities, this paper proposes a decentralized control strategy for seriesconnected single-phase two-stage PV grid-connected inverters.This strategy enables the frequency and phase of the output voltages of all series-connected units to be consistent and self-synchronized to the grid, and the amplitude of the output voltage is adjustable and proportional to their output active power.The stability of the decentralized control strategy proposed in this paper is proved via small signal analysis, and the system stability conditions are obtained.The simulation results verify the correctness and effectiveness of the control strategy.The control strategy proposed in this paper offers great potential for the application of series PV inverters in large-scale medium or high voltage grid-connected PV power generation; it is possible that this strategy could be applied to utilityscale solar farms in the future.
The following parts of this article are structured as follows.In Section 2, the configuration of the series-connected single-phase two-stage PV inverters grid-connected system is introduced, and the output power characteristic of any inverter unit is formulated.Then, in Section 3, a decentralized control strategy with MPPT ability is proposed.In Section 4, the steady-state analysis of the proposed strategy is carried out, and the possible steady points are calculated.In Section 5, the system stability is verified via the small signal analysis method.In Section 6, the simulation results are provided.In Section 7, the contributions of this paper are concluded.

Equivalent Model of Series-Connected PV Inverters Grid-Connected System
The structure of the single-phase two-stage PV inverters grid-connected system is shown in Figure 1, where N PV inverter units are series-connected.The DC pre-stage of each unit is composed of an independent PV array and a DC/DC converter, which is used to boost the PV voltage and realize MPPT control; the AC post-stage is composed of an H-Bridge inverter module and a LC filter.These PV inverters are expected to transmit their maximum active power to the power grid.From Figure 1, the output power expression of the i-th PV inverter unit is obtained as follows: where Pi and Qi represent the active and reactive power output of the i-th unit, respectively; Vi and δi represent the amplitude and phase angle of the output voltage, respectively; Vg and δg represent the amplitude and phase angle of the grid voltage, respectively; |Zline| and θline represent the amplitude and phase angle of the grid-tied line impedance, respectively, and |Zline| is usually inductive, with θline ≈ π/2; and VP and δP represent the amplitude and phase angle of the common coupling point (PCC) voltage, respectively.The PCC voltage is the sum of the output voltages of the N units, which can be expressed as follows: where N is the number of series-connected PV inverter units.
From Equations ( 1)-( 2), the expressions of Pi and Qi are derived as follows:

Design of the Decentralized Control Strategy for Series-Connected PV Inverters
To make the frequency and phase of the output voltage of each unit consistent and synchronized with the power grid, and to realize the MPPT function, this paper improves the traditional ω-P and V-Q droop controls and proposes a new droop control strategy.The principle analysis is as follows.Because the DC side voltage of the inverter dc u can reflect the power balance of the system, if dc u is greater than its reference value dc u * , it means that the inverter output power is less than the PV array output power MPPT P .On the contrary, if dc u is less than the reference value dc u * , it means that the inverter output power is greater than MPPT P .Therefore, the ωdc u droop control law can be designed as shown in Equation ( 5).Entering the steady state, there is ω = ω*, thus dc u = * dc u .As a con- sequence, the frequency self-synchronization, MPPT, and stable DC side voltage functions are achieved.To keep the phase of the output voltage consistent when each unit outputs different active powers, the V-MPPT P control law is designed as shown in Equation ( 6).It can be seen that the output voltage amplitude of each unit is adjustable and is proportional to its output active power.Because the output current of each unit is the same, their output voltage phases are also the same.
( ) where ωi and Vi represent the angular frequency and amplitude of the output voltage of the i-th unit, respectively; ω* is the angular frequency reference and is set to equal to the normal grid angular frequency, that is, ω* = ωg; dc u and dc u * indicate the amplitude and amplitude reference of the DC side voltage of the inverter, respectively; m is the droop control coefficient; V* indicates the amplitude reference of the AC side output voltage; MPPT i P − is the maximum power provided by the PV array in real-time, and its value is given by the MPPT algorithm; max P is the maximum output power of the PV array under standard test conditions (irradiance of 1000W/m 2 , PV cell temperature of 25 °C), and its value is given by the manufacturer.Obviously, − ≤

MPPT i max
P P ; M is a parameter related to system stability and the power factor, which is discussed later in Section 5.
The control block diagram of the i-th unit in the series-connected PV inverters is shown in Figure 2. The output voltage PVi u and output current PVi i of the PV array are sampled, and the driving pulse signal DC i d − of the DC/DC converter is obtained accord- ing to the MPPT algorithm and PWM modulation, to realize the real-time MPPT of the PV array.Meanwhile, the voltage on the DC side of the inverter, dci u , is sampled, and through the proposed decentralized control strategy, the output voltage reference i i V δ ∠ is generated while dci u is stabilized.Also, the output voltage i u and current i i are sam- pled, then the driving pulse signal i d of the H-Bridge inverter is obtained through volt- age and current dual closed-loop control and PWM modulation, to make sure that i u ∠ well.Finally, the goal of MPPT, frequency and phase self-synchronization, is achieved.

Steady-State Analysis
Since the output current of these series units is the same, the ratio of the apparent power of any two units is equal to the ratio of their output voltage amplitude, i.e., where At the steady state, since i ω ω * = , it can be seen from Equation ( 5) that the DC side voltage of any unit is equal to the given DC side voltage reference: At this point, the output active power of each unit is equal to the real-time power of the PV array, i.e., i M P P T i From Equation ( 6), it can be seen that the ratio of the output voltage amplitude of any two units is equal to the ratio of the maximum power output of the respective photovoltaic array, i.e., From Equations ( 7)- (10), it can be obtained that the ratio of the apparent power of any two units is equal to the ratio of their output active power, i.e., From Equation (11), there comes the conclusion that the output voltage phase of each cell is identical, i.e., δ δ = i j (12) From Equations ( 3)-( 12), the steady-state power transfer expression for each unit can be obtained: δ δ δ = − ).From Equation ( 13), it can be seen that the system has two stable operating points: ( ) ( )

Stability Analysis
To verify the stability of the proposed control strategy, a small signal analysis is carried out.First, the small-signal model of the system in Figure 2 is established.Then, an eigenvalue analysis is performed to assess the stability of the system.Finally, reasonable design rules of the system parameters are discussed to obtain a satisfactory stability margin and dynamic performance.

Small Signal Modeling
The mathematical model of the i-th unit is given as follows: ( ) where i C is the DC side capacitance; PVi P is the output power of the PV array.In the steady state there are PVi P = MPPT i P − and dci u = dc u * ; by linearizing Equation ( 14) near the steady-state operating point, the small signal model is derived as: Simplifying Equation ( 15) yields where Substituting Equation (6) into Equation (3) yields sin sin Then, linearizing Equation ( 17) yields Substituting Equation ( 18) into Equation ( 16) yields The complete small signal model is as follows: where [ ] I × is a unity matrix.

System Stabilization Conditions
Rewriting Equation (20) into the following uniform matrix forms where [ ] The eigenvalues of the system matrix A are ( ) Therefore, the sufficient non-essential condition for system stability is derived as follows: To make the stability condition (24) true, cosδ >0 (i.e., ∈ − ) must be satisfied, and M > N. According to the steady-state analysis above, only the stable point ( ) δ = meets the requirement.It can be seen from Equation (24) that to make the system stability range larger Δ should be as large as possible, so that the eigenvalues of the system matrix A will be far away from the imaginary axis.Since the power angle δ is generally small, i.e., 1 cosδ ≈ , considering the extreme case MPPT i max P P − = , M >> N is required to make Δ as large as possible.However, it can be seen from ( 13) that M >> N will make i Q << 0, causing the system to absorb more reactive power and reduce the power factor.Thus, the design of the M value is a compromise between system stability and power factor.

Simulation Settings
To verify the correctness and effectiveness of the proposed control strategy, simulation tests are carried out in MATLAB/SIMULINK.Based on Figure 1, a PV grid-connected system with the number of series units N = 3 is built, and the corresponding system parameters are shown in Table 1, where p N and s N represent the number of modules con- nected in parallel and series of the photovoltaic array, respectively, and c f is the switch- ing frequency of the inverter.The PV module cells used in the simulation are modeled according to the commercial PV panel 1Soltech-1STH-215-P, and its parameters under standard test conditions (illuminance of 1000 W/m 2 , ambient temperature of 25 °C) are shown in Table 2.The P-V characteristic curve of the PV module is obtained from the Matlab simulation component model and shown in Figure 3, which as can be seen from the figure is max P = (2) Output reactive power.It can be seen from Figure 5 that in the first stage, the reactive power output of each unit is the same, with 620 Var, which realizes the reactive power sharing between these units.In the second stage, at steady state,  (3) Frequency.As can be seen from Figure 6, the output frequencies of all series units converge to the grid frequency (50 Hz) in steady state with a deviation range of no more than 0.05 Hz.The output voltage frequency is self-synchronized with the grid frequency.6) and (10), which are consistent with the analytical conclusions of Section 3. As the output voltage amplitude of each unit decreases, the power factor of the system also decreases, which is reflected in Figure 8 as the phase difference between the output current io and voltage u1 (or u2, u3) becomes larger.(6) Power factor.To verify the relationship between the power factor of the system and the M value, a comparative test is carried out in the first stage of simulation.It can be seen from Figure 9 that when M decreases from 3.2 to 3.11, the power factor increases from 0.949 to 0.991.It shows that the value of M affects the size of the power factor, and the relationship is inversely proportional, which is consistent with the analysis conclusion in Section 5 of this paper.In summary, the simulation results are consistent with the previous analysis in this paper, which verifies the correctness and effectiveness of the proposed decentralized control strategy for series-connected inverters.

Conclusions
In this paper, a decentralized control strategy for series-connected single-phase twostage grid-connected PV inverters is proposed, which only requires local information to achieve a consistent phase and frequency of the output voltage of each unit and self-synchronization with the power grid.Each unit independently implements MPPT with efficiencies ( ) up to 98.6%.Since the output voltage amplitude of each unit is proportional to the actual output power, inverters with different generating capacities can be connected in series.
However, the disadvantage of this strategy is that the power factor of the system decreases when the light intensity is low, because the PV output power reduces at this time, and the inverter output voltage amplitude drops according to Equation (6).Similarly, another restriction is that the amplitude of the power grid cannot change too much, which also brings power factor and stability issues.As M is a constant, when the grid amplitude increases, the power factor becomes lower; when the grid amplitude decreases, the stability decreases.The next step is to study the adaptive adjustment method of the M value based on power factor feedback, and improve the system power factor by reasonably adjusting the value of M.

Figure 1 .
Figure 1.Structure of series-connected single-phase two-stage PV grid-connected system.

Figure 2 .
Figure 2. Control block diagram for the i-th unit.
the power transfer capacity and δ represents the steady-state power angle ( i g

Figure 4 .
Figure 4. Output active power of the three series-connected units.

Figure 5 .
Figure 5. Reactive power of the three series-connected units.

Figure 6 .
Figure 6.Frequency of the three series-connected units.(4) DC side voltage of the inverter.It can be seen from Figure 7 that despite the change in light intensity, the DC side voltage 1 3 dc dc u u of all series inverters can converge to the reference value (200 V) quickly, and the voltage ripple does not exceed 2 V, which verifies the effectiveness of Equation (5) for the DC side voltage regulation of the inverter.

Figure 7 .
Figure 7. DC side voltage of the three series-connected units.

( 5 )
Output voltage and current.The common output current and inverter output voltages at different stages are shown in Figure 8.It can be seen from Figure 8a that in the first stage, because the light intensity is the same, the amplitude (V1 = V2 = V3 = 95.8V) and phase of the output voltages u1, u2, and u3 are the same.In the second stage, only the light intensity of units 2 and 3 change.It can be seen from Figure 8b that the phase of u1, u2, and u3 is still the same, the amplitude V1 remains unchanged, but V2 and V3 have changed, which are given by V1 = 95.8V, V2 = 77.3V, and V3 = 67.6V.In the third stage, only the light intensity of unit 1 changes.It can be seen from Figure 8c that the phases of u1, u2, and u3 are still the same, and the output voltage amplitudes are V1 = 87.5 V, V2 = 77.3V, and V3 = 67.6V.The result data above satisfy Equations ( (a) The first stage.(b) The second stage.(c)The third stage.

Figure 8 .
Figure 8. Steady-state output current and voltage under different operating conditions.

Figure 9 .
Figure 9. Power factor in the first stage with M = 3.11 and 3.2.
s N 4