Active power control strategy based on characteristic curve fitting for photovoltaic sources

With the explosive growth of photovoltaic (PV) penetration, there is an increasingly urgent expectation for PV sources to provide a full range of ancillary services to the utility. The most critical issue for a PV source to support the power system is the flexible power regulating capability, whereas the PV sources generally operates in the maximum power point tracking (MPPT) mode to maximize the solar harvest. This paper proposes an active power control strategy for PV sources based on a characteristic curve fitting method. With the proposed control strategy, PV sources can operate in the MPPT mode or adaptively switch to the power dispatch mode to flexibly regulate its output power to track the power reference. The characteristic curve fitting method does not require an irradiation sensor, but it has a more stable performance under rapid and continuous change in irradiation with simple calculation and error control since its extremely high similarity with the P–U characteristics of the PV source. Simulation is performed based on a detailed PV dynamic model to validate the proposed control strategy.


INTRODUCTION
For recent years, the photovoltaic (PV) generation is experiencing explosive growth.Popularly, the PV sources are forced to work in the MPPT mode to maximize the economic benefits.However, as the PV penetration continuously increasing, the overall system faces a series of challenges, [1][2][3] including the overvoltage, the inertia reduction, the voltage/frequency deviation, and so forth.If the PV penetration rate is extremely high (e.g., in a microgrid or a small regional distribution network), the MPPT mode may severely weaken the system strength, even resulting in frequency or voltage instabilities.Thus, flexible power point tracking (FPPT) becomes a fundamental and essential capability for PV sources connecting to the weak grid environments.5][6][7] Furthermore, in many countries' grid codes, the PV generation is forced to meet several demands for system safety and stable operation. 8Traditionally, the energy storage system takes the role of voltage/frequency support.But the installation of energy storage involves high operation and maintenance costs.
In addition, the storage units' life is usually shorter than that of PV panels.Reference 9 points out that the FPPT is more cost-effective than that of installing of the energy storage.However, the FPPT for PV sources is usually neglected before, which may reduce the system dependence on energy storage facilities and provide fundamental control means for PV sources to provide voltage/frequency regulation support.
][17][18][19][20][21] Both of them make the PV source operate below the maximum power point (MPP) to achieve the adjustment flexibility.The PRC reserves a certain proportion of PV generation (e.g., generates 80% of its maximum available power while 20% is reserved) thus to maintain an adjustment margin for frequency/voltage regulation and inertial/damping imitation.Accordingly, the control target for PRC is the generation/reservation ratio but not the accurate power dispatch reference.The PV source in PRC deviates from the MPP to keep a certain power margin to deal with the voltage/frequency fluctuations.While for the APC, the control targets are quite different since the main purpose of it is to regulate the PV output power according to the power dispatching instruction.The power dispatching instruction may come from the superior operator or just from the outer energy control loop of the PV sources' local controller, like the f -P/V-P droop controller, which traditionally needs to be applied to adjustable resource such as the storage battery.Thus, the APC enables PV sources to behave as traditional adjustable voltage-sources to actively support the system voltage/frequency.In addition, another important difference is that the APC does not require PV sources to work in the de-loading mode all the time to keep the active support capability.This paper focuses on the APC approach.
The most critical problem for the active power control of a PV source is how to accurately determine the MPP of the PV array since there are two different operating points corresponding to one same output power in the P-U curve, except for the MPP.The two respective operating points exhibits diametrically opposed control characteristics, which makes it hard to design a stable power regulator for the PV source without the information of the MPP.There are two kinds of solutions for the aforementioned concerns: by MPP searching schemes or by MPP estimation methods.For the former, Wang et al. 16 propose a reference power tracking strategy in which a mode detection and switching module is used to decide the location of the PV operating point and accordingly switch between the proper voltage regulators.This solution is straightforward but the transition impact during the regulator switching may degrades the performance.Li et al. 17 introduce a sliding mode control based adaptive power point tracking (APPT) strategy.The sliding surface is adaptively adjusted to regulate the PV output power.However, the APPT strategy is applied to the DC/DC converters for two-stage PV sources and the sliding mode controller design is relatively complicated.Sangwongwanich et al. 18 put forward a modified P&O method to ensure a fast and smooth transition between maximum power point tracking and constant power generation.It can regulate the PV output power according to any set point, and force the PV systems to operate within the left side of the MPP without stability problems.The efficiency of the algorithm is improved compared with traditional P&O algorithms but is still need several sampling points to get convergence.
In addition to the above methods, an alternative methodology for APC is to acquire MPP by estimation, and there two major types: the sensor-based ones and the sensorless ones.For the former kind of solution, in Reference 19, a fitting function for MPP with respect to the solar irradiation and temperature is obtained by offline data training.The MPP can be calculated with the irradiation and temperature information.In Reference 20, a simplified PV array model is proposed, and measurements of solar irradiation and temperature information are substituted into the predefined model to estimate the MPP.However, these sensor-based methods increase the costs of installation and maintenance, which restricts their application, especially for distributed PV sources.For the sensorless strategies, PV voltage and current measurements are utilized as the only estimation information for the MPP.Liu et al. 21use a quadratic curve to fit the PV P-U characteristic and arbitrary operating point including the MPP can be calculated by the interpolation methodology.The algorithm has relatively poor robustness and might cause transient problems when the irradiation or loads change rapidly and dramatically since the quadratic curve is quite different from the PV P-U curve.In Reference 15, a single-diode mathematical model is employed and the MPP is estimated by the means of least squares curve fitting.However, this method required PV array modeling parameters and solution of complex Lambert-W function.
In summary, the existing APC strategies have some shortcomings: some need extra irradiation and temperature sensors and some involve complex estimation models, and some are susceptible to large disturbance.Furthermore, rapid and continuous irradiance or load changes diminish the accuracy and stability of the existing APC strategies.In this paper, a characteristic curve that is very similar with that of a PV's P-U curve is introduced to improve the transient performance.
Based on the characteristic curve fitting method, a unified APC and MPPT strategy is proposed to achieve flexible and reliable power control of PV sources.Main contributions of this paper are as follows: 1. Compared with existing sensor-based APC algorithms, the proposed method has the advantages of not requiring the irradiance or temperature sensors.2. The characteristic curve introduced in this paper enhance the algorithm's robustness and reliability.The transient performance under large disturbances is improved.3. The proposed method unified the APC and MPPT with one same control configuration and implements automatic and seamless transition between the two modes.
The rest of this paper is organized as follows: In Section 2, the principles of the proposed APC strategy are elaborated.Section 3 demonstrates the specific process of the proposed algorithm and discusses the details.Simulation studies are performed in Section 4 to validate the effectiveness and feasibility of the proposed method.Section 5 concludes the paper.

Single-stage PV generation system
The single-stage PV-inverter topology is adopted in this paper for its merits of saving power conversion stages which reduces the costs.The topology of a typical single-stage PV generation system and its control block diagram is illustrated in Figure 1.The DC/AC inverter, hereafter referred to simply as the inverter, regulates the operating point of the PV array by adjusting the PV output voltage.The upper control level (power control level) gives the PV voltage control command V pv_ref to regulate the PV output power.The classical dual-loop control scheme is adopted for the lower control level to force the inverter to track with V pv_ref .The grid voltage vector oriented dq-frame is employed for the inner dual-loop who adopts the same philosophy to control the reactive power, by which ancillary services such as voltage supporting and oscillation damping can be implemented.
For PV arrays composed of single-diode module, the mathematic model of its I-V characteristics is given by where, I and V are the PV current and voltage, respectively; N p and N s are the number of parallel and series connected PV panels, respectively; I ph is the light induced current depending on the irradiance level G and the temperature T; I 0 is the diode saturation current; R p and R s are the parallel and series equivalent resistance of the PV panel, respectively; a is the ideality constant of the equivalent diode and V t is thermal voltage of a PV panel.The dynamic equations for PV arrays present non-linear characteristics and the PV generating can be expressed as where, f pv is the function of I-V characteristic of the PV array.Figure 2 shows the P-V curves of a PV array under different irradiation and temperatures.As shown in Figure 2, P pv changes with the variety of V. When V is zero, P pv is zero, who increases with the increasing of V.At the top of P-U curves, the PV output power reaches its maximum available value, which is commonly called the MPP.The corresponding PV array voltage is called the MPP voltage.Beyond the MPP voltage, the PV output power decreases with the continuous increasing of the PV array voltage.The PV power decreases to zero when the PV voltage reaches V oc , which is often called the open-circuit voltage.This characteristic implies that we can control the PV output power by regulating the PV array voltage V.
Two quite complicated issues pertaining to the PV output power control are: (1) the P-U curve varies when the irradiation and temperature changes, which means that the PV power operating point varies under different environmental conditions; (2) there are two different operating voltages corresponding to the same PV output power.Thus, the key point for PV power control loop can be expressed as: For Arbitrary power dispatching reference P ref : If P ref is lower the maximum available PV power, then find the proper PV voltage (whether lower than the MPP voltage or higher than it but maintaining consistency) to make the PV output power P pv equal to P ref .While if P ref is higher the maximum available PV power, then find the MPP voltage.In other words: the main task for PV power control is to find a PV operating voltage V * that satisfies: where, P MPP is the maximum available PV power.As stated before, it is hard to obtain f pv so that to solve the f pv (V * , G, T) = P ref for V * .A feasible solution is to use the iterative method based on the curve fitting, such as quadratic interpolation.This paper will give a characteristic curve fitting method to achieve better performance.A more similar fitting curve may improve the robustness.

Principles of the proposed characteristic PV curve fitting method
Considering two intersections with V-axis and the maximum point in the P-V curve of a PV array, a characteristic fitting curve is given as Characteristic fitting curves with three different parameter sets.
where, a 1 , a 0 , b 1 and b 0 are the fitting coefficients.Figure 3 shows the characteristic fitting curve with different fitting parameters.
Only one point is sampled for per iteration.Consider the kth step in the iteration process and assume that there are already at least four sampling points on the P-U curve.The power and voltage values of the four sampling points are noted as . The overall strategy of the proposed method is to use the characteristic curve which passes the four sampling points to imitate the real P-U curve and by continuously iterating to make it locally converged to the target operating point, either the MPP or the power dispatch point.Once at least four sampling points are ready, the following equations are derived from Equation (4): Rewrite Equation (5) in the matrix form: where The fitting coefficients can be then solved by s = A −1 ⋅ B or by least square method if more than four sampling points are kept.
Once the valuables in s are get, substitute P ref into Equation ( 4) and rearrange it, then we get: which can be used to estimate the PV reference operating voltage.Equation ( 7) can be regarded as a quadratic equation with V being the argument.Define Δ < 0 means that P ref has no intersections with the fitting curve which indicates that the maximum available power is less than the power dispatch reference P ref .Under this circumstance, the PV source needs to operate in the MPPT mode as previously discussed and the maximum point of the fitting curve is employed for the next iteration step.The maximum point can be obtained by finding the zero point of the derivative of Equation ( 4).The MPP voltage can be approximated as Δ ≥ 0 means that P ref is less than the maximum available PV power and the PV source now needs to operate in the power dispatch mode by regulating its output power to P ref .
There are two solutions of Equation ( 7) and both of them can be easily get by solving the quadratic equation.The left-hand side solution (denoted by LHSS) is derived as whereas the right-hand side solution (denoted by RHSS) is derived as Many literatures discuss the characteristics of the two solutions for PV voltage regulating.But it is beyond the scope of this paper.The solution at any side can be selected to approximate the desired PV operating voltage for different control requirements.However, there are still some issues between the two solutions to be noted.

Selection between the LHSS and the RHSS
In References 22-24, the LHSS is looked as a non-preferred solution for PV voltage regulation for its poor performance in both steady state and transient processes.As in this paper, any side of solutions can be selected to estimate the desired PV operating voltage since the focus of this paper is on the versatility and simplicity of the proposed APC algorithm.
In addition to the above discussions, it is worthy to be noted that if the power dispatch reference is very low, the LHSS is prone to be lower than the safe DC-side voltage of the inverter, which means that the LHSS may be lower than the peak voltage of the AC-side, which leads to the inverter failure.While for the RHSS, this will never occur because the RHSS will locate within the range between the MPP voltage and the open-circuit voltage of the PV array.
No matter which of the LHSS and the RHSS is chosen for the approximation of the desired PV operating voltage, there are boundaries of the PV operating voltage.A voltage that is too close to the open-circuit voltage, as well as a voltage that is too close to the peak voltage of the AC side of the inverter can both cause control problems.To prevent these, a saturation function needs to be introduced to keep the calculated voltage within the acceptable range: where, V min and V max are the predefined constants as the lower and upper boundaries of the PV operating voltage.
The lower boundary V min can be set as the minimum DC side voltage allowed by the inverter plus proper margin, whereas the upper boundary V max can be set as the open-circuit voltage of the PV array minus proper margin.
The control performance of a PV array declines sharply when the operating voltage approaches near the open-circuit voltage.Thus, there is a compromise between control performance and the PV operating range to set up the upper boundaries.

SPECIFIC PROCESS OF THE PROPOSED ALGORITHM
For initialization, at least four different points need to be sampled.The four operating points can be carefully arranged to be evenly distributed on the both sides of the MPP, which may speed up the convergence of the algorithm since the MPP is the maximum point.Thus, the initialization sampling points that distributes on the both sides of the maximum point accelerate the stable convergence of the proposed algorithm.It is worth to mention that the proposed algorithm has no strict requirements of the four sampling points for the initialization, arbitrary four sampling points, for example, four random points within the acceptable operating range, can be employed for the algorithm initialization.
For the kth step, by the solutions of Equation ( 9), or Equation (10), or Equation (11), a new V k+1 pv_ref can be now calculated, with Equation ( 12), the reference is fed to the inner control loops of the PV inverter to track with.For the (k + 1)th step, the corresponding PV voltage and current can be sampled and P k+1 pv is calculated.By far, the information of five points, namely (V k+1 pv , P k+1 pv ), (V k pv , P k pv ), (V k−1 pv , P k−1 pv ), (V k−2 pv , P k−2 pv ), and (V k−3 pv , P k−3 pv ) have been obtained.To launch the next interpolation, one point should be removed from the four.The rule of iteration is to remove the most previous point for its presentation of the information that are sampled at the longest time before as the irradiation and the temperature may change.Then, a new V k+2 pv_ref can be calculated following the same procedure.As k increases, the iteration continues and the approximation gradually approaches the desired steady value.The iteration is considered converged if there is only small difference between P k+1 pv and P k pv , P k−1 pv , P k−2 pv , P k−3 pv .The following condition needs to be satisfied: The P-U characteristic curve varies as to the irradiance and temperature.Once the ambient environment conditions change, the algorithm quits the converged state and begins the iteration process towards the new steady state.The theoretical proof that the proposed algorithm converges to the desired operating point can be observed in literatures: when operating in the MPPT mode, the algorithm can be viewed as a maximum problem, 25 whereas when operating in the power dispatch mode, the algorithm can be viewed as a nonlinear equation solving problem with iterative methods. 26igure 4 shows the flowchart of the proposed APC strategy.The value of sampling and calculation interval T s needs to be carefully tuned in coordination with the PI parameters of the inverter's inner control loops.Specially, if T s is too small, the bandwidth among different control loops is mismatched and the controller may subject to oscillation or even failure.On the other hand, a too large T s may lower the convergence rate.A proper T s means that there is enough time for V pv to track with V pv_ref , as well as achieving an acceptable converged rate.

F I G U R E 4
Flowchart of the active power control strategy for PV system.

SIMULATION STUDY
To verify the proposed APC strategy, a detailed dynamic model of the single-stage PV power system is established in Matlab/Simulink.The PV system capacity is 100 kW, connecting to a 380 V external utility.The parameters of the DC/AC inverter, the PV array, and the inner control loops are listed in Table 1.Here, the PV module modeling is based on real commercial products and the data come from SAM distributed by the National Renewable Energy Laboratory.In order to make a comprehensive and in-dept investigation and evaluation of the proposed strategy, four different scenarios are tested, including a comparative study with the existing representative curve fitting based method proposed in Reference 21.

MPPT mode
This section is intended to test the MPPT function and the adaptive mode switching of the of the proposed control strategy.In this case, the solar irradiation is set to be changed in steps and the power dispatch reference is set to 120 kW, which is higher than the maximum available PV output power under 1200 W∕m 2 (101 kW) to force the PV source to work in the MPPT mode.As the influence of temperature on the P-U curve is trivial, it is kept constant (25 • C in this paper) and only the influence of solar irradiance is studied.Initially, the irradiation is set to 1000 W∕m 2 and it is changed to 800, 600, 500, 100, and 1200 W/m 2 by steps.100 W∕m 2 is regarded as an extreme condition and the step from 100 to 1200 W/m 2 is enough to validate the effectiveness and robustness of the proposed scheme.Figure 5 shows the output power of the PV source is nearly the same as their ideal maximum output power at all the different irradiation conditions.

Power dispatch mode under different irradiation
In this case, the power dispatch reference is fixed at 68 kW and the temperature is still set to be 25 • C. Initially, the irradiation is set to be 1000 W∕m 2 and it is changed to 900, 800, 700, and 800 W/m 2 by steps.As discussed before, the maximum PV power is 69 kW at 800 W∕m 2 .Thus, the operation of the PV source in this case can be divided into three periods.
In period I (0-17 s), the maximum available power is 85, 77 and 70 kW respectively for 1000, 900, and 800 W/m 2 , which are all higher than the power dispatching reference 69 kW.Thus, the PV source runs in the power dispatch mode whose output power is regulated at 68 kW.In period II (17-22 s), at the time of 17 s, the irradiation steps to 700 W∕m 2 and the maximum PV power is 60 kW, which is lower than 68 kW, thus the PV source switches to the MPPT mode with its output power being kept at 60 kW.In period III (22-27 s), at the time of 22 s, the irradiation recovers to 800 W∕m 2 and the PV power dispatch reference becomes re-lower than the maximum available PV power.The PV source switches to power dispatch mode automatically and regulates its output power to 68 kW. Figure 6 shows the test results.

Power dispatch mode under different power reference
In this case, the irradiation and temperature are set to 1000 W∕m 2 and 25 • C, respectively.Initially, the power dispatch reference is set to be 100 kW to make the PV source work in MPPT mode since the maximum available power at 1000 W∕m 2 is 85 kW.At the time of 7 s, this reference drops to 80 kW by step, which is obviously lower than the maximum PV power and thus the PV source switches into the power dispatch mode.The PV output power is 80 kW during this period by tracking the power dispatching reference.At the time of 12 s, the power reference further reduces to 70 kW, the PV source still operates in power dispatch mode and the output power is regulated to 70 kW.At the time of 17 s, the power dispatch reference steps back to 100 kW, the PV source comes back to MPPT mode, which are consistent with the expected results.Figure 7 shows the test results.The output power changing transient processes all end within 0.5 s and the power overshoot and disturbances are small.

Comparative study with classical PV active power control method
In this case, a comparison between the classical PV APC method and the proposed method is carried out under the same external and experimental conditions.The irradiation, temperature and the load conditions are all the same for two different controllers.Furthermore, to make the comparison mare reasonable and significant, the two controllers use the same inner PV voltage control loops and are designed to take the same sample time.In summary, only the control method is different.Figure 8 shows the dynamics of PV output power under the classical and proposed control method.The testing conditions of step-changed irradiation and PV power reference used in Sections 4.1 and 4.2 are adopted to ensure a fair comparison.Initially, the irradiation is 1000 W/m 2 , it drops to 100 W/m 2 by step at the time of 3 s.Then the irradiation raises up to 1200 W/m 2 at 5 s and then recovers to 1000 W/m 2 at 7 s.The PV active power reference value changes from 100 to 70 kW at 9 s.The simulation results indicates that the proposed method shows better transition performances since significant less disturbances are observed and smoother PV power curves are obtained, especially in the MPPT scenarios.What is more, it is worthy to note that in our test, for fairness, both methods take the same sample time of 0.02 s, which is the limit for the classical method.The control performance is unacceptable if the classical method takes 0.01 s as the sample time while for the proposed characteristic curve fitting method, shorter sampling intervals can be used to further accelerate the convergence rate.

CONCLUSION
In this paper, a novel APC strategy is introduced for a single-stage PV source to regulate its output power flexibly.With the proposed control scheme, the PV source can either work in the MPPT mode or power dispatch mode adaptively according to the power dispatch reference.The core element and novel feature of the proposed strategy is the application of a characteristic curve which is highly similar with the P-U curve of the PV sources.Therefore, a more satisfactory and stably performance is achieved.Simulation results illustrates that under rapid and continuous change in irradiation and power dispatch reference, the scheme still keeps a fast response with high accuracy, meaning that frequency and voltage supporting services may further implemented with the proposed active PV power control strategy.Although the test results demonstrate that the proposed APC strategy shows satisfactory and desirable convergence rate, four sampling points still make it require high performance in the controller's calculation capability, which could be further simplified.In addition, the application to voltage/frequency support by PV sources with the proposed APC strategy is the future research direction that needs to be further investigated.

1
Topology and control block diagram of a typical single-stage PV system.F I G U R E 2 P-V curves under different irradiation and temperature.

T A B L E 1
Parameters of the tested PV generation system.Symbol Definition ValueK i_PProportional gain of inner current loop 3Ki_I Integral gain of inner current loop 200 K v_P Proportional gain of outer voltage loop 1 K v_I Integral gain of outer current loop 10 T s Sample interval of the APC controller 0.02 s F I G U R E 5 PV output power with step-changed irradiation and fixed power reference.

F I G U R E 6 F I G U R E 8
PV output power with step-changed irradiation and fixed power reference.FI G U R E 7 PV output power with step-changed power reference.Dynamics of PV output power under different classical and proposed control method.