Grey prediction-based proportional-integral controller applied to solar energy systems

Proportional-integral (PI) controller with simple architecture as well as convenient design makes it widely adopted in the control of solar energy systems. Due to partial shading as well as fault influence, the solar panels tend to be trapped at regional extremely values, attenuating the dynamic and steady-state response. For the sake of strengthening the system control effectiveness, this paper employs a grey prediction-based PI controller to attain a global maximization of the wattage pointed following in sun power plate, providing a pure sine waveform with lower percentage of total harmonic distortion, in addition to fast transience of the solar energy system output voltage. The PI performs tracking control and simultaneously the grey prediction uses a logistic grey model to accurately predict the system state as well as to adjust the PI control parameters for establishing robust system response. The results reveal that the output of the solar system can produce fast dynamics and high efficient steady-state performance, thereby supporting the theoretical validity.


Introduction
A high performance solar energy system is always one of the most interesting events in the world, which could be done by establishing maximization of the wattage pointed following in the sunlight energies.Thereby there is a need to regulate solar modules output via a power conversion circuit equipped with the maximum power tracking control capability, allowing sunlight modules to provide maximization of the wattage with quick as well as exact tracking [1][2].Proportional-integral controllers have the merits of easy design as well as structural simplicity.Many literatures have been reported regarding the control of solar energy system [3][4][5][6][7][8][9][10][11][12][13][14][15].However, solar energy systems governed by proportional-integral controllers may be affected with uncertainties, like partial shading problem.This prevents quick tracking in wattage pointed maximization, and has a degradation of wattage output, nonzero steady-state error, instability, slow convergence, and the poor performance.Several works have proposed maximization of the wattage pointed following approaches trying to strengthen the solar energy system performance, such as H-infinity control, mu-synthesis method, and repetitive control [16][17][18].These solutions are time-consuming, need lots of sampling data as well as sophisticated computations.Grey logistic prediction shows great success in a variety of fields, where it characterizes and analyzes the system response for future tendency development on the basis of historical as well as existing data [19][20][21][22][23][24][25][26][27][28][29][30].Grey logistic prediction just depends on the system output signal as well as a few numerical data to structure the grey logistic model without complexity of computation and sophisticated mathematical modeling.Grey logistic modeling features simple computation, less information request, which has the ability to improve the forecast accuracy for non-linear data types.Therefore, a mathematically simple as well as computationally effective grey logistic model can address the problem of uncertainties in the solar energy system and permit appropriate parametric adjustment in PI methodology.The union of the grey logistic model together with the proportional-integral controller can lead to the solar energy system with high-quality as well as highefficiency AC output voltage (root-mean-square voltage of 110 V with a frequency of 60 Hz).Numerical simulation results have been presented to validate the strength in proposed technical theory.

System modelling and control method design
As depicted in Figure 1, the power electronic circuit can be used to regulate the solar panel voltage.Through the state-space averaging method, the dynamical equation displayed in (1) of the noninverting buck-boost circuit leads to the following [31][32]: .As the control method is well-designed, (1) is robust where the desired solar output voltage corresponds to the wanted reference voltage.The follow function enables swift as well as precise although sunlight energy may be partly obscured.The control law u by using proportional-integral controller can be formulated as where p K represents the proportional gain, and i K symbols the integral gain.To suppress the phenomenon of nonlinear saturation, the output of the proportional-integral controller is limited as follows: , then it is u .Here the max Ψ shows the limit value, and the ) (⋅ sign stands for signum function.Then as presented in Figure 2, the DC is converted to AC via a full-bridge DC-AC inverter for supplying the load ac R .The error equation of the DC-AC inverter can be expressed by using KVL and KCL, and giving the requested reference sinusoidal voltage as r υ :  Then, the grey logistic prediction is employed to forecast the system state, and adjust the PI parameters.The modeling steps of the grey logistic prediction are represented by the following: Step 1: An initial set of records Step 2: The accumulative generation operation (AGO) sequence can be obtained by executing AGO on where Step 3: The differentiation formula constructed for the grey logistic (also called verhulst) model becomes where The whitening differential equation can be presented as where κ and ρ are the developing and grey influence coefficients, respectively.
Step 4: The estimated parameters κ as well as ρ can be resolved through the least square method in the following way. where Step 5: The solution can be derived as Then with the inverse accumulated generating operation, the data sequence can be estimated as The modeling process for grey logistic prediction can plotted in Figure 4.

Results and discussion
With TRIAC (Triode for Alternating Current) loading as well as encountering partially shaded scenario, a resultant AC wave shape from presented solar energy system is displayed in Fig. 5.While under the same loading circumstance, Figure 6 depicts a resultant AC wave shape from classic PI controlled solar energy system.It is observed that the proposed solar energy system can yield lower transient voltage drop as well as a faster compensation of a voltage dropping to reach referenced level as found in Figure 5 and Table 1.The total harmonic distortion of the proposed solar energy system AC output-voltage is much lower than that of the classic PI method, establishing good steady-state response, as shown in Table 2. Regarding the high performance of the proposed solar energy system, it can be summarized as follows: As the temperature of the environment rises, a maximization of the wattage in sun power plate becomes lower, but more illumination makes maximization of the wattage higher.Due to the fact that the output power of the solar cell depends on variations in illumination as well as temperature conditions, which also influences the efficiency of the system.It is therefore imperative to possess a maximum power tracking function which maximizes the efficiency of the solar cell system.By operating the solar power output at the maximum point with the suggested power electronic circuit is one of the keys.Furthermore in the case of partially shaded solar arrays, the power-voltage curve is subject to multiple local extremes.So the proposed method leads to an effective as well as quick solution in finding the global optimum, avoiding premature trapping of parameters in local optimum.In the closing remarks, the power of the inverter reaches one kilowatt at a load resistance of twelve ohms.For the future, the circuit framework can be extended and modified to raise the output voltage (e.g.above two hundred volts) and wattage (e.g. more than three kilowatt) [33][34][35].

Conclusions
A grey prediction-based PI control method is adopted to construct a solar energy system capable of producing good AC output voltage.The classic proportional-integral control displays higher susceptibility during partially shaded solar arrays or external load perturbation of the system, making the AC output response of the solar energy system unsatisfactory.Grey prediction could forecast the system output and get the appropriate proportional-integral control parameters for a highly accurate tracking control.The results suggest that by combining grey prediction and proportional-integral control, a solar energy system with global maximization of the wattage pointed following is achievable, validating the efficacy of the proposed method.

Figure 1 .
Figure 1. Circuit of power electronics.Consider d v for a voltage referential level at maximized wattage location, and let sun v follow d v ; the error voltage can be written as d sun v v − = 1 d.As the control method is well-designed, (1) is robust where the desired solar output voltage corresponds to the wanted reference voltage.The follow function enables swift as well as precise although sunlight energy may be partly obscured.The control law u by using proportional-integral controller can be formulated as equivalent gain.The control law it u is designed with the proportional-derivative (PD) control written by convergence to zero, in which P k denotes the proportional gain as well as D k means the derivative gain.Figure3displays a full controlled configuration.

Table 1 .
Comparison of voltage drop.