(PV) systems. The proposed MPPT algorithm is a multi-stage algorithm combined with the golden section search (GSS) method and two other commonly used ...
A Multi-Stage MPPT Algorithm for PV Systems Based on Golden Section Search Method Riming Shao, Rong Wei and Liuchen Chang Department of Electrical and Computer Engineering University of New Brunswick Fredericton, NB Canada E3B 5A3 Abstract—This paper proposes a complete maximum power point tracking (MPPT) solution for grid-connected photovoltaic (PV) systems. The proposed MPPT algorithm is a multi-stage algorithm combined with the golden section search (GSS) method and two other commonly used methods: the perturb and observe (P&O) and the incremental conductance (INC). The GSS method is used for its fast convergence time and robustness. The GSS mechanism, as well as the whole multistage MPPT strategy, is described in this paper. Finally, the results of simulations and experiments are given to verify the good performance of the system.
I. INTRODUCTION Due to the nonlinear voltage-current characteristics of a PV array, there is a unique maximum power point (MPP) at a given set of irradiance and temperature conditions. To increase its energy production efficiency, a maximum power point tracking (MPPT) technique is necessary for a PV system. In fact, many MPPT algorithms have been proposed in the literature [1]-[10]. One of the common used MPPT method is the perturb and observe (P&O), which is well known as hill-climbing [1]-[2]. The P&O operates by periodically perturbing the PV array voltage or the duty cycle of the power converter, and observing the array power. If a given perturbation leads to an increase (decrease) in array power, the subsequent perturbation is made in the same (opposite) direction. This method has the advantages as ease of implementation and independence from PV characteristics. However, a major problem for P&O is that it is hard to choose a proper and optimized perturbation step. A big step leads to fast convergence time, but also to large oscillations around the MPP in steady state and thus compromising energy efficiency. Conversely, a small step can reduce the oscillations and improve the tracking accuracy, but it also can lead to a long search process and result in low efficiency under rapidly changing conditions. Another drawback of P&O algorithm is that it can fail under rapidly changing atmospheric conditions [3]. To overcome this drawback, the incremental conductance (INC) algorithm is introduced in [3]. The INC method is based on the fact that the derivative of the array power P with respect to the array voltage V is equal to zero at the MPP, positive on the left of the MPP, and negative on the right. Since the derivative 1 where I is the array output current, the INC method can determine the relative location of the MPP by evaluating the
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incremental conductance of the array. Hence, the INC algorithm has superior performance to P&O in that it can determine the proper direction of the perturbation, and identify whether the MPP is reached [3]-[4]. However, like P&O, there is a tradeoff when choosing the increment (step) size. For fast tracking, large increments must be used, with which, however, the system might not operate exactly at the MPP and oscillate about it instead [11]. Variable step size P&O or INC strategies have been proposed in [2][5]-[6]. These MPPT algorithms present better performances than the conventional fixed step size strategies with respect to the tracking speed and accuracy. However, deliberate tuning is required for those optimization parameters which are dependent on the system characteristics. References [1][7] provide the guidelines and rules for determining the optimal values. There are still many other MPPT algorithms. For example, the fractional open-circuit voltage or fractional short-circuit current algorithms simply assume that the MPP is located at the constant fraction of the open-circuit voltage or shortcircuit current [11]. These algorithms are very simple. However, they cannot track the true MPP and require periodically disconnection or short-circuit of the PV array to measure the open-circuit voltage or short-circuit current for reference, resulting in more power loss. The intelligent control technologies like fuzzy logic and neural networks have been used as the good alternatives for the MPPT control because of their capabilities of dealing with the nonlinear PV characteristics [8]-[9]. However, the effectiveness of fuzzy control depends a lot on the knowledge of the control engineer in choosing the right error computation and coming up the rule base table [11]. Similarly, the performance of neural networks MPPT depends a lot on the designing of the neuron layers and the training of the weights. In addition, the neural networks MPPT usually needs the measurements of the solar irradiance and the temperature. Hence, the versatility of these methods is limited. References [11]-[13] compare different MPPT algorithms and provide general reviews. It is difficult to determine which method is the best. Nevertheless, in this paper, the golden section search (GSS) method is integrated in the proposed multi-stage MPPT as a straightforward variable step size optimizing algorithm. Provided an initial bracketing interval in which the MPP is located, the GSS algorithm can converge to the MPP by repeatedly narrowing the width of the interval with the rate of the golden ratio [14]. Equivalently, the search step size is repeatedly reduced. So the GSS can provide a balance between the dynamic response and the accuracy. It is pointed that the GSS requires an initial interval where the
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MPP is located to start searching. Therefore, because of the ease of implementation and fast speed, the P&O is applied to locate the MPP in a vicinity area as the initial interval for GSS. A relative large perturbation can be used in P&O here to speed up the search. Moreover, as discussed in [1], the P&O algorithm is less likely to be confused by the rapidly changing atmosphere conditions with a large perturbation. As a result, the proposed multi-stage MPPT involves the P&O in the first stage and the GSS in the second stage. In addition, it is important to have the found MPP by GSS being verified in the third stage, because the system needs to know the MPP drift when the conditions change and start over the tracking for the new MPP. The INC technique is the appropriate MPP verification method used in the third stage. The details of the proposed multi-stage MPPT are presented in Section II.
selection of the new evaluation point x4 must satisfy: width of (x1, x4) = width of (x3, x2), that is
The remainder of this paper is organized as follows. Section II discusses the proposed multi-stage MPPT algorithm. Section III provides the system configuration and modeling. Section IV presents the simulation results and the experimental results. Finally, Section V provides the conclusion.
where φ is the golden ratio. This is the reason GSS is so named.
II. PROPOSED MULTI-STAGE MPPT ALGORITHM A. Principle of GSS The GSS is a technique for locating the extremum of a unimodal function inside a bracketing interval by repeatedly narrowing the width of the interval. As shown in Fig. 1, it is true for a unimodal function that f(x3) > f(x4) if the maximum lies on the left side of x3 (dashed line), and similarly, f(x3) < f(x4) if the maximum lies on the right side of x4 (solid line). The contrapositive statements are also true that the maximum must lie on the right side of x3 if f(x3) ≤ f(x4), and the maximum must lie on the left side of x4 if f(x3) ≥ f(x4). Therefore, the maximum point can be located in one of the two reduced intervals (x1, x4) and (x3, x2) after f(x4) is compared to f(x3). A new point then can be inserted into the reduced interval to form a new search interval for next iteration. After the search repeats for a certain number of times or the search interval shrinks into a desired tolerance, the latest inserted point can be considered to be the maximum point [15].
Fig. 1. Search scheme diagram for GSS.
As shown in Fig. 1, for each iteration there are two possible search sections: (x3, x2) and (x1, x4). Only one of them is selected to be the next search section. It is required that the two possible sections are equally wide. Otherwise, the convergence speed is decreased if wider sections are taken more frequently in certain cases. Therefore, the
a+b=b+c
(2)
When this relation is satisfied, all the sections of the iterations can have the same proportion of spacing between the three points such that the algorithm converges at a constant and optimal speed. As shown in Fig. 1, section (x1, x2) and the next iteration section (x1, x4) or (x3, x2) have the similarity of spacing, that is a / b = c / b = (b + c) / a = φ
(3)
Combining (2) and (3) yields: φ = (1
√5 /2 = 1.61803398…
(4)
Hence, the convergence rate of the GSS is R = φn
(5)
where n is the number of iterations. Using (5), it can be shown that only 15 iterations are required for GSS to shrink the search interval to less than 0.1% of its original length. This means, for a digital control system with 0.1% resolution corresponding to a 10-bit analog-todigital converter (ADC), no more than 15 iterations are needed for the best applicable solution. Hence, the GSS converges quickly. In addition to its fast search speed, the GSS has more merits that make it appropriate for MPPT in PV systems. The GSS does not require any derivatives; as a result, it is robust and has noise-resistive capabilities as the derivative computation is easily disturbed by fluctuating signals and noises. Moreover, the GSS mechanism ensures that the evaluation points are spread out from each other in every search interval. Therefore, when comparing the values in order to find the maximum, the GSS algorithm has good tolerance for measurement errors due to inherent noises and power ripples caused by switching mode converters. Fig. 2 shows the flowchart of the GSS algorithm used in the PV application. V, I, and P represent the PV voltage, current, and power respectively, and n counts the search iterations. As V is used for the search variable, the PV voltage is regulated to the reference Vref every iteration by the converter. When the search section is shrunk to a value less than the tolerance ε, the MPP is considered to be reached and the GSS ends. B. Multi-stage MPPT strategies In this paper, P&O, GSS, and INC are combined for use in a multi-stage MPPT algorithm for PV systems. The MPPT procedure is divided into three stages, where these three different methods are used, respectively. First, the P&O is applied with relative large perturbation steps to quickly obtain a fast search speed. Once the MPP is located in a relatively narrow range by P&O, MPPT switches to the second stage where the GSS takes over. Using GSS,
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1) Stage 1 with P&O:
Start
The perturbation steps are counted by number n = 0, 1, 2 …. Starting with the initial point n = 0 with PV power P0 and voltage V0, the P&O search steps are:
Initial search interval V(n-3) V(n-1) * V(n-2) |_________|_____|_________|
a) Perturb the system by adding a perturbation ΔV to V0, that is, V1 = V0 + ΔV. ΔV can either be positive or negative (so the PV voltage can be increased or decreased).
Insert new point V(n) V(n-3) V(n-1) V(n) V(n-2) |_________|_____|_________|
b) Observe the PV steady state output power P1 at the new operating point after a short time. c) If P1 > P0, keep the same sign of ΔV, otherwise change the sign of ΔV and swap the point 0 and point 1 to reestablish P1 > P0; then continue with the perturbations, that is, Vn = Vn-1 + ΔV.
Vref=V(n)
Sampling time?
No
d) Observe Pn. If Pn > Pn-1, repeat the perturbation and the observation by adding 1 to n. The P&O continues as long as Pn > Pn-1. Hence, the power values exhibit ascending order with Pn > Pn-1 > Pn-2 > …. > P0.
Yes P(n)=V(n)×I(n)
Yes
e) P&O ends when Pn < Pn-1. The new MPP is now known to lie inside section Vn-2 and Vn, since the power value of the middle point Vn-1 is greater than both Pn and Pn-2 as shown in Fig. 1. From now on, if P&O continues, the PV operation will oscillate about the MPP. However, the GSS is now able to refine the MPP since the MPP is known to be in a defined interval.
No
P(n)
