Global Maximum Power Point Tracking based on new

Global Maximum Power Point Tracking based on new Extremum Seeking Control scheme Nicu Bizon1)2) 1)

University of Pitesti, 1 Targu din Vale, Arges, 110040 Pitesti, Romania

2)

University Politehnica of Bucharest, 313 Splaiul Independentei, 060042 Bucharest, Romania

Abstract: A new Perturbed-based Extremum Seeking Control (PESC) scheme is proposed in this paper to track the Global Maximum Power Point (GMPP). The PESC scheme has two control loops based on power of the Photovoltaic (PV) array: the first loop operates as usually to track the Maximum Power Point (MPP) and the second sweeps all Local MPPs (LMPP) to locate the GMPP. Once the GMPP is located based on its uniqueness (after the PV pattern is quickly scanned many times, depending on the PV pattern’s profile), the GMPP is accurately tracked based on first control loop. The used PV patterns have the profile of the PV power characteristics obtained for PV array under Partially Shaded Conditions (PSCs). This PESC scheme is proposed to track the GMPP in the PV applications, but also in other multimodal problems from industry, being a good motif to revive the specialists’ interest for the Extremum Seeking Control (ESC) field. The results obtained here are very promising for both search speed and tracking accuracy performances of the GMPP under different PSCs simulated on the PV array. Thus, the energy efficiency of PV array controlled with the proposed PESC scheme will increase with more than 1.2% in comparison with that obtained with the other MPP algorithms due to better performance shown by this PESC scheme. A 99.6% tracking accuracy is obtained here in comparison with a maximum 98.4% tracking accuracy reported in the literature. Furthermore, 100% hit and high search speed is obtained here for the GMPP localization.

Keywords: Extremum Seeking Control (ESC), Photovoltaic (PV), Partially Shaded Conditions (PSCs), Global Maximum Power Point Tracking (GMPPT), multimodal PV patterns.

List of abbreviations: AC – Alternative Components; ACS - Ant Colony Systems; ANN - Artificial Neural Network; aPESC - asymptotic Perturbed-based Extremum Seeking Control;

ASA - Averaging Stability Analysis; BPF - Band Pass Filter; DE - Differential Evolution; dPESC - derivative Perturbed-based Extremum Seeking Control; EA - Evolutionary Algorithms; ESC - Extremum Seeking Control FLC - Fuzzy Logic Controller; GA - Genetic Algorithm; GMPP - Global Maximum Power Point; GMPPT - Global Maximum Power Point Tracking; G2PESC - Global PESC scheme based on two band-pass filters HC - Hill Climbing; HPF - High Pass Filter; H1 – first Harmonic; IC - Incremental Conductance; ICDA - Initialization and Control of the Dither Amplitude; LMPP - Local Maximum Power Point; LmPP – Local minimum Power Point; LPF - Low Pass Filter; mPP - minimum Power Point; MIMO - Multi-Input Multi – Output; MPP - Maximum Power Point; MPPT - Maximum Power Point Tracking; MV - Mean Value; PESC - Perturbed-based Extremum Seeking Control; P&O - Perturb and Observe; PSC - Partially Shaded Condition; PSO - Particle Swarm Optimization; PV - Photovoltaic; RCC - Ripple Correlation Control; SISO - Single-Input Single-Output;

SIDO - Single-Input Double-Output;

1. Introduction It is known that the PV power generated by a PV array depends on PV panel’s orientation, but even if PV panel is fixed or sun orientated, the PV power still varies according to irradiance and temperature profile during a day. Therefore, the output PV power characteristics are of multimodal type, having in general multiple LMPPs. The PV patterns used to test the GMPP tracking algorithms are obtained for an array of PV panels connected in series and/or parallel under PSCs (obtained by using different irradiance levels for each PV panel). The objective of a GMPP tracking algorithm is to always track, quickly and accurately, the GMPP (not a LMPP with power level lower than the GMPP level). Thus, the maximum PV power generated by a PV array will be extracted based on GMPP tracking algorithm during a sunny-cloudy day, when many PSCs occur. Also, it is known that popular Maximum Power Point Tracking (MPPT) algorithms, such as Perturb and Observe (P&O) [1], Incremental Conductance (IC) [2], and Hill Climbing (HC) [3] and other algorithms included in recent reviews [7-12], cannot track the GMPP. It is worth mentioning that improved variants of the P&O [4], IC [5], and HC [6] algorithms, and other MPPT algorithms proposed in the literature, such as Ripple Correlation Control (RCC) [7,13], sweep current or voltage methods [8,14], load current and load voltage minimization [9,15], fractional methods based on Short Circuit Current or Open Circuit Voltage [10,16], dP/dV or dP/dI feedback control [11,17], slide control methods [12,18], and so on (note that more than fifty conventional MPPT algorithms are identified in reviews [7-12]), have demonstrated good performance (search speed and tracking accuracy) under constant irradiance level. During the last years, the subject of GMPP tracking algorithms became of major interest, and this was extensively presented in reviews [19-23], where the main two-stages based Global Maximum Power Point Tracking (GMPPT) methods proposed in the literature have been presented. The conventional MPPT algorithms, including the ESC methods [23], can be used in the second stage, after the GMPP was located using firmware-based algorithms [19,20] or hardware architecture-based algorithms [21,22]. It is worth mentioning that ESC methods have been omitted for brevity in reviews [19-22], even if some simulation [25] and laboratory

experiments [24,26] were recently reported based on ESC schemes. Reported results show an increase with about 5% in the PV power generated if a GMPPT algorithm is used instead of a conventional MPPT algorithm [27]. The GMPPT algorithms were classified in two main classes in [23]: firmware-based and hardware architecture-based algorithms. The topology and design of PV hybrid system are main issues explored in the hardware-based algorithms [28]. More than forty hardware-based algorithms have been identified in the reviews [21-23]. The types of common PV system architectures are of decentralized (where each panel incorporates a microinverter [29], centralized (central inverter [30]), or hybrid [28] (PV array of series and/or parallel connection of strings of PV panels, using a DC-DC converter for each string). The soft computing techniques are mainly used in the first stage to locate the GMPP due to its natural adaptability to multimodal PV patterns that could appear in large PV array under PSC. More than forty firmwarebased algorithms have been identified in the reviews mentioned above [19,20,23] based on Fuzzy Logic Controller (FLC) [31], Artificial Neural Network (ANN) [32,33], Evolutionary Algorithms (EAs) (genetic algorithms (GAs) [34], differential evolution (DE) [35], particle swarm optimization (PSO) [36], or Ant Colony Systems (ACSs) [37]), and chaotic search [38]. Also, in review [23] is mentioned that ESC method could be used in the second stage of GMPP tracking process if the performance will be improved based on advanced ESC schemes such as the one proposed in [39], which eliminates the power ripple. In this study, the idea from [39] is improved at the topological level with an additional feedforward control loop for the dither gain based on the first derivative of the PV power. Note that the derivative operator will be practically approximated by a Band Pass Filter2 (BPF2), besides the BPF1 used in the main control loop. So, the proposed PESC scheme can search the GMPP quickly as the two-stage searching methods based on two BPFs. The search speed and tracking accuracy are evidently comparable with the results reported in [39]: 2 kW/s search speed and 99.9% tracking accuracy during steady-state regime. These values of the performance indicators are comparable or higher than other results reported in [19-23] and this two BPFs-based PESC scheme is new in existing literature based on the author’s knowledge. So, this paper is focused on the following aspects: (1) to briefly classify and compare the PESC schemes from the topological point of view; (2) to present and model the two BPFs-based PESC method proposed, highlighting the main topological differences in comparison with other PESC

methods; (3) to identify the generic PV patterns under PSCs to appropriately test the two BPFsbased PESC method; (4) to evaluate the performance of the two BPFs-based PESC method based on generic PV patterns selected. Consequently, the paper is structured as following. In Section 2, the main PESC schemes were presented and compared from the topological point of view. The proposed two BPFs-based PESC are analyzed in Section 3 based on Averaging Stability Analysis (ASA) technique. The PV patterns generated by a PV array under PSCs are shown in Section 4 in order to select the generic PV patterns used in simulation. In Section 5, the results obtained highlight the promising performance of the two BPFs-based PESC method proposed to track the GMPP. Last Section concludes the paper. 2. Perturbed-based Extremum Seeking Control schemes It is known that the ESC method is of adaptive close-loop type, being used to search the extreme values of a nonlinear map based on analog [40] or discrete-time search [41]. Thus, it is worth to mention some references on the above classification of the ESC schemes: (1) ESC-based analog optimization: sinusoidal perturbation [40], sliding mode-based analog optimization [42], gradient feedback [43], model-based methods [44], and so on [40,41]); (2) ESC-based numerical optimization [40, 45]. This paper is focused on the PESC schemes, so next subsections will present the scalar and asymptotic PESC schemes proposed and their variants, besides few PESC-based GMPPT applications. 2.1. The scalar and asymptotic PESC schemes The scalar PESC schemes have been proposed since 1922, and this control was very popular in the 1950s and 1960s for different industrial application [41]. The interest on PESC applications increased after publication of the mathematical systematic approach of the local [46,47] and global stability [48]. The ESC problem is defined below for the basic PESC schemes [40] and their variants, but also for new PESC schemes proposed in the last decade [39,49]. The Multi-Input Multi -Output (MIMO) nonlinear system is defined by (1) [61]: dx dt

f x(t ), u (t ) ,

y

h x(t )

(1)

where f(x,u) and h x) are smooth functions that define the system dynamic and nonlinear map, and x Rn , u Rm , and y R are the state variables, system inputs and system output, respectively.

The GMPP can be located and tracked if all three assumptions mentioned in [48,50] are valid. To stabilize the nonlinear system it can be defined a smooth control law, u(t)=g(x(t),p), where p is a parameter vector that defines a unique GMPP, x e(p), based on the smooth function x e, x e:Rl Rn : f x, g ( x, p )

0

x

xe p

(2)

So, at the equilibrium point the parameter-output map can be represented as: y

hx

h xe ( p )

h p

(3)

The derivatives of both sides of relation (3) give: dy dt

h dp p dt

(4)

where h p

h h ,..., p1 pl

T

,

dp dt

dp dp1 ,..., l dt dt

T

(5)

Thus, the y vector converges to the global extremum defined by (3) based on the seeking vector (p). Note that the MIMO [51], Single-Input Single-Output (SISO) [50,52] and Single-Input DoubleOutput (SIDO) [53,54] nonlinear systems are considered to show the performance of both scalar and asymptotic PESC schemes proposed in the literature. The basic scalar PESC scheme is presented in Figure 1 [40,46,47].

Figure 1. Scalar PESC scheme

The structure of the filtering block usually includes a High-Pass Filter (HPF) to remove the DC part of the probing signal. One variant of the scalar PESC scheme uses a Low-Pass Filter (LPF) after the demodulation of the Alternative Components (AC) part of the probing signal with the sinusoidal dither, sin(ωt) (see the product block shown in Figure 1). The series connection of the HPF and LPB defines an equivalent BPF [52], having the cut-off frequencies of the LPF and HPF related to dither frequency as ω l=

l

and ω h =

h

, where 0< h (100 tracking accuracy)[%]. Note that the highest level of the LMPP differs from GMPP by 0.001, so the searching resolution is 0.0238% (=0.001/4.2). Consequently, the GMPP may be tracked by the G2PESC scheme (because the tracking accuracy is higher than 99.99%, so the searching resolution is limited to 0.01%). The results shown in Figure 26 prove this tracking accuracy mentioned above. The structure of Figure 24 is the same as Figure 23 with exception of adding a supplementary plot to better show the dynamic of the GMPP search. The searching signal (p) and its components given by (17) are shown in the last two plots. Note that sweeping signal ( pˆ 2 ) has high harmonics during the stage of GMPP localization, which decay to zero during the tracking of the GMPP. Sweeping of the LMPPs during the stage of GMPP localization is shown in Figure 25. The GMPP is swept during both the localization and tracking stages instead the LMPPs are swept only during the GMPP localization phase (see Figure 27).

Figure 24. The GMPP search

Figure 25. The LMPPs’ sweeping stage 5.1.3. Robustness of the GMPP search to design parameters

The design parameters of the G2PESC scheme, which will be considered to test the control robustness, are the following: the normalization gain (Figure 26), loop gain (Figure 27), dither gain (Figure 28-30), BPF1 parameters (Figure 31), BPF2 parameters (Figure 32), and initial condition (Figure 33). Note that all Figures have the same structure of plots: the PV pattern is shown on top; the output (y), dither gain (Gd), and input (p) and its components are shown in next plots. The normalization gain was set to be five times lower (kN=0.1) or higher (kN=2.5) than kN=2, which is the value set in range of the GMPP values obtained for the (l, m, r) triplet having the vector components in range [1,5]. It can be observed that the maximum values of the dither gain (Gd) are almost proportional with the values of the normalization gain. A high value of the dither gain means a large sweeping range, while an optimal kN value means a sweeping range covering at limit the search range. Thus, the search time is higher in case kN=2.5, being about four times higher than in case kN=0.1 (see Figure 26). It is obvious that the optimal kN value depends on the loop and dither gain set and vice versa. The values of loop gain were set at the lower and upper limit of the choosing range by setting sd=4

and

sd=40

(see Figure 27). The case

sd=40

give a kL value that is close to the maximum

value shown in Figure 18. So, a high frequency oscillation appears on the probing signal (see first plot in Figure 27). The probing signal is filtered and averaged by the BPF and MV block, but some oscillations still appear on the dither gain (see second plot in Figure 27). Thus, the searching time is a bit higher in case

sd=40,

being about 1.2 times higher than in case

sd=4

(see

last plots in Figure 27). The effect of the dither gain on the searching time was shown in Figures 28-30 for the pattern shown on top of each Figure. For k2 =15 and

sd=4

is obtained a kL value that is lower than the

maximum value shown in Figure 18. Consequently the G2PESC scheme operates in the stable region. It is worth mentioning that the searching time is two times higher in case k 2 =5 in comparison with k2 =15 (see Figures 28-30 for three PV patterns considered in this study and the step responses shown in Figure 19 as well). Note that the dither gain is two times lower in case k2 =5 in comparison with k2 =15. So, the rule kL=k1 k2 k3 < kL(max) give high flexibility in designing the G2PESC scheme. The searching gradient depends on k1 and k2 parameters based on (18), and k3 parameter, which is an uncertain constant that depends on the curvature of the nonlinear map. If the range for the k3 parameter can be set, then kL(max)/k3(max) gives the upper limit for the

parameters’ product, k1 k2 . The lower limit for the product k1 k2 is related to the searching range that must be swept (k2 ) and dither persistence (k1 ) [56] The dither persistence depends on the frequency band of the BPF1 and the levels of the first harmonic evaluated in [56]. Note that the BPF1cut-off frequency (fl1 =

l1 fd)

are also important in

setting of the kL(max) value (see Figure 18). The GMPP search is shown in Figure 31 for and

l1 =3.5.

l1 =3.5,

Note that the harmonics’ level is higher in case

l1 =5.5

l1 =5.5

in comparison with the case

but the searching time is a bit higher as well.

The BPF2 cut-off frequencies (fh2 =

h2 fd

and fl2 =

l2 fd)

are used in this study to approximate the

magnitude of first harmonic (H1) of the probing signal. The GMPP search is shown in Figure 32 for

h2 =0.1

and

l2 =1.9,

and

h2 =0.2

and

l2 =1.8.

It can be observed that a narrow band of the

BPF2 reduces the searching time. It would be of interest to show the effect of using a BPF2 of higher order on the searching time, but this may be suggested by the results shown in Figure 32 as well. Note that the PV pattern used in simulations shown in Figure 31 and 32 has the GMPP located in its right side, but the search starts from the left side, p(0)=0. The starting point is considered on the right side, p(0)=9, in Figure 33 in order to show that the choice of the initial condition is not important on the GMPP search proposed and tested here under different PV patterns.

Figure 26. Robustness of the GMPP search to normalization gain for PV patterns (4.1,4.2,4)

Figure 27. Robustness of the GMPP search to loop gain for PV patterns (4.2,4,4.1)

Figure 28. Robustness of the GMPP search to dither gain for PV patterns (4.2,4,4.1)

Figure 29. Robustness of the GMPP search to dither gain for PV patterns (4.1,4.2,4)

Figure 30. Robustness of the GMPP search to dither gain for PV patterns (4.1,4,4.2)

Figure 31. Robustness of the GMPP search to BPF1 parameters for PV pattern (4.1,4,4.2) and sd=4

Figure 32. Robustness of the GMPP search to BPF2 parameters for PV patterns (4.1,4,4.2)

Figure 33. Robustness of the GMPP search to initial condition, p(0)=9, for PV patterns (4.2,4,4.1) and (4.1,4.2,4), and

sd=40

5.1.4. Robustness of the GMPP search to noise Robustness of the GMPP search to noise was tested for different peak-to-peak (p-p) levels of random noise added to the PV pattern. The parameters of random noise are the following: the sampling period is of 10 milliseconds, and noise amplitude is of ±0.1p-p (Figure 34a) and ±0.3p-p (Figure 35). The PV pattern (4.1,4,4.2) is used in both cases. The noisy PV pattern is shown on bottom of each Figure. The output (y), dither gain (Gd), and input (p) and its components are shown in next plots of Figures 34 and 35.

Note that the original GMPP (without noise) are still tracked by the G2PESC scheme for PV pattern with ±0.1p-p noise, but the current GMPP (with noise) is continuously tracked for PV pattern with ±0.3p-p noise. The stationary phase of GMPP tracking is not reached in this case (±0.3p-p noise) due to the high level of noise. The Gd dither gain reacts to perturbations (noise, load changes, and PSCs) [39]. The Gd level decay to zero if the perturbation disappears, but will be again high if the perturbation appears (see the second plot on Figure 35).

Figure 34. Robustness of the GMPP search to ±0.1 random noise added on the PV pattern (4.1,4,4.2)

Figure 35. Robustness of the GMPP search to ±0.3 random noise added on the PV pattern (4.1,4,4.2) 5.2. Tracking of the GMPP on other multimodal patterns The 6th-order polynomial (25) was used as a counter-example in [63] for the aPESC scheme proposed in [50]: y

( p 1) 6

1 ( p 1) 5 10

623 ( p 1) 4 400

659 ( p 1) 3 4000

11287 ( p 1) 2 20000

259 ( p 1) 4000

637 100 20000

(25)

This polynomial has three maxima at u =1−0.8985=0.1015, u = 1+0.05=1.057, and u = 1+0.8951=1.8951, with u=0.1015 being the global maximum (see Figure 36) [63]. The pattern

(25) and zooms of the three maxima are shown on the bottom of Figure 36. The starting point p(0)=0 was considered in Figure 36, but the searching and tracking of the GMPP were obtained for different starting points as well. The plots on Figure show the output (y), dither gain (G d), and the searching signal (p). The tracking accuracy is of 99.997% (=6.7945/6.7947). Note that the level of the LMPP differs to GMPP with about 0.5503, so the searching resolution is of 8.1% (=0.5503/6.7947). Thus, the GMPP can be accurately tracked based on G2PESC scheme for different starting points. The second example from [63] will be used for testing the G2PESC scheme under pattern with multiple local extremes (see Figure 29). The pattern considered here is: y

p 0.3 sin(10 p)

(26)

Note that this example has a discontinuous derivative at the global optimum and the local extremes are symmetrically located to global optimum. If the initial condition |p(0)|>1, then this pattern has a lot of local extremes between the start point and GMPP. The searching and tracking of the GMPP were obtained for different starting points. For example, the starting point p(0)=-10 was considered in Figure 37, where the plots show the output (y), dither gain (Gd), and the searching signal (p). The tracking accuracy is of 99.6% (=100%-0.4%) during the stationary phase when p

p3

Am sin( t ) (see last plot on Figure 37).

Figure 36. Tracking of the GMPP on the PV pattern (25)

Figure 37. Tracking of the GMPP on the PV pattern (26) Conclusions

The main objective of this paper was to present a new perturbed-based extremum seeking scheme based on two band-pass filters, named as the G2PESC scheme. The G2MPP can search and track accurately the global optimum on different multimodal pattern. The obtained performances are the following: (1) the tracking accuracy during the stationary phase is higher than 99.99% or 99.6% for patterns that have a continuous or discontinuous derivative at the global optimum; (2) the current GMPP is continuously tracked for noisy PV patterns; (3) the searching time is about 100 milliseconds if the dither frequency is of 100 Hz; (4) the lower searching resolution is about 0.01% (which means a LMPP level very close to GMPP level: y LMPP(max) / yGMPP

99.99% ); (5) the design conditions are not restrictive (which means a large

range where the design parameters of the G2PESC scheme may be chosen); (6) the searching process to locate the GMPP is not dependent on the starting point if the normalization gain is appropriately chosen considering the sweeping range,

p=|pGMPP -p0 |.

The performance reported above for the G2PESC scheme will be better than those obtained for the popular P&O and IC algorithms based on the European Efficiency Test, EN 50530 [69], where the average of tracking accuracy is around 98.4% for both algorithms, in comparison with the lowest value obtained here of 99.6%. Furthermore, the oscillation around MPP is avoided in the G2PESC scheme due to the asymptotic decrease of the dither's magnitude [70]. Some design rules were given for the main design parameters of the G2PESC scheme: the normalization gain (kN), loop gain (k1 ), and dither gain (k2 ). The simulations to validate that the design conditions are not restrictive to normalization gain, loop gain, and dither gain are shown in Figures 26, 27, and 28-30, respectively. Also, the effect of changing the cut-off frequencies of the BPF1 and BPF2 filters is shown in Figure 31 and 32. The G2PESC scheme proposed and analyzed here is a new topological scheme of perturbedbased extremum seeking type. This G2PESC scheme can be successfully applied in large PV array, where the PV pattern has multiple local extremes, but in different multimodal optimization problems as well. This work may open new research in order: (1) to find the design relation of the G2PESC scheme based on mathematical model; (2) to improve performances of the G2PESC scheme based on optimal values of the design parameters; (3) to enhance the stability of G2PESC-based control loop using advanced compensators; (4) to apply the G2PESC scheme in different multimodal optimization problems.

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