Two-level control of photovoltaic systems using global perturbation-based extremum seeking control and model reference adaptive control

Abstract

One of the main renewable energy sources for the future is photovoltaic (PV) energy. Hence, working of the PV systems at maximum efficiency is taken into consideration in recent years. In this paper, for improving the performance of the global maximum power point tracking under partial shading conditions and uncertainty in parameters of DC-DC converter, a two-level adaptive control scheme is proposed. The proposed controller is capable of efficiently handling the uncertainties in the PV systems and the perturbations in the environment. The first level is global perturbation-based extremum seeking control (GPESC), and the second level is model reference adaptive control (MRAC). GPESC is used to find global maximum power point and MRAC is utilized to handle the dynamics of the DC-DC converter. Adequate difference in the time constants of control levels, causes decoupled control levels, which in turn makes it easy to design the controller. The performance of the proposed control scheme is evaluated through simulation based on four indicators: tracking accuracy, tracking efficiency, tracking speed and searching resolution for different irradiance patterns. The results are compared with GPESC and GPESC with PID controller.

Keywords

Model reference adaptive control (MRAC)global perturbation-based extremum seeking control (GPESC)global maximum power point tracking (GMPPT)photovoltaic (PV) systems partial shading conditions (PSCs)

Introduction

Nowadays, performance improvement of photovoltaic (PV) power systems under variations in solar insolation and environmental temperature has attracted the attention of many researchers (Necaibia et al., 2015). Since the variations of irradiation are much faster than the temperature changes, partial shading conditions (PSCs) has been more considered in the literature. The only way to make the PV systems continuously hand over their theoretical optimal power in different climate conditions is by using a maximum power point tracking (MPPT) algorithm (Khanna et al., 2014). MPPT means that automatic tuning of the PV systems load enables them to get the maximum possible power output. There is a complex relationship between current, voltage, and output power of PV systems making it difficult to extract the maximum power (Koutroulis and Blaabjerg, 2015). A comparative study of MPPT methods for PV power systems can be found in Koutroulis and Blaabjerg (2015), D’Souza et al. (2010) and Yang and Blaabjerg (2015). Perturb and observe (P&O) and modified P&O (D’Souza et al., 2010; MohdZainuri et al., 2014), incremental conductance (IC) and modified IC (Faraji et al., 2014; ; Hsieh et al., 2013; Sekhar and Mishra, 2014), ripple correlation control (Barth and Pilawa-Podgurski, 2015; Moo and Wu, 2014), fuzzy logic, neural network (Yau and Chen, 2012; Zhang and Bai, 2008) and particle swarm optimization (PSO; Duan et al., 2015) are the other popular techniques for extraction of the maximum power in the PV systems. These methods vary in complexity, range of effectiveness, convergence speed, implementation hardware, cost and popularity. On the other hand, uniform illumination intensity in a PV array is not almost satisfied because of buildings or trees shades, atmosphere fluctuation, existence of clouds and daily sun angle changes. As it is seen in Figure 1, partial shading across a PV string can lead to multimodal P–V characteristics. The global MPPT (GMPPT) strategy under such conditions is a complex and challenging task. The conventional MPPT algorithms remain trapped in one of multiple local maximum power points (LMPPs) of the PV pattern (Bizon, 2016a; 2016b).

Figure 1.

The PV characteristics for the seven SX60 modules in series connection under different irradiance sequences.

Tracking a varying extremum (maximum or minimum) of a performance (output, cost) function is called the extremum seeking control (ESC; Zhang et al., 2016), which has two layers of meaning: first, the need for seeking an extremum of the performance function; second, the need for stabilizing the system and driving the performance output toward that extremum. The ESC is trying to accomplish real-time optimization for dynamic systems containing unknown dynamics (Ariyur and Krstic, 2003; Zhang and Ordóñez, 2012). Perturbation-based and model-based methods are two important classes of the ESC approaches. Some improvements of the ESC-based numerical optimization have been reported in Ghaffari et al. (2015). Perturbation-based extremum seeking control (PESC) method is a model-free optimization algorithm, so it is suitable for applying to various applications such as MPPT in PV systems (Ahmadi and Zargarzadeh, 2015; Ghaffari et al., 2015; Lei et al., 2010; Moura and Chang, 2013). But ESC methods are not popular as the other conventional MPPT methods mentioned before (Bizon, 2016a). The PESC has the advantages of fast convergence and guaranteed stability over a range of environmental conditions. In addition, it is simple to implement, which, in turn, makes it very cost effective in terms of processing/hardware requirements (Ghaffari et al., 2015). In the last decade, modified versions of PESC algorithm have been used for GMMPT in PV systems. In order to deal with the multimodal $P - V$ characteristics for PV array with variable shading, Lei et al. (2011) proposed a sequential ESC-based GMPPT control strategy, based on approximate modelling and analysis for the $P - V$ characteristics under variable-shading circumstances. Elnosh et al. (2014) proposed an ESC-based method to track the global power peak under non-uniform irradiance conditions. It was based on the measurements of power and estimation of the power gradient to iteratively determine the segment of the $P - V$ characteristics in which the global peak lies, without converging to the other local maxima. The review of GMPPT and ESC-based GMPPT has been addressed in Eltawil and Zhao (2013), Ishaque and Salam (2013), Bidram et al. (2012), and Liu et al. (2015). A good survey of global PESC (GPESC) methods and their applications in PV systems under PSCs for GMPPT has been presented in Bizon (2016a). The new ESC scheme which has two adaptive control loops has been presented in Bizon (2016b) for GMMPT based on scalar PESC scheme in Tan et al. (2009). This method has shown very good performance in comparison with other advanced ESC. But in that paper, only a static nonlinear map has been considered as the model of PV system without considering the dynamics of PV system.

In PV systems, regulating the voltage or current of the solar panel is done by a DC-DC converter. Three famous standard types of DC-DC converters are buck converter, boost converter, and buck-boost converter. In this work, the PV system has been integrated with the boost converter in order to deliver optimal power to the load. Conventionally, proportional integral (PI) or PI derivative (PID) control schemes are proposed for the boost converter via root-locus method or pole-placement technique (Alvarez-Ramirez et al., 2001; Cortes et al., 2004; Ding et al., 2007). However, when system parameters are uncertain, these methods cannot afford to determine the suitable PI (or PID) control gains. Therefore, various kinds of adaptive control that are able to deal with the plants with uncertain parameters have been proposed (Konstantopoulos and Alexandridis, 2015; Salhi et al., 2015). Among the various kinds of adaptive control, model reference adaptive control (MRAC; Black et al., 2014; Ioannou and Fidan, 2006; Sastry and Bodson, 1989) is the most popular and effective ones. MRAC is a control system structure in which the desired performance of the system with unknown parameters is expressed in the terms of the reference model that gives the desired response to the command signal. The command signal is fed to both the reference model and the actual system. Then the controller adapts the gains such that the actual system will respond like the reference system and the error between their outputs will be minimized.

In this paper, a two-level GMPPT control has been constructed. At the first level, global perturbation based extremum seeking (GPESC; Bizon, 2016b) has been used to determine the optimal duty cycle of the boost converter for reaching to the GMPP in the steady state (when the dynamics of the DC-DC converter has been neglected). The main contribution of this work in comparison to Bizon (2016b) is that unlike that research, a dynamic model for PV system has been considered: a second order transfer function has been derived as the model of the boost converter; while the relation of the output power and voltage contains a nonlinear map similar to Bizon (2016b). Considering the dynamic model of the converter requires the addition of adaptive control level to capture the converter’s dynamics in order to reach the desired transient performance. Consequently, we have utilized GPESC proposed by Bizon (2016b) only as the first control level. In Bizon (2016b), there is no second control level, and dynamic behaviour of the converter has been neglected. In other words, regardless of the uncertain dynamic model for the boost converter, GPESC without MRAC yields high performance in GMPPT; but, considering the uncertain dynamic model for the boost converter causes inability of GPESC without MRAC in GMPPT. Therefore, the GPESC with MRAC has been proposed to succeed in dealing with this difficulty. Owing to the unknown parameters of the PV array, the response of the converter to the previous mentioned duty cycle maybe undesired. Therefore, at the second level, the desired response of the boost converter has been achieved by the MRAC scheme. During adaptation, the error between the plant and the reference model has been utilized to tune the parameters of the controller. The two-level structure of the GMPPT algorithm reduces the complexity of the design and implementation by GPESC handling the slow dynamics (steady state behaviour) and MRAC handling the fast dynamics (transient behaviour). In this manner, there are two stable systems: GPESC-level has a large time constant and MRAC-level has a small time constant. Thus, the proposed two-level structure can effectively decouple GPESC and MRAC levels in the design and analysis. It is also worth mentioning that connection of two stable systems will not necessarily lead to a stable system; but, if their time constants are so far from each other, they can be considered decoupled. To the best of our knowledge, it is for the first time that GPESC and MRAC have been joined together for GMPPT in PV systems to reach the desired response. Four indicators have been used for evaluating the performance of the proposed control scheme in different irradiance patterns.

The paper has been organized as follows: In Section 2, the problem formulation for GMPPT of the PV system as well as the dynamic model of boost converter have been presented. A two-level adaptive control algorithm for GMPPT has been described in Section 3. Simulations and results have been presented in Section 4 to illustrate the effectiveness of the proposed method. Finally, a conclusion remark has been given in Section 5.

Problem formulation

PV characteristics

In this work, the one diode model of the PV cell has been considered. The model has been constructed with an ideal current source $I_{s c}$ that has been paralleled with a diode, which has the voltage $V_{D}$ , and parallel resistance $R_{p}$ . They are all in series with resistor $R_{s}$ , which has been used for the modelling of contactor and semiconductor material resistance. The ideal current source that is a function of temperature T delivers current in proportion to solar flux G. Totally, the PV module model equations are as follows

\begin{array}{l} I_{s c} = (I_{s c}^{r} + k_{I} (T - T_{r})) (\frac{G}{G_{r}}) \\ V_{D} = V_{c e l l} + I_{P V} R_{s} \\ I_{P V} = I_{s c} - I_{0} [\exp (\frac{V_{D}}{N V_{t}}) - 1] - \frac{V_{D}}{R_{p}}, V_{t} = \frac{k T}{q} \\ I_{0} = I_{0}^{r} {(\frac{T}{T_{r}})}^{3} \exp [\frac{E_{s i}}{N V_{t}} (\frac{1}{T_{r}} - \frac{1}{T})] \\ V_{P V} = n_{c e l l} V_{c e l l} \end{array}

(1)

where $I_{s c}^{r}$ , $T_{r}$ , $G_{r}$ and $k_{I}$ are, reference short-circuit current, reference temperature, reference irradiation and short-circuit temperature coefficient, respectively. The average terms of the voltage and current of PV module have been shown by $V_{P V}$ and $I_{P V}$ . $n_{c e l l}$ is the number of series PV cells in the module, $I_{0}^{r}$ the diode reference reverse saturation current, $E_{s i}$ the semiconductor band gap energy (barrier height) and $N$ is the emission coefficient. The later three are cell material/construction dependent. $V_{t}$ has been used to determine the thermal cell voltage; $k = 1.38 \times 10^{- 23} J / K$ , and $q = 1.6 \times 10^{- 19} C$ are Boltzmann’s constant and the charge on an electron, respectively. In this paper, the temperature has been considered to be constant and the PSCs have been focused on. Also, the PV model error does not affect to the GMMPT. For performing simulations, the PV module SX60 type has been picked up, with datasheet parameters presented in Table 1. Usually, to increase the power generation, the PV array is constructed using two or more PV panel in series and/or parallel connections. For simulation, a PV array with seven SX60 panels in series connection has been considered. The $P - I$ and $P - V$ curves for this PV array under different irradiance sequences are shown in Figure 1. The temperature T has been assumed equal to $25^{o} C$ . The experimental values for voltage, current and power at maximum power point (MPP) have been given in Table 2 for SX60 panel under $500 W / m^{2}$ irradiance. The vector ( $G_{1}$ , $G_{2}$ , $G_{3}$ , $G_{4}$ , $G_{5}$ , $G_{6}$ , $G_{7}$ ), with $G_{i}$ values in $W / m^{2}$ , $i = 1 - 7$ is the irradiance sequence for each panel. One can see in Figure 1, the number of LMPPs depends on the irradiance sequence used. Therefore, generic PV pattern, $p = y (v)$ , will be used in simulations performed as

\begin{matrix} p (v) = s a t (M_{l} - {(v - l)}^{2}) + s a t (M_{m} - {(v - m)}^{2}) \\ + s a t (M_{r} - {(v - r)}^{2}) \end{matrix}

(2)

where $s a t (.)$ denotes the saturation function with zero as its lower limit and infinite as its upper. According to Figure 1, the GMPP may be located in the left, middle, or right side of the power characteristic corresponding to the different voltages. The (l, m, r) triplet represents the PV array voltages in which GMPP occurs. Although this triplet varies with the changes in the irradiance sequences, but (20, 50, 80) triplet has been selected for simulations. The ( $M_{l}, M_{m}$ , $M_{r}$ ) triplet represents the maximum power points, which are corresponded to the (l, m, r) triplet. For different irradiance sequences this triplet varies, as well. Therefore, in the simulations the ( $M_{l}, M_{m}$ , $M_{r}$ ) triplet has been selected as (430, 420, 410), (410, 430, 420), and (410, 420, 430), which means that the GMPP is located in the left (see Figure 6(a)), middle (see Figure 7(a)), and right side (see Figure 8(a)) of the respective PV pattern.

Table 1.

Parameters of PV module SX60.

Parameter	Value	Unit	Parameter	Value	Unit
$G_{r}$	1000	$W / m^{2}$	$R_{s}$	3	$m Ω$
$T_{r}$	298	K	$I_{s c}^{r}$	3.8	A
$E_{s i}$	1.12	eV	$I_{0}^{r}$	$2 \times 10^{- 9}$	A
$k_{I}$	0.0025	$A / K$	$n_{c e l l}$	34	—
$R_{p}$	10	$Ω$	$N$	1.3	—

Table 2.

Values at the MPP under $500 W / m^{2}$ irradiance for PV module SX60.

Parameter	Description	Value	Unit
$I_{M P P}$	MPP current	3.5	A
$V_{M P P}$	MPP voltage	14.63	V
$P_{M P P}$	MPP power	51.2	W

Converter dynamics

Illustrated by Figure 2(a), the boost converter transfers the power produced by the PV array to the power consuming components (load) at higher voltage level. It is worth mentioning that other kinds of power converter may also be used in place of the boost converter according to the application. Based on Figure 2(a), the duty cycle d that is tuned by GMPPT controller has been applied to switching device Q. The relationship between the duty cycle of the switch Q and voltage of PV array is (Khanna et al., 2014)

v_{P V} = i_{P V} R_{O} {(1 - d)}^{2}

(3)

in which $R_{O}$ is the load resistance, $v_{P V} = V_{P V} + {\hat{v}}_{P V}$ and $i_{P V} = I_{P V} + {\hat{i}}_{P V}$ are the voltage and current of PV array, respectively. The average terms have been shown by $V_{P V}$ and $I_{P V}$ and the ripple terms by ${\hat{v}}_{P V}$ and ${\hat{i}}_{P V}$ , as well. The main goal of the GMPPT controller is computing the optimal duty cycle to deliver the maximum power to the load, and in turn achieving the desired transient response.

Figure 2.

(a) A typical PV energy system based on the boost converter. (b) Small signal equivalent circuit of PV array connected to the boost converter. The battery storage set has been considered as the load.

For the design of the MRAC, it is needed to connect a dynamic model of the boost converter to the output of the PV array. The current-source of the PV Array has an effect on the input-current-based feedback controller such that it may easily be saturated under the climatic conditions changes. Because of that, it is recommended that the input-voltage-based feedback control is used (Messo et al., 2012). In this work, a small signal equivalent circuit, as suggested in Khanna et al. (2014) and Femia et al. (2005), has been considered. This model is shown in Figure 2(b). A resistor $R_{I}$ has been used to model the solar array with a small signal array voltage ${\hat{v}}_{P V}$ and small signal array current ${\hat{i}}_{P V}$ across its terminals. The value of $R_{I}$ for the PV array can be determined by the slope tangential to the operating point of the system. The parasitic components of the elements of the boost converter have been not considered in this analysis.

According to Figure 2(b), the load of the boost converter, which is a storage battery set, yielding the voltage over the output capacitor is kept constant by the load voltage in each switching interval. This will cause only the output impedance to be affected by the output capacitor. Based on Figure 2(b), the transfer function from the control signal $d (t)$ to the PV array voltage has been derived in the small signal operation about an operating point. This transfer function is used for designing of MRAC in the Section 3.2.

In the steady state condition, the relationship between the operating duty cycle, D, and the steady-state DC input voltage of the boost converter $V_{P V}$ is

V_{P V} = (1 - D) V_{O}

(4)

in which $V_{O}$ is the steady-state DC output voltage of the boost converter. By defining (4) as $f (D)$ , the following relation in the frequency domain is established (Khanna et al., 2014)

\frac{{\hat{v}}_{P V} (s)}{R_{I}} + s {\hat{v}}_{P V} (s) C_{I} = \frac{f' (D) \hat{d} (s) - {\hat{v}}_{P V} (s)}{s L_{O}}

(5)

where s is the Laplace variable, $\hat{d}$ represents the small signal variation around the converter’s duty cycle D at the operating point, $\hat{d} (s)$ and ${\hat{v}}_{P V} (s)$ are the Laplace transforms of $\hat{d} (t)$ and ${\hat{v}}_{P V} (t)$ , respectively. $f' (D) = - V_{O}$ is the derivative of $f (D)$ with respect to the duty cycle at the operating point D. The minus sign in $f' (D)$ signifies that decreasing the duty ratio will cause to increase the panel voltage and vice versa. From (5), the following relation is derived

\frac{{\hat{v}}_{P V} (s)}{\hat{d} (s)} = \frac{- \frac{1}{L_{O} C_{I}}}{s^{2} + \frac{1}{R_{I} C_{I}} s + \frac{1}{L_{O} C_{I}}}

(6)

which is actually a second order linear model of Figure 2(b) around the operating point. The values of $C_{I}$ and $L_{O}$ are known, but value of $R_{I}$ is unknown. On the other hand, the operating point of the system will change by variation of the weather conditions. This, in turn, causes to change the effective value of $R_{I}$ . In (6), the natural frequency is $ω_{n} = \frac{1}{L_{O} C_{I}}$ and the damping ratio is $ζ = \frac{1}{2 R_{I}} \sqrt{\frac{L_{O}}{C_{I}}}$ . As these relations show, the variations of $R_{I}$ affect the damping ratio, which may result in undesired system response such as oscillation response or slow response. To overcome this problem, a MRAC has been proposed to regulate the dynamics of the closed-loop system. Applying the MRAC enables us to tune the damping ratio of the controlled plant to reach the suitable response of the closed loop system that has the appropriate velocity without much oscillating. In the constant operating point, the value of $R_{I}$ can be tuned for the desired response, but this tuned value of $R_{I}$ is not efficient for the other operating points. Therefore, an adaptive controller is required for tuning of the damping ratio of the closed loop system.

Proposed GMPPT algorithm

In this section, a two-level control structure is proposed for GMPPT. This structure is shown in Figure 3. In the first control level, GPESC has been used to compute v for delivering the maximum available power of the PV array to the load in the steady state. Converting v to the duty cycle $d (t)$ has been performed by a block with $d (t) = \frac{v - v_{p v}}{v}$ as its relation. Next, in the second control level, the MRAC has been utilized for handling the dynamics of the boost converter in response to the duty cycle calculated from the first control level. This stage leads us to achieve the appropriate response of the closed loop system. The GPESC level is responsible to capture variations in weather conditions. Thus, the tuning process of GPESC should be fast enough to catch up the changes in weather conditions. Therefore, the time constant of GPESC is required to be smaller than that of the dynamics of the weather changes. On the other hand, the tuning process of MRAC must be fast enough to catch up with the changes in the operating point of the boost converter in response to the GPESC. Therefore, MRAC must be much faster than the GPESC level. In other words, the time constant of MRAC should be much smaller than that of the GPESC. The substantial difference between these two time constants has been utilized to decouple the analyses for GPESC and MRAC. This decoupling greatly reduces the complexity of the overall control design.

Figure 3.

Proposed two-level control structure.

Remark 1: The dynamic behaviour of DC-DC converter is very important and without consideration of dynamic response of the boost converter, the GMPPT algorithm does not yield the favourable results, or in the worst case fails in finding the GMPP. The basis of our approach is decoupling the GPESC and MRAC. This means that the MRAC must be much faster than GPESC and it could be achieved via the selection of a very fast reference model for MRAC. In other words, the boost converter may have the slow dynamics, but choosing a very fast reference model can meet the requirements for this decoupling.

Global perturbation-based extremum seeking control

In this subsection, the GPESC based on Bizon (2016b) is presented. The diagram of the GPESC is shown in Figure 4. Specifications of the GPESC parameters are presented in Table 3. The mean value (MV) block computes the MV of the filtered probing signal during a dither period to produce extra smooth signal. The $v_{3}$ dither with minimum amplitude ( $A_{m}$ ) has been used to start the searching on a static nonlinear map, $p = h (v)$ . The equations of the signal processing based on the GPESC scheme are

\begin{matrix} v = h (p), p_{N} = k_{N_{p}} . p \\ {\overset{\cdot}{p}}_{f} = - ω_{h} . p_{f} + ω_{h} . p_{N}, p_{H P F} = p_{N} - p_{f}, \\ {\overset{\cdot}{p}}_{B P F} = - ω_{l} . p_{B P F} + ω_{l} . p_{H P F} \\ p_{D M} = p_{B P F} . s_{d}, s_{d} = \sin (ω t) \\ {\overset{\cdot}{p}}_{I n t} = p_{D M} \\ G_{d} = | p_{M V} |, p_{M V} = \frac{1}{T_{d}} \int p_{B P F} d t \\ v_{1} = k_{1} . p_{I n t}, k_{1} = γ_{s d} . ω \\ v_{2} = k_{2} . G_{d} . s_{d} \\ v_{3} = A_{m} . s_{d} \\ v = k_{N_{v}} (v_{1} + v_{2} + v_{3}) + v_{0} \end{matrix},

(7)

where, (7) represents the static map, normalization, band pass filtering, demodulation, integration, computation of the gain dither ( $G_{d}$ ), and $v_{1}$ , $v_{2}$ , and $v_{3}$ components of the searching signal (v). Note that $p_{f}$ is an intermediate variable related to high pass filter operating, and the MV block uses the Pade approximation. The MV block (the moving average filter) has been used to further smooth the $p_{B P F}$ signal.

Figure 4.

The GPESC diagram for GMPPT in the PV system.

Table 3.

Value and description on GPESC parameters.

Parameter	Description	Value	Unit
$f_{d}$	dither frequency	100	Hz
$f_{h}$	cut-off frequency of the HPF	10	Hz
$f_{l 1}$	cut-off frequency of the LPF1	190	Hz
$A_{m}$	minimum amplitude of dither	0.001	V
$k_{N_{p}} \geq 1 / p_{m a x}$	normalization gain	$1 / 300$	—
$k_{N_{v}} \geq v_{N}$	normalization gain	30	—
$k_{2} \geq Δ v$	gain of the dither amplitude	3	—
$k_{1} = 2 π f_{d} γ_{s d}$	loop gain	$300 π$ (with $γ_{s d} = 1.5$ )	—

Stability analysis of the above GPESC scheme is based on averaging stability analysis technique that has been completely presented in Bizon (2016b) and it has been removed here for the sake of time.

Model reference adaptive control

In the previous section, the first control objective was achieved and GPESC was utilized to compute the duty cycle for GMPPT in the steady state. The second control aim is to reach the response of the overall system with appropriate characteristics. For this reason, the MRAC has been applied to achieve the desired response. In MRAC, the output of the actual system is continuously compared with the output of the reference model. The reference model is the one that gives the desired response to the command signal. Both the actual system and the reference model are motivated by the same command signal. The goal is to decrease the differences between the two outputs, it means that the actual system responds like the reference model (desired) system. For this purpose, the controller adapts the gains such that the output tracking error will be minimized. The proposed scheme of MRAC is shown in Figure 5. The plant model in Figure 5 is the boost converter of which its transfer function is described by (6). The input to overall system $d (t)$ is the duty cycle computed by GPESC. $u (t)$ is the duty cycle of the switching device Q in the boost converter, which is the output of MRAC. $y_{p} (t)$ is the output voltage of the boost converter. For convenience, the model of the plant has been rewritten as

G_{p} (s) = \frac{y_{p} (s)}{u (s)} = \frac{k_{p}}{s^{2} + a_{p} s + b_{p}}

(8)

where $k_{p} = - \frac{V_{O}}{L_{O} C_{I}}$ , $a_{p} = \frac{1}{R_{I} C_{I}}$ and $b_{p} = \frac{1}{L_{O} C_{I}}$ . The equation below represents the transfer function of the reference model in Figure 5

G_{m} (s) = \frac{y_{m} (s)}{d (s)} = \frac{k_{m}}{s^{2} + a_{m} s + b_{m}}

(9)

in which $k_{m}$ is a positive gain. $a_{m}$ and $b_{m}$ are chosen so that the reference model generates the desired response. As mentioned, the control objective is to tune $u (t)$ so that $y_{p} (t)$ asymptotically tracks $y_{m} (t)$ .

Figure 5.

(a) The structure of proposed MRAC in the second control level. (b) The controller structure in the proposed MRAC.

According to Figure 5(b) and based on Ioannou and Fidan (2006), and Sastry and Bodson (1989), the adequate controller structure for the objective control is

\begin{matrix} u = θ_{0} d + θ_{1} \frac{1}{s + λ} u + θ_{2} \frac{1}{s + λ} y_{p} + θ_{3} y_{p} \\ = θ_{0} d + θ_{1} u_{f} + θ_{2} y_{f} + θ_{3} y_{p} = θ^{T} ω \end{matrix}

(10)

In above controller structure, $ω \equiv {[d u_{f} y_{f} y_{p}]}^{T}$ is a regressor vector with $u_{f} = \frac{1}{s + λ} u$ and $y_{f} = \frac{1}{s + λ} y_{p}$ . $\frac{1}{s + λ}$ are arbitrarily stable filters with $λ > 0$ . So, the dynamic models of $u_{f}$ and $y_{f}$ cab be expressed as

\begin{matrix} {\overset{\cdot}{u}}_{f} = - λ u_{f} + u \\ {\overset{\cdot}{y}}_{f} = - λ y_{f} + y_{p} \end{matrix}

(11)

The vector $θ \equiv {[θ_{0} θ_{1} θ_{2} θ_{3}]}^{T}$ in the controller is not known. Actually, the art of MRAC is to fit suitable controller parameters, which, in turn, would be obtained by finding worthy adaptive laws. The adaptive laws must be able to estimate the unknown parameters of the controller besides guaranteeing the closed loop stability. To reach this objective it is convenient to use the certainty equivalence principle and Lyapunov stability theorem. The following procedure will be carried out to find the adaptive laws.

Suppose that the nominal parameter values are $θ^{*} \equiv {[θ_{0}^{*} θ_{1}^{*} θ_{2}^{*} θ_{3}^{*}]}^{T};$ so, the close loop transfer function matches the reference model with the below equations

\begin{matrix} θ_{0}^{*} = \frac{k_{m}}{k_{p}} \\ θ_{1}^{*} = a_{p} - a_{m} \\ θ_{2}^{*} = \frac{(a_{p} - a_{m}) (- λ^{2} + λ a_{p} - b_{p})}{k} \\ θ_{3}^{*} = \frac{b_{p} - b_{m} + (a_{p} - a_{m}) (λ - a_{p})}{k_{p}} \end{matrix}

(12)

Let ${A_{p}, B_{p}, C_{p}}$ be a minimal realization of the plant $G_{p} (s)$ and $x_{p} \in R^{2}$ .

\begin{matrix} {\overset{\cdot}{x}}_{p} = A_{p} x_{p} + B_{p} u \\ y_{p} = C_{p} x_{p} \end{matrix}

(13)

With an augmented state vector $Y_{c} = {[x_{p}^{T} u_{f} y_{f}]}^{T}$ , a composite state space representation of the plant and controller is

\begin{matrix} {\overset{\cdot}{Y}}_{c} = A_{c} Y_{c} + B_{c} θ_{0}^{*} d + B_{c} (u - {θ^{*}}^{T} ω) \\ y_{p} = C_{c} Y_{c} \end{matrix}

(14)

where matrices $A_{c}$ , $B_{c}$ and $C_{c}$ are defined as

\begin{matrix} A_{c} \equiv [\begin{matrix} A_{p} + B_{p} θ_{3}^{*} C_{p} & B_{p} θ_{1}^{*} & B_{p} θ_{2}^{*} \\ θ_{3}^{*} C_{p} & - λ + θ_{1}^{*} & θ_{2}^{*} \\ C_{p} & 0 & - λ \end{matrix}] \\ B_{c} \equiv {[\begin{matrix} B_{p}^{T} & 1 & 0 \end{matrix}]}^{T} \\ C_{c} \equiv [\begin{matrix} C_{p} & 0 & 0 \end{matrix}] \end{matrix}

(15)

According to (14) the state vector $Y_{c}$ is bounded if $A_{c}$ is a stable matrix and $d (t)$ is a bounded signal (Ioannou and Fidan, 2006). Using the matching condition $u = {θ^{*}}^{T} ω$ , realization of the reference model is as

\begin{matrix} {\overset{\cdot}{Y}}_{m} = A_{c} Y_{m} + B_{c} θ_{0}^{*} d \\ y_{m} = C_{c} Y_{m} \end{matrix}

(16)

where $Y_{m} \in R^{4}$ .

Now, to obtain the dynamic of the output error, consider the following steps

\begin{matrix} \overset{\cdot}{e} = A_{c} e + B_{c} (u - {θ^{*}}^{T} ω) = A_{c} e + B_{c} {\tilde{θ}}^{T} ω \\ e_{1} = C_{c} e = G_{m} (s) ρ^{*} (u - {θ^{*}}^{T} ω) \end{matrix}

(17)

where $ρ^{*} \equiv \frac{1}{θ_{0}^{*}}$ . In addition, e, $e_{1}$ and $\tilde{θ}$ are state error, output error and error of the control parameter, respectively

\begin{matrix} e \equiv Y_{c} - Y_{m} \\ e_{1} \equiv y_{p} - y_{m} \\ \tilde{θ} = θ - θ^{*} \end{matrix}

(18)

Since a necessary condition for state error equation is being strictly positive real (SPR), which is not satisfied here, the identity $(s + p_{0}) {(s + p_{0})}^{- 1} = 1$ is used for some $p_{0} > 0$ to rewrite (17) as

\begin{matrix} \overset{\cdot}{e} = A_{c} e + B_{c} (s + p_{0}) (u_{p} - {θ^{*}}^{T} ϕ) = A_{c} e + B_{c} (s + p_{0}) {\tilde{θ}}^{T} ϕ \\ e_{1} = C_{c} e = G_{m} (s) (s + p_{0}) ρ^{*} (u_{p} - {θ^{*}}^{T} ϕ) = G_{m} (s) (s + p_{0}) ρ^{*} {\tilde{θ}}^{T} ϕ \end{matrix}

(19)

where $u_{p} \equiv \frac{1}{s + p_{0}} u$ and $ϕ = \frac{1}{s + p_{0}} ω$ . The term $(s + p_{0})$ is chosen so that $G_{m} (s) (s + p_{0}) ρ^{*}$ is SPR. Therefore, it can be shown that $G_{m} (s) (s + p_{0}) ρ^{*}$ is SPR for any $p_{0}$ satisfying $0 < p_{0} < a_{m}$ . We know that $u_{p} = θ^{T} ϕ$ and $\overset{\cdot}{ϕ} + p_{0} ϕ = ω$ . So, the control law can be expressed as $u = (s + p_{0}) u_{p} = {\overset{\cdot}{θ}}^{T} ϕ + θ^{T} \overset{\cdot}{ϕ} + p_{0} θ^{T} ϕ = {\overset{\cdot}{θ}}^{T} ϕ + θ^{T} (\overset{\cdot}{ϕ} + p_{0} ϕ) = {\overset{\cdot}{θ}}^{T} ϕ + θ^{T} ω$ . Since $\overset{\cdot}{θ}$ is made available by the adaptive law, this control law can be implemented without the use of differentiators.

Now, consider the following transformation

\bar{e} \equiv e - B_{c} {\tilde{θ}}^{T} ϕ

(20)

Using that, the following relations are established

\begin{matrix} \overset{\cdot}{\bar{e}} = A_{c} e + B_{1} {\tilde{θ}}^{T} ϕ \\ e_{1} = C_{c} \bar{e} + C_{c} B_{c} {\tilde{θ}}^{T} ϕ = C_{c} \bar{e} \end{matrix}

(21)

where $B_{1} = A_{c} B_{c} + B_{c} p_{0}$ . In the second equation of (21), $C_{c} B_{c} = 0$ , because the relative degree of the reference model is two.

To find the suitable adaptive laws, consider the Lyapunov function as

V (\bar{e}, \tilde{θ}) = \frac{{\bar{e}}^{T} P_{c} \bar{e}}{2} + \frac{{\tilde{θ}}^{T} Γ \tilde{θ}}{2} | ρ^{*} |

(22)

where Γ is an arbitrary symmetric positive definite matrix and $P_{c}$ is a symmetric positive definite matrix determined using Meyer–Kalman–Yakubovich (MKY) Lemma. The time-derivative of the Lyapunov function (22) along the solution of (21) is given by

\overset{\cdot}{V} (\bar{e}, \tilde{θ}) = - \frac{1}{2} {\bar{e}}^{T} q_{c} q_{c}^{T} \bar{e} - \frac{ν_{c}}{2} {\bar{e}}^{T} L \bar{e} + {\bar{e}}^{T} P_{c} B_{1} {\tilde{θ}}^{T} ϕ + {\tilde{θ}}^{T} Γ^{- 1} \overset{\cdot}{\tilde{θ}} | ρ^{*} |

(23)

where the vector $q_{c}$ and scalar $υ_{c} > 0$ have been determined using MKY Lemma

\begin{matrix} P_{c} A_{c} + A_{c}^{T} P_{c} + q_{c} q_{c}^{T} + υ_{c} L = 0 \\ P_{c} B_{1} = C_{c}^{T} \end{matrix}

(24)

for any given symmetric positive definite matrix L. Since ${\bar{e}}^{T} P_{c} B_{1} = {\bar{e}}^{T} C_{c}^{T} = e_{1}$ , the adaptive law can be chosen as

\overset{\cdot}{θ} = \overset{\cdot}{\tilde{θ}} = - Γ e_{1} ϕ s g n (ρ^{*})

(25)

to have the derivative of the Lyapunov function negative.

\overset{\cdot}{V} (\bar{e}, \tilde{θ}) = - \frac{1}{2} {\bar{e}}^{T} q_{c} q_{c}^{T} \bar{e} - \frac{ν}{2} {\bar{e}}^{T} L \bar{e} \leq 0

(26)

This implies that $\bar{e}, e_{1}, \tilde{θ} \in L_{\infty}$ and $\bar{e}, e_{1} \in L_{2}$ . Since $e_{1} = y_{p} - y_{m}$ , it is concluded that $y_{p} \in L_{\infty}$ .

Totally, the equations of the overall MRAC scheme are

\begin{matrix} {\overset{\cdot}{u}}_{f} = - λ u_{f} + u \\ {\overset{\cdot}{y}}_{f} = - λ y_{f} + y_{p} \\ \overset{\cdot}{ϕ} = - p_{0} ϕ + ω, ω \equiv {[d u_{f} y_{f} y_{p}]}^{T} \\ u = {\overset{\cdot}{θ}}^{T} ϕ + θ^{T} ω = - ϕ^{T} Γ e_{1} ϕ s g n (ρ^{*}) + θ^{T} ω \\ \overset{\cdot}{θ} = - Γ e_{1} ϕ s g n (ρ^{*}) \end{matrix}

(27)

Based on Ioannou and Fidan (2006), all signals in the closed-loop system are bounded and $e_{1} (t) \to 0$ as $t \to \infty$ .

Simulation results and discussion

In this section, the proposed control structure has been compared with two methods: the GPESC which is proposed in Bizon (2016b), and with GPESC with PID for GMPPT in PV array in different PV patterns.

For the switch Q of the boost converter, the pulse width modulation (PWM) technique with switching frequency $f_{s} = 10 K H z$ has been utilized. Also, sampling period of simulations has been selected as $T_{s} = 1 μ s$ . Therefore, in each switching interval, 100 samples of the variables are existed. For real-time implementation, this sampling period is so small. But, if the sampling period for real-time implementation will be selected 10 times greater than ones used in the simulation ( $T_{s} = 10 μ s$ ), in each switching interval, there are 10 samples of the variables, yet. Therefore, it seems that $f_{s} = 10 K H z$ is a suitable choice for switching frequency.

The equation (6) has been used for designing of MRAC, not for simulation. Indeed, the boost converter has been simulated based on its real model, using Simscape (Electrical) Toolbox of Simulink®. Therefore, the dynamic behaviour of the controller for the real model of the boost converter with all parasitic components has been studied in the simulations.

The boost converter specifications, the parameters of the reference model and MRAC used in simulations have been presented in Table 4. The other parameters used in the simulations were mentioned before in Table 3. But the normalization gains should be modified for each PV pattern. If the maximum speed changes in shaded conditions is 10 times per second, then $f_{d} = 100 H z$ (dither frequency in GPESC) can capture the changes in the weather conditions. Consequently, the time constant of GPESC is $\frac{1}{f_{d}} = 10 m s$ . As it was mentioned earlier, in the transfer function of the plant, the value of $R_{I}$ is not known. In the simulations, the value of $R_{I}$ has been assumed $R_{I} = 0.16 Ω$ . The aim of choosing the big capacitance value ( $4500 μ F$ ) of the input capacitor in the boost converter is the oscillation reduction in the electrical input variables. This choice has been resulted in the natural frequency $ω_{n} = \sqrt{\frac{1}{L_{O} C_{I}}} = 1005 r a d / s$ and damping ratio $ζ = \frac{1}{2 R_{I}} \sqrt{L_{O} / C_{I}} = 0.7$ . This damping ratio yields underdamped or slow response of the boost converter. But, using MRAC and selection of very fast reference model, the boost converter is forced to have the small time constant and fast response. Consequently, the parameters of reference model in MRAC have been selected as in Table 4. The reference model has been designed to deliver maximum power with a critically damped step response ( $ζ_{m} = 1$ ). According to the proposed method, we need to decouple GPESC and MRAC. GPESC has large time constant and slow response, inherently. So, we need very fast MRAC. Although the boost converter has a slow response individually, the parameters of the reference model have, however, been chosen such that results in very fast system with a critically damped step response ( $ζ_{m} = \frac{a_{m}}{2 \sqrt{b_{m}}} = 1$ ) and very fast natural frequency $\sqrt{b_{m}}$ in comparison to the first level of control structure, namely GPESC loop. According to the step response of the reference model (when the output reaches to 63.2% of its final value.), its time constant is $71.5 μ s$ . This time constant is small enough for MRAC to be so faster than GPESC. $K_{m}$ is chosen to lead the unit DC gain for the reference model (when $s = 0$ ). Also, it should be noted that there are not any limitations in selecting excessive values for the parameters of reference model, practically. This is because, the reference model runs only as the software and not as a real circuit. In addition, the stable filters have been designed such that having efficiently fast responses.

Table 4.

The parameters of the boost converter, reference model and MRAC.

Parameter	Description	Value	Unit
$L_{O}$	boost converter inductance	220	$μ H$
$C_{O}$	output capacitor of boost converter	200	$μ F$
$C_{I}$	input capacitor of boost converter	4500	$μ F$
$λ$	time constant of stable filter in MRAC	100	—
$p_{0}$	positive constant	100	—
$Γ$	coefficients of MRAC	$d i a g {10, 10, 10, 10}$	—
$k_{m}$	reference model gain	$9 \times 10^{8}$	${(r a d / s)}^{2}$
$a_{m}$	$2 ζ ω_{n}$	$6 \times 10^{4}$	$r a d / s$
$b_{m}$	square of natural frequency of reference model	$9 \times 10^{8}$	${(r a d / s)}^{2}$

Four indicators have been utilized: searching resolution ( $R_{s}$ ), tracking accuracy ( $T_{a c c}$ ), tracking efficiency ( $T_{e f f}$ ), and tracking speed for evaluating the performance of the proposed control scheme. Respectively, these indicators are defined as

R_{s} = \frac{m i n_{i} | p_{G M P P} - p_{L M P P_{i}} |}{p_{G M P P}} \times 100 %

(28)

T_{a c c} = \frac{p_{P V}}{p_{G M P P}} \times 100 %

(29)

T_{e f f} = \frac{\int_{0}^{t} p_{P V} d t}{\int_{0}^{t} p_{G M P P} d t} \times 100 %

(30)

where $p_{p v}$ is the PV array power that is theoretically available and $p_{G M P P}$ is the power extracted by the GMPP algorithm. Tracking speed is measured by the number of iterations to locate the GMPP. An iteration for the PESC-based GMPPT algorithms means a dither period that is a step of scanning.

In the all scenarios of the simulations, GPESC is unsuccessful to track the GMPP for the dynamical system with uncertainties and the output power of PV system is converged to zero. But because of page limitation, the results of this approach have not been presented and for a fair comparison, the GPESC with a PID controller used to handle the dynamic behaviour of the boost converter has been considered. Also, the proposed scheme has been compared to GPESC with PID controller. The PID coefficients have been tuned as $P = 100$ , $I = 10$ and $D = 1$ for the all patterns. It is important to note that for using the different converters, the tuning of PID controller is necessary, but one of the main features of MRAC is that it requires no adjustment for the different converter.

The simulation results are shown in Figures 6 to 8 for the different PV patterns. Based on these figures, one can see that the proposed control scheme can track GMPP in the different scenarios. Although GPESC with PID controller is also successful in tracking the GMPP, the better performance of the proposed scheme is obvious in all the scenarios. As is illustrated in Figures 6 to 8, the steady state, the ripple of voltage and power is so small in both methods. But, in the transient, these components have ripple. It should be noted that the transient time in the proposed method is very short. So, voltage and power reach to steady state with no ripple in a very small time. In Table 5, the comparison of the proposed structure with GPESC with PID in the three patterns has been presented based on the indicators that were explained before. It should be explained that based on (28), the searching resolution for all PV patterns is $R_{s} = \frac{10}{430} = 2.33 %$ for the all patterns. Based on Figures 6 to 8, the tracking time for the different PV patterns is about $0.23 s$ , $0.27 s$ and $0.26 s$ for the proposed controller; and is $60 s$ , $78 s$ and $69 s$ for GPESC with PID. The tracking speed is a good alternative instead of the tracking time for evaluating the performance of GMPPT schemes. Regarding $f_{d} = 100 H z$ , the tracking speed has been computed for two schemes and can be seen in Table 5. It should be noted that increasing the dither frequency yields to increasing the tracking speed, linearly. On the other hand, increasing the dither frequency can decrease the search phase, and as a result leads to increasing the tracking efficiency. But using the very high frequency for the dither signals may yield to inability of the filters in the separation of the dither signals and noises. Based on the results presented in Table 5, it is clear that GPESC with MRAC can improve the tracking accuracy slightly, and it can increase the tracking efficiency considerably in the comparison to GPESC with PID controller. Also, enhancement of the tracking speed in the proposed method is very obvious based on Table 5.

Figure 6.

Comparison of GMPP searching in the GPESC with MRAC scheme and the GPESC with PID controller: (a) PV pattern with GMPP in left, (b) PV power, (c) PV voltage.

Figure 7.

Comparison of GMPP searching in the GPESC with MRAC scheme and the GPESC without MRAC: (a) PV pattern with GMPP in middle, (b) PV power, (c) PV voltage.

Figure 8.

Comparison of GMPP searching in the GPESC with MRAC scheme and the GPESC without MRAC: (a) PV pattern with GMPP in right, (b) PV power, (c) PV voltage.

Table 5.

Comparison of the performance of the proposed control scheme and GPESC with PID controller.

Location of maximum	Controller	$R_{s} [%]$	$T_{a c c} [%]$	$T_{e f f} [%]$	Tracking Speed [dither periods]
Left	GPESC with PID	2.33	99.97	92.74	60
Left	GPESC with MRAC	2.33	99.99	98.72	23
Middle	GPESC with PID	2.33	99.97	90.07	78
Middle	GPESC with MRAC	2.33	99.99	98.42	27
Right	GPESC with PID	2.33	99.97	91.56	69
Right	GPESC with MRAC	2.33	99.99	98.45	26

Remark 2: In this work, the proposed method, namely GPESC with MRAC, has been compared with GPESC, and GPESC with PID controller. As was mentioned in Remark 1 in Section 3, one of the main targets of the paper is showing the importance of consideration of the DC-DC converter’s dynamics. Without consideration of the boost converter’s dynamics, the GMPPT algorithm does not yield the favourable results, or in the worst case fails in finding of GMPP. Also, when there is uncertainty in the boost converter, simple controllers such as PID controller are not satisfactory in order to capture the converter’s dynamics. On the other hand, in the proposed method, the decoupling of GPESC and MRAC has been achieved via selection of fast reference model in MRAC. In other controllers, this option there does not exist. Therefore, it seems that combination of GPESC with other controllers cannot yield the desired performance as illustrated in the combination of GPESC with PID controller. The basis of the proposed method is the difference between the time constants of the control levels, but this property has not been satisfied in the other control methods. Therefore, comparison with other controller combined with GPESC will lead to the same results as GPESC and PID controller.

Conclusion

In order to deliver the maximum available power from the PV array to the load, the two-level GMPPT control structure, was proposed. In the first level, GPESC was utilized to calculate the setpoint for GMPPT in the steady state. In the second level, the dynamic model for the boost converter was constructed; then, MRAC was used to achieve the desired dynamical performance in the response to the setpoint calculated by GPESC in the previous level. This level compensated the underdamped characteristics of the boost converter. Owing to the satisfactory difference in time constant of each level of the control structure, the design and analysis of those levels was decoupled. Therefore, the difficulty of designing and implementation was reduced. The simulation results demonstrated that owing to the converter dynamics, the GPESC cannot reach the GMPP. The performance of the proposed method was evaluated based on four indicators: searching resolution, tracking accuracy, tracking efficiency and tracking speed. Based on the simulation results, one can see that GPESC with MRAC yields the better performance in comparison with GPESC with PID controller. For future work, stability analysis of combined GPESC and MRAC will be studied.

Footnotes

Declaration of conflicting interest

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

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