MIMO nonlinear modeling and robust control of standalone dc photovoltaic systems

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MIMO nonlinear modeling and robust control of standalone dc photovoltaic systems lu Onur Deveci*, Cos‚ku Kasnakog TOBB Economics and Technology University, Electrical & Electronics Engineering Department, Turkey

article info

abstract

Article history:

Multiple-input multiple-output (MIMO) modeling and simulation techniques are utilized to

Received 15 November 2016

boost the usefulness of a common DC-output photovoltaic (PV) system. Vast studies in the

Received in revised form

literature are mostly based on single-input single-output (SISO) methods. As a result they

3 April 2017

suffer numerous unpredicted features should they be exposed to particular temperature,

Accepted 6 April 2017

irradiation and load changes. These features stem from parameter errors and unmodelled

Available online 29 April 2017

dynamics of actual circuit components. Another issue is that common linearization may fail due to substantial nonlinear behavior. These are caused by switching converters run by

Keywords:

means of pulse width modulation (PWM). As a remedy a different method relying on

Photovoltaic

simulated input/output data is put to use to compute a working condition and then

PV

obtaining a linear model of the entire system around it. Multiple SISO compensators are

Voltage regulation

replaced with a single MIMO robust controller based on the MIMO model, after which the

MPPT

controlled system is tested in simulations with varying irradiation, temperature and load

Hammerstein-Wiener

resistance values. Evaluating the outcomes unveils the fact that newly designed autono-

MIMO robust control

mous system can run the PV panel at the maximum power point over a wide envelope of atmospheric and load combinations. Moreover the voltage value fed to a sensitive load is constant and the extra power obtained from the PV panel is used to charge the battery. © 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Introduction The United Nations predict that the current world population of nearly 8 billion will reach to about 10 billion in 35 years [1]. Such population growth along with industrialization will increase the energy demand considerably. The International Energy Agency projects that the global energy consumption will grow about one-third and the electricity demand of the world will increase more than 70% until 2040 [2]. Oil, gas and coal-based energy generation is not sustainable due to their contribution to global warming as well as expected depletion of reserves by years 2040, 2042 and 2112

respectively [3,4]. Researchers are looking to develop better ways of utilizing sustainable and inexhaustible renewable energy technologies for electricity generation which is expected to generate about half of all growth over the period to 2040 [2,5]. Different technologies are developed to obtain energy from different renewable resources including solar, wind, hydropower, biofuel, geothermal, tidal, biomass and wave power systems. Among these, solar energy is a remarkable type of renewable energy with its low operational costs and maintenance requirements, no moving parts, no carbon emissions and long lifetimes (more than 20 years) [6,7]. Moreover, photovoltaic (PV) systems have long lifetimes which makes

* Corresponding author. E-mail address: [email protected] (O. Deveci). http://dx.doi.org/10.1016/j.ijhydene.2017.04.033 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

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them economically and technically feasible [8]. Although the amount of power from the sun that end up on the surface of the Earth is low, it is quite enough to satisfy the global energy demand [9]. But high capital costs for system installation and the intermittency of the energy source e Sun e affect adversely the widespread use of solar energy [10]. Research on system components having lower-cost and higher efficiencies and better control strategies to capture more energy and much better regulated outputs are in progress to overcome these drawbacks. Typical stand-alone PV systems are composed of PV panels, electric batteries, battery chargers, maximum power point trackers (MPPT), DC/DC converters, and inverters [11,12]. Diesel generator and ultracapacitor based hybrid renewable power generation systems are also available in the literature [13,14]. Several MPPT methods perturb and observe, like hill climbing, incremental conductance, and fuzzy logic controllers are used to adapt PV panel against negative effects of changing atmospheric conditions so that higher energy conversion efficiency can be achieved with less number of PV panels or to overcome negative effects of PV panel shading on energy output [15,16]. There are also some studies regarding non-linear model based MPPT controller design for PV systems in the literature [17,18]. Other than MPPT, controllers for stand-alone PV systems with DC load can be designed for the purposes of battery charging/discharging and output voltage regulation. Duryea et al. [19], Glavin et al. [20], Pacheco et al. [21], Hong et al. [22] and Huang et al. [23] developed control systems based on battery management and/or load regulation without MPPT that prevents effective utilization of the solar energy. Matsuo et al. [24], Liao et al. [25], Abouda et al. [26], Hasan et al. [27], Wang et al. [28] and Rani et al. [29] included MPPT algorithm together with battery charging/discharging control and load voltage regulation but such studies do not include long-term results under variable atmospheric conditions and load changes. Moreover, controllers utilized in these studies are typically based on single-input single-output (SISO) methods. But due to substantial nonlinearities in PV systems attributable to pulse-width-modulation-(PWM)-operated DC/DC converters, traditional linearization techniques may not be directly applicable to the design of such controllers. PV systems within the scope of this study have a multipleinput multiple-output (MIMO) structure susceptible to these difficulties, making it valuable to investigate robust MIMO controllers that can withstand varying atmospheric conditions and load levels. In this study, a novel modeling and control structure for PV systems are considered. The modeling approach in this study is based on identifying a nonlinear MIMO HammersteinWiener (HW) plant [30,31]. While these models are widely utilized in many fields, only a few applications have been reported for PV systems (e.g. Ref. [32]) none of which consider a full system model and control design for simultaneous DC output and PV panel voltage regulation. In contrast, our model represents the whole system consisting of the DC/DC converters, PV panels, electric battery and the resistive load. Moreover, we consider an innovative extension to the method by constraining the input/output nonlinearities to be invertible and derive the required conditions. This allows for a

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strategy where the inverse functions are utilized to carry out controller design on the linear block of the HW model and later map it to the full nonlinear HW system. With the help of this technique, H∞ loop shaping and Linear-QuadraticGaussian (LQG) MIMO robust controllers based on Hammerstein-Wiener nonlinear MIMO model of the PV system are designed. Performance of these robust MIMO controllers are compared with previously designed ProportionalIntegral (PI) and PI/LQG hybrid SISO controllers found in literature [33,34] under different irradiation/temperature/load conditions. Results regarding performance and robustness of output voltage regulation, as well as sensitivity to uncertainties in system parameters are discussed.

System configuration Overall system structure A typical PV system configuration, based on an existing study [33], is given in Fig. 1. A bidirectional boost converter for controlling charge and discharge cycles is present between the battery and the load. For controlling MPPT a unidirectional boost converter is located between the solar panels and the load. In literature, the PV system given in Fig. 1 is typically controlled in three stages: PV power controller, PV Unidirectional Converter Controller and Battery Bidirectional Controller. Each of the stages is based on PI controllers. In a related study, PI/LQG hybrid control structure is utilized to improve system performance [34]. These approaches employ SISO design methods based on linearization techniques and suffer numerous problems in the presence of multiple inputs/ outputs as well as nonlinearities. In this work, the standard SISO controllers (composed of four separate PI controllers based on four linearized models) are to be replaced with a single MIMO controller based on a nonlinear MIMO model. This is achieved with the configuration given in Fig. 2 that enables collecting identification data to be used for obtaining a MIMO model through system identification. Fig. 3 shows the inside of the Whole PV System block. S1 port represents the duty cycle Dpv, and S2 represents the duty cycle Db, which are the system inputs. S3 is the inverse of S2 due to bidirectional converter structure. The outputs are Vdc and Vpv, which are the DC and panel voltages. The input/ output data collected from simulations is shown in Fig. 4 as variations (D values) around a typical operating point (Db,op, Dpv,op, Vdc,op, Vpv,op) ¼ (0.4, 0.4, 400 V, 193 V). That is, Db ¼ Db;op þ DDb Dpv ¼ Dpv;op þ DDpv Vpv ¼ Vpv;op þ DVpv Vdc ¼ Vdc;op þ DVdc These variations will be used for system identification to obtain a MIMO model of the system, which will be described next.

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Fig. 1 e PV system configuration [33].

Fig. 2 e Configuration for numerical data generation.

Fig. 3 e Inside structure of the “Whole PV System” block.

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Fig. 4 e Variations around the operating point (Db,op, Dpv,op, Vdc,op, Vpv,op) ¼ (0.4, 0.4, 400 V, 193 V) for inputs (DDb and DDpv) and outputs (DVdc and DVpv). These are the data for system identification.

Modeling through system identification Once the inputeoutput data is obtained, nonlinear MIMO system identification is performed. A HW nonlinear model structure (Fig. 5) which consists of a linear block and input/ output nonlinearities [30,31] is utilized to obtain an accurate estimation based on simulated data. According to Fig. 5,  G(s) is a linear transfer function,  f is a nonlinear function that maps the input of the system u(t) to the input of G(s),  h is a nonlinear function that maps the output of G(s) to the output of the system. Many numerical computing packages (including MATLAB used here) have tools to build HW models with typical input/ output nonlinearities, but building a controller based on these models are quite difficult when f and g are not invertible and high-degree. So several constraints over the identification operation should be imposed to profit the MIMO compensator

design [31]. Here the nonlinear functions are restricted to be polynomials of degree two. In addition, they need to have an inverse on the intervals of interest, which is [0.25, 0.25] for the inputs and [80, 80] for the outputs in line with the actual physical attributes of the PV system. Specifically, the input polynomial f has the structure fi ðui Þ ¼ ai2 u2i þ ai1 ui þ ai0

(1)

where ai2 ; ai1 ; ai0 2ℝ and i ¼ 1,2 represents the respective input. The only extremum point of this function is at ui ¼ ai1 =ð2ai2 Þ. Therefore invertibility will be guaranteed if this point is not inside the region [0.25, 0.25], i.e. ai1 ai1  0:25 or   0:25  2ai2 2ai2

(2)

Equivalently,


ai1
a2i1

 0:252
2a
 0:25⇔ i2 ð2ai2 Þ2 which implies

Fig. 5 e Hammerstein-Wiener model.

(3)

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a2i1  0:0625a2i2  0

min JðpÞs:t:ci1 ðpÞ  0 and cj2 ðpÞ  0

(4)

Likewise the output nonlinear function h has the structure   hj xj ¼ bj2 x2j þ bj1 xj þ bj0

where J is given in (10). The inequality constraints obtained from (4) and (8) are

(5)

where bj2 ; bj1 ; bj0 2ℝ and j ¼ 1,2 represents the respective output. The only extremum point of h is at u ¼ bj1 =ð2bj2 Þ so making sure it sits outside [80, 80] requires bj1 bj1  80 or   80  2bj2 2bj2

ci1 ðpÞ ¼ a2i1  0:0625a2i2

(15)

cj2 ðpÞ ¼ b2j1  6400b2j2

(16)

(6)

To solve the problems in Eqs. (15) and (16), Sequential Quadratic Programming (SQP) method is applied [35,36]. We first formulate a Quadratic Programming (QP) task approximating the Lagrangian function as

(7)

L ðp; s11 ; s12 ; s21 ; s22 Þ ¼ JðpÞ  s11 c11 ðpÞ  s12 c12 ðpÞ  s21 c21 ðpÞ

Equivalently,


b
b2j1
j1
2

 80⇔ 2  80
2bj2
2bj2

 s22 c22 ðpÞ

which implies b2j1



6400b2j2

(17)

0

(8)

where sij are Lagrange multipliers. The QP task given below is solved for the search direction dk at each iterate pk

As to the linear block, we employ a transfer function of order n XðsÞ bn sn þ bn1 sn1 þ … þ b2 s2 þ b1 s1 þ b0 ¼ GðsÞ ¼ WðsÞ an sn þ an1 sn1 þ … þ a2 s2 þ a1 s1 þ a0

   T 1 min dT V2 L pk ; s11;k ; s12;k ; s21;k ; s22;k d þ VJ pk d dk 2

(9)

(18)

subject to

where an ; an1 ; …; a2 ; a1 ; a0 ; bn ; bn1 ; …; b2 ; b1 ; b0 2ℝ. The goal is to minimize cost function J characterized by the integral of the squared error (ISE)

 T    T   Vci1 pk d þ ci1 pk ¼ 0; Vcj2 pk d þ cj2 pk ¼ 0

e2 dt

(19)

The problem above can be handled using any QP algorithm as these are well established. A new iterate is formed using he solution dk

Ztend JðpÞ ¼

(14)

p

(10)

tstart

pkþ1 ¼ pk þ ak dk

where p ¼ ½an an1 …a1 a0 bn bn1 …b1 b0 a12 a11 a10 a22 a21 a20 b12 b11 b10 b22 b21 b20  (11) is the parameter vector, [tstart, tend] ¼ [0s, 12s] is the time interval for estimation eðtÞ ¼ ydata ðtÞ  ymodel ðtÞ

(12)

is the error representing how well the estimation data (ydata) fits the HW model output which is ymodel ¼ hðGðf ðudata ÞÞÞ

(13)

The estimation process could thus be expressed as a constrained nonlinear optimization in the following common form

2 6 6 6 6 6 6 6 6 6 A¼6 6 6 6 6 6 6 6 6 4  C¼

T

where ak is attained through line search to ensure that a sufficient reduction in the merit function below is accomplished JðpÞ ¼ JðpÞ þ r11 c11 ðpÞ þ r21 c21 ðpÞ þ r12 c12 ðpÞ þ r22 c22 ðpÞ

0:925 0

0 0

0 0:764

0 0:5

0 0

1 0

0:49 0

0 0

(21)

The penalty parameters rij are positive numbers. Implementing the operations above, a HW model for the whole PV system is obtained. For controller design, the linear block of the system is written as xðk þ 1Þ ¼ AxðkÞ þ BuðkÞ yðkÞ ¼ CxðkÞ þ DuðkÞ

(22)

where x represents 12 states, u represents the inputs DDb and DDpv, y represents outputs DVdc and DVpv and the matrices A, B, C, D are as follows

3 0:906 0:9838 0:895 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 7 7 0 1 0 0 0 0 0 0 0 0 0 0 7 7 0 0 0 2:806 1:316 0:825 0 0 0 0 0 0 7 7 0 0 0 2 0 0 0 0 0 0 0 0 7 7 0 0 0 0 0:5 0 0 0 0 0 0 0 7 7; 0 0 0 0 0 0 2:79 1:301 0:811 0 0 0 7 7 0 0 0 0 0 0 2 0 0 0 0 0 7 7 0 0 0 0 0 0 0 0:5 0 0 0 0 7 7 0 0 0 0 0 0 0 0 0 2:795 1:306 0817 7 7 0 0 0 0 0 0 0 0 0 2 0 0 5 0 0 0 0 0 0 0 0 0 0 0:5 0 1 0

(20)

0 2:438

0 1:218

   0 0 0 ; D¼ 0 0 0

2

1 60 6 60 6 61 6 60 6 60 B¼6 60 6 60 6 60 6 60 6 40 0

3 0 07 7 07 7 07 7 07 7 07 7 17 7 07 7 07 7 27 7 05 0

(23)

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The nonlinear functions for input and output nonlinearities have been estimated as: yin1;DDb ¼ 5:62x106 x2 þ 1:047x  8:93x105 yin2;DDpv ¼ 0:0001x2 þ 0:99x  0:00047 yout1;DVdc ¼ 0:001659x2 þ 1:678x  18:7 yout2;DVpv ¼ 0:0008937x2 þ 0:6814x  5:91 For comparison with the constrained second order polynomial fit, an unconstrained optimization with various function types have been performed, and the best results have been obtained with wavelet networks [37]. Both types of nonlinearity functions are plotted in Fig. 6. Clearly the second order constrained fit provides an acceptable invertible approximation within the region of interest. Finally, a comparison of the outputs estimated by the HW model and to measured output data are shown in Fig. 7. The percent mean squared error for the fits are 97.1% for DVdc and 96% for DVpv which are judged to be of enough accuracy in representing the PV system.

With the help of Figs. 5 and 8, and using the fact that the functions f and h have been forced to be invertible by design, the equivalent closed-loop system for the whole HW model is given in Fig. 9 [38]. Based on this structure, the actual MATLAB/Simulink implementations of the controlled system for LQG and H∞ loop shaping design are shown in Figs. 10 and 11. PV panel reference voltage is generated by MPPT controller which includes the incremental conductance MPPT algorithm and PI controllers that limits the MPPT against overcharging of battery and overcurrent on the battery during charging. The design of these blocks is standard and the reader interested can refer to Ref. [34].

Controller design by H∞ loop shaping Utilizing the loop-shaping approach, a MIMO robust controller K is designed for G [39]. For robust multivariable control design, the desired specifications are commonly expressed as


sðSðjuÞÞ 
W1 1 ðjuÞ

 or equivalently

1  jW1 ðjuÞj sðSðjuÞÞ



(24)

Control design




sðTðjuÞÞ 
W1 3 ðjuÞ

Main approach to the control of the HW system

where s and s denote minimum and maximum singular values respectively. Here, the sensitivity function S(s) is

(25)

The idea is to first design a controller K(s) for the linear block G(s) only, as illustrated in Fig. 8.

Fig. 6 e Nonlinearity functions obtained through constrained polynomial optimization (brown) and unconstrained wavelet network optimization (blue). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

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Fig. 7 e Measured (black) and estimated (red) model outputs for DVdc (left) and DVpv (right). (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

Fig. 8 e Controller design for the linear block.

Fig. 9 e Nonlinear closed loop control structure for the whole system.

Fig. 10 e MIMO H∞ Loop Shaping controller model in MATLAB/Simulink.

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Fig. 11 e MIMO one-degree-of-freedom (1-DOF) LQG controller model in MATLAB/Simulink.

1

SðsÞ ¼ ðI þ LðsÞÞ ;

(26)

and the complementary sensitivity function T(s) is TðsÞ ¼ LðsÞðI þ LðsÞÞ1 ;

where diag stands for diagonal matrix. Carrying out the procedure described above for the G(s) block of the HW model yields the controller  GH∞ ;e1/DDb ðsÞ ¼ 3:72s15 þ 1:68  107 s14 þ 1:13  1012 s13 þ 3

(27)

 1016 s12 þ 3:85  1020 s11 þ 2:42  1010 s10 þ 6:95  1027 s9 þ 1:21  1031 s8 þ 1:39  1034 s7 þ 1:08

and the loop transfer function L(s) is LðsÞ ¼ GðsÞKðsÞ; (28)

1
W ðjuÞ
is the desired disturbance attenuation factor and 1 jW3 ðjuÞj is the largest anticipated uncertainty of the plant expressed as a multiplicative perturbation. S(s) is the sensitivity function from disturbance d to output y. T(s) is the complementary sensitivity function. At this point one computes a stabilizing H∞ controller K for plant G to make the singular values of L have a preferred loop shape Gd with error g. The conditions for disturbance attenuation and stability margins in Eqs. (24) and (25) can be written in terms of singular values by making the following approximation for sðLðsÞÞ[1

 106 s6 þ 5:59  105 s5 þ 1:82  1042 s4 þ 3:34   1044 s3 þ 2:65  1046 s2 þ 1:5s  2:23  1035  16  s þ 4:39  105 s15 þ 3:81  1010 s14 þ 1:53  1015 s13 þ 3:44  1019 s12 þ 4:63  1023 s11 þ 3:7  1027 s10 þ 1:68  1031 s9 þ 4:08  1034 s8 þ 6:07  1037 s7 þ 5:68  1040 s6 þ 3:19  1045 s5 þ 9:46   1045 s4 þ 1:09  1048 s3 þ 6:23  1048 s2  GH∞ ;e1/DDpv ðsÞ ¼  0:67s15  3:03  106 s14  1:98  1013 s13  4:99  1015 s12  5:82  1019 s11  2:92  1023 s10  3:46  1026 s9  4:72  1029 s8 þ 2:28  1033 s7 þ 3:58  1036 s6 þ 3:18  1039 s5 þ 1:66  1042 s4

1

1

SðsÞ ¼ ðI þ LðsÞÞ zLðsÞ ;

(29)

and the one below for sðLðsÞÞ≪1

þ 4:64  1044 s3 þ 5:35  1046 s2 þ 2:72s  1:73   16 s þ 4:39  105 s15 þ 3:81  1010 s14  1034 þ 1:53  1015 s13 þ 3:44  1019 s12 þ 4:63  1023 s11

1

TðsÞ ¼ LðsÞðI þ LðsÞÞ zLðsÞ:

(30)

Hence if uc is the 0 dB crossover frequency of Gd(ju), the specifications can be stated as s ðGðjuÞKðjuÞÞ 



1 s ðGd ðjuÞÞ; cu < uc g

(31)

þ 3:7  1027 s10 þ 1:68  1031 s9 þ 4:08  1034 s8 þ 6:07  1037 s7 þ 5:68  1040 s6 þ 3:19  1045 s5  þ 9:46  1045 s4 þ 1:09  1048 s3 þ 623  1048 s2  GH∞ ;e2/DDb ðsÞ ¼ 0:75s15 þ 3:43  106 s14 þ 2:32  1011 s13 þ 6:15  1015 s12 þ 7:91  1019 s11 þ 5  1023 s10 þ 1:46

sðGðjuÞKðjuÞÞ  gsðGd ðjuÞÞ; cu > uc

(32)

 1027 s9 þ 2:59  1030 s8 þ 2:95  1041 s4 þ 2:22  1036 s6 þ 1:05  1039 s5 þ 2:97  1041 s4 þ 4:38   1043 s3 þ 2:61  1045 s2 þ 1:27s þ 1:91  1035   s16 þ 4:39  105 s15 þ 3:81  1010 s14 þ 1:53

Thus, when the system model is more accurate, high tracking performance is achieved at low frequencies, and when the system model is less accurate and noise impacts are stronger, high robustness is achieved at high frequencies. For the PV system at hand, the desired loop-shape is chosen as

 1027 s10 þ 1:68  1031 s9 þ 4:08  1034 s8 þ 6:07

  10 10 ; Gd ðsÞ ¼ diag s s

 1037 s7 þ 5:68  1040 s6 þ 3:19  1045 s5 þ 9:46   1045 s4 þ 1:09  1048 s3 þ 6:23  1048 s2

(33)

 1015 s13 þ 3:44  1019 s12 þ 4:63  1023 s11 þ 3:7

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GH∞ ;e2/DDpv ðsÞ ¼

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 7:55s15  3:65  106 s14  2:29  1011 s13  6

 1015 s12  7:83  1019 s11  5  1023 s10  1:72  1027 s9  3:59  1030 s8  5:03  1033 s7  4:93  1036 s6  3:33  1039 s5  1:49  1042 s4  3:87  1044 s3  4:21  1046 s2  2:13  1047 s  3:09   16 s þ 4:39  105 s15 þ 3:81  1010 s14  1035 þ 1:53  1015 s13 þ 3:44  1019 s12 þ 4:63  1023 s11 þ 3:7  1027 s10 þ 1:68  1031 s9 þ 4:08  1034 s8 þ 6:07  1037 s7 þ 5:68  1040 s6 þ 3:19  1045 s5 þ 9:46  10 s þ 1:09  10 s þ 6:23  10 s 45 4

48 3

48 2



For this controller it is seen from Fig. 12 that L(s) approximates Gd(s) within reasonable tolerance bounds. The gain of L(s) is high for low frequencies and vice versa with bandwidth 10 rad/s. This ensures low frequency performance and high frequency robustness as targeted earlier. Since L(s) z Gd(s), this suggests that TðsÞ ¼ LðsÞðI þ LðsÞÞ1 zGd ðsÞðI þ Gd ðsÞÞ1

(34)

from where one obtains 3

2

10   6 s þ 10 10 10 ; ¼6 TðsÞzdiag 4 s þ 10 s þ 10 0

0

7 7: 10 5 s þ 10

(35)

From here we observe the following: i. The individual transfer functions for the diagonal channels are approximated by

Vpv ðsÞ Vdc ðsÞ 10 ¼ z Vdc;ref ðsÞ Vpv;ref ðsÞ s þ 10

(36)

so the closed loop system will be able to track all references successfully with minimal overshoot and a settling time of

1 approximately ts ¼ 5t ¼ 5 10 ¼ 0:5 seconds which matches the settling time in Fig. 16.

ii. The off-diagonal entries of T(s) are roughly zero, which indicates that the coupling between different commanderesponse pairs is eliminated.

LQG controller design A 1-DOF LQG controller uses the error e which is the difference of the set point commands r and measurements y to produce the control signals u as illustrated in Fig. 13. It has integral action to ensure that the outputs y track the commands r, w and v are process and measurement noises respectively. Z t eðxÞdx. The LQG controller minimizes the cost Let, xi ¼ 0

function based on the expected value E as follows [40]. 8 <

1 J ¼ E lim : t/∞ t

9    = Zt 

T T x x ; u Qxu þ xTi Qi xi dt : u ;

(37)

0

Here Qxu and Qi are the weighting matrices. Since the PV system in this study is a two-input two-output MIMO system, r, y and xi also have length two. The control law u ¼ K½x xi T is built using a gain vector K computed to minimize J. As state measurements x are not available, an estimate xe converging to x is constructed by a Kalman filter so that o n P ¼ lim E ðx  xe Þðx  xe ÞT t/∞

(38)

is minimized. The Kalman filter is a standard method that produces good estimates under practical conditions such as input/output noises, parameter uncertainties and unmodelled dynamics. The 1-DOF LQG controller designed for the G(s) block of the HW MIMO model is as follows  GLQG;e1/DDb ðsÞ ¼ 5:8  104 s12 þ 7:63  1012 s11 þ 3:31  1019 s10 þ 1:14  1024 s9 þ 1:2  1028 s8 þ 3:97  1031 s7 þ 7:47  1034 s6 þ 8:91  1037 s5 þ 7:05  1040 s4 þ 3:64  1043 s3 þ 1:16  1046 s2 þ 2:03  1048 s   13 s þ 4:49  108 s12 þ 4:08 þ 1:49  1050  1016 s11 þ 3:63  1021 s10 þ 1:29  1026 s9 þ 2:36  1030 s8 þ 2:28  1034 s7 þ 1:08  1038 s6 þ 2:09  1041 s5 þ 2:27  1044 s4 þ 1:35  1047 s3 þ 4:03   1049 s2 þ 4:671051 s

Fig. 12 e H∞ Loop-shaping controller performance and robustness.

Fig. 13 e One-Degree-of-Freedom LQG controller block diagram.

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Fig. 14 e SISO controllers applied to SISO plants.

Fig. 15 e SISO controller applied to MIMO plant.

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Fig. 16 e Step responses of MIMO controllers applied to MIMO plant.

 GLQG;e1/DDpv ðsÞ ¼  1:87  106 s12  6:03  1014 s11  3:62

 GLQG;e2/DDpv ðsÞ ¼  4:15  106 s12  1:34  1015 s11  8:02

 1019 s10  7:52  1023 s9  6:35  1027 s8  2:02

 1019 s10  1:67  1024 s9  1:42  1028 s8  4:57

 10 s  3:76  10 s  4:39  10 s  3:33

 1031 s7  8:51  1034 s6  9:95  1037 s5  7:57

 1040 s4  1:57  1043 s3  4:11  1045 s2  4:55   13 s þ 4:49  108 s12  1047 s  2:31  1048

 1040 s4  3:58  1043 s3  9:38  1045 s2  1:03   13 s þ 4:49  108 s12  1048 s  5:27  1048

þ 4:08  1016 s11 þ 3:63  1021 s10 þ 1:29  1026 s9

þ 4:08  1016 s11 þ 3:63  1021 s10 þ 1:29  1026 s9

38 6

þ 2:36  10 s þ 2:28  10 s þ 1:08  10 s

þ 2:36  1030 s8 þ 2:28  1034 s7 þ 1:08  1038 s6

þ 2:09  1041 s5 þ 2:27  1044 s4 þ 1:35  1047 s3  þ 4:03  1049 s2 þ 4:671051 s

þ 2:09  1041 s5 þ 2:27  1044 s4 þ 1:35  1047 s3  þ 4:03  1049 s2 þ 4:671051 s

31 7

34 6

30 8

37 5

34 7

 GLQG;e2/DDb ðsÞ ¼ 179:1s12 þ 8:87  1010 s11 þ 3:92  1017 s10 þ 8:49  1020 s9  1:71  1026 s8  1:64  1030 s7  4:96  10 s  8:47  10 s  8:95  10 s  6:08 33 6

36 5

39 4

 1042 s3  2:51  1045 s2  5:6  1047 s  5:04   13 s þ 4:49  108 s12 þ 4:08  1016 s11  1049 þ 3:63  1021 s10 þ 1:29  1026 s9 þ 2:36  1030 s8 þ 2:28  1034 s7 þ 1:08  1038 s6 þ 2:09  1041 s5 þ 2:27  1044 s4 þ 1:35  1047 s3 þ 4:03  1049 s2  þ 4:671051 s

Stability and robustness analysis Robustness of a system can be defined in many ways. For the SISO controller designs, gain and phase margins (GM and PM) are standard tools for stability and robustness analyses. Parameter variations, noises and uncertainties disturb the nominal model leading to gain and phase variations. It is important that a robust system withstand such problems. The higher GM and PM are, the more disturbances the system can tolerate [34]. GM and PM are evaluated for the PI controller approach [33] and for the PI/LQG hybrid controller approach [34] since these

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Table 1 e Comparison of GM and PM for SISO controllers. Gain Phase Gain Phase margin margin margin margin for PV PV boost battery boost battery boost converter boost converter converter converter PI PI/LQG Hybrid

39.2 dB ∞ dB

44.8 85.3

33.4 dB 44 dB

48.1 89.9

Fig. 17 e Sigma plots of the designed H∞ loop shaping and LQG controllers.

are SISO methods and tabulated in Table 1. It is seen that PI/ LQG hybrid controller is superior to PI controller since it has higher margins. The step responses of the SISO controllers are shown in Fig. 14. A key assumption in SISO controller design is that each of the SISO models utilized are independent of each other. This implies that the model from input DDb to DVdc and the model from DDpv to DVpv do not affect each other in any way, as a result of which a reference command for DVdc should not affect DVpv and vice versa. Regrettably this assumption in not always a safe one to make, e.g. from the input/output data in Fig. 4, some amount of cross coupling is clearly visible. Such cross channel excitations could hinder performance and robustness or even destabilize the system. To illustrate we apply the SISO controllers PI and PI/LQG to the HW MIMO model for the PV system and show the step response in Fig. 15. It is seen that the cross channel affects are large enough to cause instability. This is yet another justification to derive an accurate MIMO model and design a robust MIMO control where the system is treated as a whole and not partitioned into presumably independent channels. The closed-loop responses of the MIMO methods, namely LQG controller design and H∞ loop shaping designs are shown in Fig. 16. It can be seen that reference tracking is achieved while minimizing unwanted excitation of the other output. H∞

Fig. 19 e Load variation from 500 U (low load) to 111 U (high load) during simulations.

is better at this as expected since the loop shape Gd is designed explicitly. The off-diagonal entries for Gd are set to zero, corresponding to the desire to eliminate crosstalk. For MIMO systems sigma-plots are a valuable tool for robustness and performance analysis. These display the singular values of the system and provide a means to evaluating the loop gain on a single plot. Sigma plots of H∞ loop shaping and LQG controllers are provided in Fig. 17. The former controller is clearly superior to the latter as it has higher lowfrequency gain (i.e. better tracking), higher 0 dB crossover frequency (i.e. higher bandwidth) and lower low-frequency gain (i.e. better robustness).

Results First, a high fidelity numerical model of the PV system is constructed as [34]. Utilizing the discretized transfer functions of the H∞ loop shaping and LQG controllers and the inverse

Fig. 18 e Daily irradiation and temperature data for Ankara on a sunny and cloudy day [27].

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Fig. 20 e Simulation results for H∞ loop shaping control on a sunny day. input/output nonlinearities, a complete control system is built as shown in Figs. 10 and 11. The entire system is simulated for actual historical environmental data for a cloudy and clear day for Ankara, Turkey. The data is shown in Fig. 18 and was gathered from the Solar Radiation Data (SoDa) and the Photovoltaic Geographical Information System (PVGIS). The load is also varied as in Fig. 19. Simulation results are shown in Figs. 20e23. The positive values for battery current indicate discharging of the battery

and negative values indicate charging. As expected, the battery current stays more at negative values for sunny days, indicated that the battery is charged more in these scenarios. For H∞ loop shaping control on a sunny day in Fig. 20, the peak-to-peak voltage fluctuation is 3 Vdc,pp and percent change with respect to the desired DC voltage is 0.3%. For H∞ loop shaping loop shaping control on a cloudy day in Fig. 21, the peak-to-peak voltage fluctuation is again 3 Vdc,pp and percent change with respect to the desired DC voltage is again

Fig. 21 e Simulation results for H∞ loop shaping control on a cloudy day.

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Fig. 22 e Simulation results for LQG control for a sunny day. 0.3%. For LQG control on a sunny day in Fig. 22, the peak-topeak voltage fluctuation is 17 Vdc,pp and percent change with respect to the desired DC voltage is 2.1%. For LQG control on a cloudy day in Fig. 23, the peak-to-peak voltage fluctuation is again 15 Vdc,pp and percent change with respect to the desired DC voltage is again 1.8%. It can therefore be stated that both H∞ loop shaping and LQG MIMO controllers show better results than SISO controllers in Refs. [33,34] from the aspect of load voltage regulation against irradiation, temperature and

load changes. For H∞ loop shaping control, load voltage is always within the desired design limits 390 VDC and 410 VDC against changing atmospheric and load conditions throughout the day both in clear and cloudy days. For LQG controller, load voltage fluctuates more and slightly exceeds design limits. For both controllers voltage peaks (up to 300 VDC) that were observed under SISO controllers are prevented. This is important as such peaks risk undesirable

Fig. 23 e Simulation results for LQG control for a cloudy day.

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Table 2 e Comparison of output voltage fluctuations for methods in current literature (PI [33], PI/LQG [34]) and those suggested in this work (LQG, H∞ Loop Shaping). Plant model

SISO MIMO

Control method

Peak-to-peak voltage fluctuation on a sunny day

PI [33] PI/LQG Hybrid [34] LQG H∞ Loop Shaping

300 Vdc,pp 190 Vdc,pp 17 Vdc,pp 3 Vdc,pp

Percentage change w.r.t desired DC voltage

damage on a critical load. The key metrics from this discussion are summarized in Table 2. An uncertainty analysis is carried out as well by varying the system parameters at random. The battery parameters varied are the nominal voltage and capacity, which in turn affect many other internal characteristics such as nominal discharge current, maximum capacity, exponential zones and internal resistance. The PV panel parameters varied are the ideality factor of the diode, reverse saturation current of the diode, series and shunt resistance. While certain studies in

400 400 400 400

Vdc ± Vdc ± Vdc ± Vdc ±

37.5% 23.7% 2.1% 0.3%

Peak-to-peak voltage fluctuation on a cloudy day 8 Vdc,pp 24 Vdc,pp 15 Vdc,pp 3 Vdc,pp

Percentage change w.r.t desired DC voltage 400 400 400 400

Vdc Vdc Vdc Vdc

± 1% ± 3% ± 1.8% ± 0.3%

literature (e.g. Ref. [41]) also carry out uncertainty analysis, out approach differs in that we perform random simulations of the overall system, directly changing parameters in the simulations for observing controller performance. Perturbing the parameters up to %20 percent of the nominal values, we carry out ten simulations to test the robustness, representing the fact that in an actual system the real parameters will undoubtedly be different from the design ones. Figs. 24 and 25 show simulations with ten random parameter combinations up to ±20% of their nominal values used in design. Both

Fig. 24 e Effect of uncertainty analysis on load voltage of H∞ loop shaping controlled system.

Fig. 25 e Effect of uncertainty analysis on load voltage of LQG controlled system.

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controller methods succeed in keeping the load voltage around the desired value but the H∞ loop shaping controller shows better invariance to parameter changes.

[5]

Conclusions

[6]

In this paper, MIMO controllers for standalone PV systems, including DC/DC converters, PV panels, battery and a resistive load are designed and evaluated to enhance the effectiveness of a common PV system producing DC output. System identification is performed and Hammerstein-Wiener nonlinear model of the system is obtained. According to this nonlinear model, a novel control methodology for PV solar systems is proposed based on H∞ loop shaping and LQG controllers. As such, the typical SISO control design approach (such as those described in Refs. [33,34]) is abandoned in favor of a single robust MIMO robust controller based on the MIMO model. Simulations on the controlled system are performed under irradiation, temperature and load variations. The simulations unveil that H∞ loop shaping MIMO control system is able to provide the most stable output voltage against changing atmospheric and load combinations while assuring the maximum power point tracking. LQG MIMO controller has the second best performance and robustness. Both MIMO controllers are substantially better than their classical SISO counterparts. It should be pointed out that for this study data is collected around a single operating point (Db,op, Dpv,op, Vdc,op, Vpv,op) ¼ (0.4, 0.4, 400 V, 193 V) but the variation around it is large enough for our purposes. The variations are ±0.2, ±0.2, ±100 V and ±50 V respectively. While a purely linear model could represent small variations around the operating point, it would not be adequate for such a large neighborhood. The input and output nonlinearities defined in the model remedy this situation, and the fact that we force them to be invertible makes the HW model suitable for common MIMO robust control techniques. If the main goal were modeling accuracy for all possible conditions, several excellent models in the literature of power converters exist which are valid whatever the point of operation [17,18]. Future works include investigation of alternative control strategies, construction of an experimental systems and the implementation of the proposed strategies of other real-life problems.

[7]

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