Optimal regional pole placement for sun tracking control of high concentration photovoltaic (HCPV) systems case study

DOI: 10.1002/oca.971Optimal regional pole placement for sun tracking control of high-concentration photovoltaic HCPV systems: case study Chee-Fai Yung1, Hong-YihYeh2, Cheng-Dar Lee2, Jen

Published online 11 November 2010 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/oca.971

Optimal regional pole placement for sun tracking control of high-concentration

photovoltaic (HCPV) systems: case study

Chee-Fai Yung1, Hong-YihYeh2, Cheng-Dar Lee2, Jenq-Lang Wu1,∗, †, Pei-Chang Zhou1,

Jia-Cian Feng1, Hong-Xun Wang1 and Sheng-Jin Peng1

1Department of Electrical Engineering , National Taiwan Ocean University, Keelung 202, Taiwan

2Institute of Nuclear Energy Research , Longtan 325, Taoyuan, Taiwan

SUMMARY This paper proposes an optimal regional pole placement approach for sun tracking control of high-concentration photovoltaic systems A static output feedback controller is designed to minimize an LQG cost function with a sector region pole constraint The problem cannot be solved by LMI approach since it is a non-convex optimization problem Based on the barrier method, we instead solve an auxiliary minimization problem to obtain an approximate solution Simulation results show the benefit of our approach Copyrightq 2010 John Wiley & Sons, Ltd

Received 26 May 2010; Revised 27 August 2010; Accepted 5 October 2010

KEY WORDS: regional pole placement; LQG optimal control; barrier method; Lagrange multiplier method; sun tracker

1 INTRODUCTION

Recently, the problems of shortage of fossil fuel source and global warming effects have become more and more severe People begin to seek various possible solutions to those problems One of the potential options is the use

of the sun energy, which not only provides an alternative energy source but also improves environmental pollution Therefore, sun tracking systems have attracted much attention in recent years

In the literature, common sun tracking systems consist of open-loop and closed-loop types In[1], a lookup table was pre-established to obtain the position of the sun at any time and then the direction of tracking mechanism is adjusted to point the direction of the sun In[2] a type of on–off control was utilized for two-axis sun tracking On the other hand, common closed-loop control methodologies include robust proportional (P) control, proportional–integral (PI) control, derivative (D)-like control, proportional–integral–derivative (PID) control[3–8], fuzzy control [9–12], LQG control[13] or H∞ control[14, 15] Various controllers have individual advantages and disadvantages [14] For instance, PID control and fuzzy control could be good options when accurate model of tracking system is

absent, while LQG or H∞ control are preferred if higher accuracy tracking performance or tracking robustness

against to exogenous disturbance, like wind gusts or cloud effects, is the main concern[16–18]

∗Correspondence to: Jenq-Lang Wu, Department of Electrical Engineering, National Taiwan Ocean University, Keelung 202, Taiwan.

†E-mail: wujl@mail.ntou.edu.tw

Most recently, the Institute of Nuclear Energy Research (INER) has developed a high concentration photovoltaic (HCPV) high power generation system with III-V solar cells, as an alternative source to the application of solar PV and as a dependable energy source to the mankind[19, 20] The main purpose of the present paper is to develop

an accurate sun tracking control strategy for the HCPV power generation system implemented and installed at INER An optimal regional pole placement approach is proposed for designing static output feedback sun tracking controllers The minimization of quadratic cost functions can improve systems’ static responses but cannot guarantee good transient responses Good transient responses can be ensured by properly assigning closed-loop poles to some particular regions In[21–25], the authors determined feedback controllers to assign the closed-loop poles to some particular regions Moreover, a quadratic cost function being minimized by the resultant controller is found Nevertheless, for a given cost function, how to find the optimal controller subject to the regional pole’s constraint was not discussed In[26], the authors solved a modified Lyapunov equation to obtain a controller which minimizes

a function, which is an upper bound of the original cost function, and guarantees that the resultant closed-loop poles lie in a desired region Recently, based on the barrier method, Wu and Lee[27–29] have developed a novel approach for solving optimal regional pole placement problems Wu and Lee[27, 28] considered the state feedback case and[29] considered the output feedback case Different to the approach in [26], in barrier method a solution arbitrarily close to the infimal solution of the constraint optimization problem can be obtained In this paper we employ a similar approach to solve static output feedback optimal regional pole placement problem for sun tracking systems The considered cost function is quadratic and the closed-loop poles are required to locate on a sector region For sector constraint region, in[29] three constraint matrix equations must be included but in this paper only two constraint matrix equations should be considered This is a constrained optimization problem and its minimum point may not exist It often happens that its infimum point lies on the boundary of the admissible solution set and

it is not a stationary point Therefore, the Lagrange multiplier method cannot be employed to derive the necessary conditions for optimum Moreover, this problem cannot be solved via linear matrix inequality (LMI) approach since the admissible solution set may be non-convex It is known that static output feedback control problems are difficult

to solve [30] In this paper, based on the barrier method (see [31]), we instead solve an auxiliary minimization problem to obtain an approximate solution of the original problem The new cost function is the sum of the actual cost function of the original problem and a weighted ‘barrier function’ If the admissible solution set is non-empty, the minimal solution of the auxiliary minimization problem exists and is a stationary point Therefore, the Lagrange multiplier method can be used to derive the necessary conditions for optimum The minimal solution of the auxiliary minimization problem converges to the infimal solution of the original problem if the weighting factor of the barrier function approaches zero

Notations

In this paper, E (.) denotes the expected value, (M) is the spectrum of matrix M, Tr(M) means the trace of matrix

M, MT(M∗) is the (conjugate) transpose of matrix M, M>0(0) means that the matrix M is positive (semi)definite,

and ¯ is the complex conjugate of ∈C.

2 PROBLEM FORMULATION AND PRELIMINARIES

Based on the technology of the semiconductor radiation detector, Institute of Nuclear Energy Research (INER) of Atomic Energy Council (AEC), Executive Yuan in Taiwan, has started the R&D projects on the 100 kW HCPV systems The detailed architecture of the HCPV power system is shown in Figure 1[19] The HCPV system is composed of the III–V solar cell, concentrating solar module, solar tracker, inverter, and the tracking control system (Figure 2) This tracker, an azimuth-elevation tracker, consists of two axes One axis is a vertical pivot shaft that allows the device to be swung to a compass point The other axis is a horizontal elevation pivot mounted upon the

Figure 1 100 kW HCPV Power System Architecture.

Sensor

Sun

y

+

-Figure 2 The sun tracking control system

azimuth platform A photo sensor mechanism oriented to sun direction is mainly composed of four photo detectors, located at 90 degrees apart from each other and oriented to the cardinal points Two differential signals between east and west detectors, and south and north detectors are sent to the tracking controller A CCTV system was implemented for remote monitoring of the tracking system The DC power output was connected to the charge controller, which tracks the maximum peak power point to keep the HCPV modules output power in the maximum condition There are two kinds of power measurement design implementation in the system One is the measurement

of DC current and voltage for HCPV modules output power, and the other is the measurement of AC current and voltage for consumption power of the load The PC controller collects those signals from power measurement devices through interface modules and Ethernet network The control function of the tracker is implemented into PC-based controller by high-level programming language The controller collects signals from the photo-sensor mechanism, and sends commands to control the tracker motion

The main purpose of the present paper is to develop an optimal regional pole placement method for designing sun tracking control strategy (see Figure 2) for the HCPV power generation system at INER Suppose that the considered azimuth/elevation tracking system is modeled as

˙x(t) = Ax+Bu

Re

Im

α

π/4

Figure 3 The constraint region 

where x∈ R n is the state, u∈ R is the control input (the voltage to the DC motor), and y∈ R is the measured output

(the tracking error); A, B, and C are constant matrices of appropriate dimensions Define

H (,) ≡ {s ∈C|Re[ei (s −)]<0 and Re[e−i (s −)]<0}

W () ≡ {s ∈C|Re[s]<−}

where 0/2 and  ,∈ R Note that W(0) is the complex left half plane.

The design goal is to find a static output feedback controller

for the azimuth/elevation tracking system to achieve the infimum of the cost function

J (F)=E

 ∞

0 (yT(t)Sy(t)+xT(t)Qx(t)+uT(t)Ru(t))dt



(3)

where S>0, R>0, Q=DTD0 with (A,D) being observable, subject to the constraints that

It is clear that is the sector region shown in Figure 3 The reason for introducing the term yT(t)Sy(t) in the

cost function (3) is to align the solar panel to the direction of the sun as soon as possible The selection of weighting

matrices S, Q, and R depends on practical considerations If we want to align the solar panel to the direction of the sun faster, we can choose larger S In contrast, if we want to minimize the dissipation power of the system, we can choose larger R.

Let

s ≡ {F∈ R|(A+BFC)⊂ W(0)},

r ≡ {F∈ R|(A+BFC)⊂}.

The sets is the collection of all gains F∈ R such that the resultant closed-loop system is stable; the set  r is the

collection of all gains F∈ R such that the poles of the resultant closed-loop system lie in the region .

As in[27–29], it is clear that the objective function J(F) is equivalent to

J (F)=



Tr (PX0) if F∈ s

where X0=E{x(0)xT(0)} and P=PT0 is the unique solution of

Suppose that X0>0 Two useful lemmas are introduced in the following.

Lemma 1 (Wu and Lee [29])

All the eigenvalues of a real matrix ˆA lie in the region W () if and only if for any given positive definite matrix

ˆQ, the Lyapunov equation

has a unique solution ˆP>0.

Lemma 2

All the eigenvalues of a real matrix ˆA lie in the region H (,/4)∪ H(,5/4) if and only if for any given positive

definite matrix ˆQ, the matrix equation

−( ˆA−I)2 T

has a unique solution ˆP>0.

Proof

Sufficiency: Suppose that ∈( ˆA) and v is an associated eigenvector, i.e ˆAv=v Pre- and post-multiplying (8) by

v∗ and v respectively yields

−v∗( ˆA−I)2 T

ˆPv−v∗ˆP( ˆA−I)2v +v∗ˆQv=0

That is,

(¯−)2v∗ˆPv+(−)2v∗ˆPv=v∗ˆQv

By the fact that ˆQ>0 and ˆP>0, we have

(¯−)2+(−)2=v∗ˆQv

v∗ˆPv>0

This is equivalent to∈ H(,/4)∪ H(,5/4) since

(¯−)2+(−)2=2((Re()−)2−(Im())2).

Necessary: Let ∈( ˆA) and let − be expressed as −=|−|e i Then,∈ H(,/4)∪ H(,5/4) is

equiv-alent to−/4/4 or 3/45/4 It is clear that −|−|2ei 2 =|−|2ei 2(+/2) ∈(−( ˆA−I)2) By the

fact that−/4/4 or 3/45/4, we have /22(+/2)3/2 This implies that (−( ˆA−I)2)⊂ W(0).

Since−( ˆA−I)2is Hurwitz, for any positive definite matrix ˆQ the following equation

−( ˆA−I)2TˆP− ˆP( ˆA−I)2+ ˆQ=0

has a unique positive definite solution ˆP This completes the proof.  The result in Lemma 2 is a generalization of some results in[27]

3 THE AUXILIARY MINIMIZATION PROBLEM

The problem under consideration is a constrained optimization problem To solve this problem analytically is difficult since its minimal solution may not exist In fact, its infimal solution may lie on the boundary of the setr; and furthermore, it may not be a stationary point In this paper, motivated by the barrier method (Luenberger[31]),

we instead solve an auxiliary minimization problem to obtain an approximate solution of the original problem The auxiliary cost function Jaux(F) is the sum of the actual cost function J(F) and an additional barrier function

Jpole(F) The auxiliary minimization problem is formulated as follows: Find F, over  r , to minimize the auxiliary

cost function

Jaux(F)= J(F)+· Jpole(F)

where the term J (F) is defined in (3),  is a weighting factor, and

Jpole(F)=



Tr (P1)+Tr(P2) if F∈ r

with matrices P1>0 and P2>0 being the solutions of

and

−(A+BFC−I)2TP2−P2(A+BFC−I)2+Q2=0 (11)

for given positive definite matrices Q1 and Q2

As shown in[31], a barrier function must satisfy: (1) it is continuous, (2) it is nonnegative over the set r, and (3)

it will approach infinity as F approaches the boundary of the setr Now we will show that the function Jpole(F)

satisfies these three conditions

Lemma 3

The function Jpole(F) defined in (9) satisfies

(1) Jpole(F) is continuous in the set  r,

(2) Jpole(F)>0 over the set  r, and

(3) Jpole(F) approaches infinity as F approaches the boundary of the set  r

Proof

The proof is similar to that in[29] and therefore is omitted here 

Although the auxiliary minimization problem is, from a formal viewpoint, a minimization problem with inequality constraints; from a computational viewpoint it is unconstrained[31] The advantage of the auxiliary minimization problem is that it can be solved by unconstrained search techniques

Remark 1

It is shown in[31] that the optimal solution of the auxiliary minimization problem converges to the solution of the original problem as the weighting factor→0+ This suggests a way to approximate the infimal solution of the

original problem in our approach

As we have shown in[29], if the set r is non-empty, the auxiliary cost function Jaux(F) has a minimum point

in the setr Since the minimum point of the auxiliary cost function Jaux(F) lies in the interior of the admissible

solution set, it must be a stationary point The Lagrange multiplier method can be employed to derive the necessary

conditions for local optimum of cost function Jaux(F).

Theorem 1

Let F∈r minimize Jaux(F) Then there exist P0, P1>0, P2>0, L>0, L1>0, and L2>0 satisfying

−(A+BFC−I)2TP2−P2(A+BFC−I)2+Q2= 0 (16)

−(A+BFC−I)2L2−L2(A+BFC−I)2 T

and

Fgrad(U) ≡ 2(BTPL +RFCL+BTP1L1−BTATP2L2−BTP2L2AT

−BTP2L2CTFTBT−BTCTFTBTP2L2+2BTP2L2)CT=0 (18)

Proof

The Lagragian H am is defined as

H am = Tr(PX0)+·(Tr(P1)+Tr(P2))

+Tr(L((A+BFC)TP+P(A+BFC)+CTFTRFC +Q+CTSC))

+Tr(L1((A+BFC−I)TP1+P1(A+BFC−I)+Q1))

+Tr(L2(−(A+BFC−I)2TP2−P2(A+BFC−I)2+Q2)).

The necessary conditions for local optimum are *H am /*F=0, *H am /*L=0, *H am /*P=0, *H am /*L1=0,

*H am /*P1=0, *H am /*L2=0, and *H am /*P2=0 After some manipulations, we have (12)–(18)  The above theorem provides not only a necessary conditions for optimum, but also a method to calculate the

gradient direction of Jaux(F) at a given point F The gradient of Jaux(F) at a fixed point F is Fgrad(F) In the solution

algorithm, this gradient direction is used as the searching direction

4 AN ILLUSTRATIVE EXAMPLE

In order to aim a collector aperture toward the sun during daytime, the two-axis movement of sun tracker is always required Two independent tracking systems (azimuth and elevation) are designed

Case 1: For azimuth tracking system, by applying system identification method on practical experimental data

of the tracking system, we have the following dynamic equation:

˙x =

⎡

⎢

⎢

⎢

⎢

⎣

−10.65 −15.63 −8.938 −7.602 −1.294

⎤

⎥

⎥

⎥

⎥

⎦

x+

⎡

⎢

⎢

⎢

⎢

⎣

1 0 0 0 0

⎤

⎥

⎥

⎥

⎥

⎦

u

y= [0.2533 0.2054 0.2534 0.1888 0.5181]x

Suppose that E {x(0)xT(0)}=X0=I5×5

The design goal is to find a static output feedback gain F such that the controller u =Fy achieves the infimum

of the cost function

J (F)=E

 ∞

0 (yTSy +xTQx +uTRu)dt

subject to the constraints that(A+BFC)∈≡{s ∈C|s ∈ H(7,/4)∩W(−1)}, where S=1, Q=I5×5, and R=2 Choosing different values for parameters and  will lead to different responses In general the tracker will have

faster response for small However, for sun tracking systems, very fast response is not necessary and therefore

we let =−1 We find that, for the azimuth tracking system with =−1, no solutions can be found if <6.3.

Therefore, we consider three different values of for comparison (=6.4, 7, and 8) In general we choose a very

small weighting factor Here we let =0.001 and choose Q1=Q2=I5×5 From the discussions in Section 3, solving the corresponding auxiliary minimization problem yields the results shown in Table I:

It can be verified that, in all cases, all the closed-loop poles are located in  The responses of the resultant azimuth tracking system are shown in Figure 4

Table I Optimal solutions for the azimuth tracking control problem

−1.0118±i×6.6436 −1.1053±i×6.5744 −1.3155±i×6.4314

−7.1905±i×13.5875 −6.6205±i×13.6174 −5.5950±i×13.5922

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 -1.5

-1 -0.5 0 0.5

-10 0 10 20 30 40

t

Figure 4 The responses of the azimuth tracking system (dotted line:=6.4; dashed line: =7; solid line: =8).

Case 2: For elevation tracking system, by applying system identification method on the tracking system, we have

the following dynamic equation:

˙x =

⎡

⎢

⎣

−11 −8.325 0.1623

0 0.25 0

⎤

⎥

⎦x+

⎡

⎢

⎣

2 0 0

⎤

⎥

⎦u

y= [0.818 0.01719 0.6088]x Suppose that E {x(0)xT(0)}=X0=I3×3

The design goal is to find the static output feedback gain F such that the controller u =Fy achieves the infimum

of the cost function

J (F)=E

 ∞

0 (yTSy +xTQx +uTRu)dt

subject to the constraints that(A+BFC)∈≡{s ∈C|s ∈ H(0,/4)∩W(−1)}, where S=1, Q=I3×3, and R=2 Here we let=−1 and =0 The choices of S, Q, and R depend on practical consideration In general larger R

will lead to less power dissipation and larger S will lead to faster output response Here we let Q =I and choose several different values for S and R for comparison (R=0.0001, S=1000; R=0.0001, S=10; and R=1000,

S =10) To construct barrier function, let Q1=Q2=I3×3 and=0.01 From the discussions in Section 3, solving

the corresponding auxiliary minimization problem yields the results shown in Table II:

It can be verified that, in all cases, all the closed-loop poles are located in the desired region The responses

of the resultant elevation tracking system are shown in Figure 5

Table II Optimal solutions for the elevation tracking control problem.

R=0.0001, S=1,000 R=0.0001, S=10 R =1000, S=10

−1.1515±i×1.1145 −1.5043±i×0.4517 −2.4133

−1.0024

-1.5 -1 -0.5 0 0.5

-10 0 10 20 30 40

t

Figure 5 The responses of the elevation tracking system (dotted line: R=0.0001, S=1,000; dashed line: R=0.0001, S=10;

solid line: R=1,000, S=10).

5 CONCLUSION

Based on barrier method, an optimal static output feedback law to minimize an LQG cost function as well as satisfy

a sector region pole constraint has been proposed for sun tracking control of HCPV systems at INER Simulation results have also been given to serve as evidence

ACKNOWLEDGEMENTS The work was financially supported by the Institute of Nuclear Energy Research (INER) under Grant 982001INER035

REFERENCES

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1592–1595.