Robust Controller Design For Load Frequency Control Of Non

Robust Controller Design for Load Frequency Control of Non-minimum Phase Hydro Power Plant using PSO Enabled Automated Quantitative Feedback Theory B.Satpati E E Dept HIT,Haldia,W.B [email protected]

I Bandyopadhyay EE Dept HIT,Haldia,W.B [email protected]

Abstract—This paper presents the design of a robust PID controller for load frequency control of non-minimum phase Hydro power plant using Particle Swarm Optimization (PSO) enabled automated Quantitative Feedback Theory (QFT). The plant model considered here is a Dynamic Model of Power System that includes the turbine, governor, load and machine dynamics subjected to control the load frequency in accordance with power input to the governor. In the present contribution, a proposal is being presented to automate the loop shaping phase in QFT design method to synthesize a robust load frequency controller that can undertake the exact amount of plant uncertainty and can ensure a proper trade off between robust stability specifications and plant input disturbances over the entire range of frequencies. In this article the PSO technique has been employed to tune the controller automatically that can greatly reduce the computational effort compared to manual graphical techniques. It has also been demonstrated that this methodology not only automates loop-shaping but also improves design quality and, most usefully, improves the quality with a reduced order controller. Index Terms— Dynamic Model, Hydro Power Plant, Load Frequency Control, Quantitative Feedback Theory, Particle Swarm Optimization.

I. INTRODUCTION

L

OAD frequency control (LFC) or Automatic generation control (AGC) is an emerging issue in electric power systems, the objective of which is to maintain the system frequency and the power exchange between areas within specified limits irrespective of sudden change in load. The prime mover governing system provides a means of controlling power and frequency; and this function is commonly called Load Frequency Control or Automatic Generation Control. The main function of the Hydroelectric Power Plant Controller (governor) is to regulate turbine speed, and hence frequency and active power. The generators' speed governor system utilizes load frequency controllers to operate the water gates of the turbines; hence the mechanical power on the shaft of the generators is controlled. Normally, hydroelectric plants are subjected to two kinds of control action: the first one is the frequency control action, where a change in the electric power consumption of the system will result in deviation of the frequency from its steady state value, which is dependant on controller regulation characteristics and frequency sensitivity with respect to sudden change in mechanical load.All generating units with a primary speed control contribute to compensate the deviation by changing the total generation independently where load is changing. The 978-1-4244-2746-8/08/$25.00 © 2008

G. Das E E Dept HIT,Haldia, W.B [email protected]

C Koley E E Dept NIT,Durgapur, W.B [email protected]

final complete restoration of the system frequency to its nominal value needs a supplementary control action, which adjust the reference generated power load value. As the system load is varying all the time it is necessary to vary automatically the output power of the generators. Load Frequency Controller design is always a challenging task due to inherent non-linear model dynamics of hydroelectric power plant and also subject to parametric variations, and disturbances caused by sudden change in load. In the present work, a linearized non-minimum phase process model is constructed following the article of Khodabakhshian et al [1]. Various robust control design technique such as, “Hinfinity”[2], “Sliding Mode Control”[3], “Fuzzy logic control” [4] have been applied recently to handle such problems. In article [1] Khodabakhshian et al have presented a robust PID controller design method considering a NMP Hydro turbine plant cascaded with an uncertain Load machine dynamic model and subjected to external disturbances due sudden change in load. Designing controller for a NMP system is always a challenging task because of the fact that NMP system stabilizes for a small amount of gain and open loop crossover frequency of a NMP system has an upper bound, i.e. the amplitude of the loop transmission at frequencies below the crossover frequencies are also bounded. Considering the complexities of NMP system design Khodabakhshian et al have manually synthesized a PID controller using QFT method. But the potential of the PID controller can be efficiently utilized only if it is tuned properly. The graphical loop shaping performed by using QFT tool box involves a trial and error procedure and also the success of the design depends, to a large extent, on the experience of the designer. That is why in recent years researcher are more concentrating on the development of automatic loop shaping algorithm. One of the first papers to address the automatic loop shaping problem is Gera and Horowitz [5]. This work use Bode’s gain–phase integral to derive a nominal loop shape in an iterative fashion. There was no guarantee of convergence and rational function approximation was ultimately needed to obtain an analytical expression for the loop. This approach was automated in a QFT tool box (Ballance and Gawthrop)[6] which specified the iteration process and allowed for higher order approximation of the integral. Thompson and Nwokah [7] proposed that automatic loop shaping was achieved using nonlinear programming techniques where the QFT bounds were over bounded by disks. With the continuation of this work,

Borguesani et al [8] proposed an automatic loop shaping technique via linear programming. Actually this method is continuation of convex optimization problem. Chen et al [9] developed a Genetic Algorithm enabled automatic designed method based on the fact that performance, stability and high frequency gain is optimized by a population based search technique which ensure good robust stability and appreciable tracking performance with the use of minimum control effort. In the present article, the objective function has been defined to get optimized considering the design constraints for NMP system.

II. QFT OVERVIEW QFT is a unified frequency domain technique utilizing the Nichols chart (NC) for achieving the desired robust design over a specified region of plant uncertainty. This method was created and developed by Prof.I.M.Horwitz [5]. It is now recognized as a well established method for design of robust controllers for the plant with large classes of uncertainties, output/input disturbances and noises. This method was successfully implemented in process control, flight control, marine control, missile control, power systems and power electronics applications, robot manipulator control, to name a few. The true importance of feedback is in 'achieving desired performance despite uncertainty'. If so, then the actual design and the cost of feedback should be closely related to the extent of the uncertainty and to the narrowness of the performance tolerances. In short, it should be quantitative. Basically, the QFT design is based upon: • Specifying the tolerances in frequency domain (time domain tolerances should be converted into corresponding frequency domain tolerances) by means of set of plant transfer functions and closed loop control ratios, and • Determining the loop transmission functions and prefilter functions to satisfy the various resulting bounds corresponding to the tolerances. A single-loop two-degree of freedom (DOF) feedback control structure (providing freedom to shape the feedback and tracking responses independently) is shown in Figure.1. Here, P is the set of transfer functions {P(s)}, which describe the region of plant parameter uncertainty, C(s) is the cascade compensator, and F(s) is an input pre-filter.

specification requirements [8],[10]. III. INTRODUCTION TO PSO The particle swarm optimization (PSO) is a population based optimization tool, which is developed by Kennedy et al [11] motivated by the social behavior of bird flocking and fish schooling. Population is formed by a predetermined number of particles; each particle is a candidate solution to the problem. In a PSO system, particles fly around in a multi-dimensional search space until relatively unchanging positions have been encountered or until computational limits are exceeded. During the flight, each particle adjusts its position according to its own experience and experience of its neighboring particles [11]. In PSO algorithm every particle remembers its best solution (local best, ‘ J pbest ’) as well as the group best solution (global best, ‘ J gbest ’). In PSO each particle adjusts its flight according to its own and its companion’s flying experience. Let X and V represent the particle position and flight velocities in the given search space respectively. Therefore the i th particle is represented as X i = (xi1 , xi 2 ,......, xim ) in the mdimensional search space. The base previous position of the particle is recorded and represented as i th J pbest = J pbesti1 , J pbesti 2 ,...........J pbestim .The index of the best

(

)

particle among all the particles in the group is represented by J gbest .The rate of the velocity for particle i is represented as

Vi = (vi1 , vi 2 .........vin ) .The modified velocity and position of each particle can be calculated using the current velocity and distance from J pbest and J gbest using following: V

t +1 i

= QVi t + K 1 rand 1 ()( X Pbest − X it ) + K 2 rand 2 ()( X gbest − X it )

(1) X it +1 = X it + γ .Vi t +1

(2)

Where K1 and K2 are two positive constants, rand1 ( ) and rand2 ( ) are random numbers in the range [0, 1], and Q is the inertia weight. X it represents current position of the i th particle and Vi t is its current velocity. The position of the particles are updated using (2), where X it +1 is the new position of the i th particle at m-dimensional search space. The weight Q is updated using the following equation ⎡ Q − Qmin ⎤ Q = Qmax − ⎢ max ⎥iter ⎣ itermax c ⎦ (3) Where ‘iter’ is the iteration count IV. PSO ALGORITHM

Figure.1. 2-DOF Feedback Control System

The output Y(s) is required to track the command input R(s) and to reject the external disturbance D(s). The compensator C(s) is to be designed so that the variation of R(s) to the uncertainty in the plant P is within allowable tolerances, the robustness criteria is ensured and the disturbance-rejection requirement is met. In addition, the pre-filter properties of F(s) must be designed to tailor the responses to meet the tracking

The PSO algorithm comprises of several steps [12] which are discussed in the flow chart of Figure. 2. V. DEFINITION OF OBJECTIVE FUNCTION FOR AUTOMATED LOOP-SHAPING IN QFT In the proposed approach the controller structure is predetermined and is given by [9,12]:

C (s) =

br s r + ......b1 s + b0 a n s m + ......... + a1 s + a 0

(4) The coefficients br ,...., b1 , b0 and a m ,...., a1 , a 0 are searched

⎧⎪0 if J gcf = ⎨ ⎪⎩1 otherwise

L( jω gcf ) = 1

by the PSO algorithm to satisfy the constant equations. am can be set to 1. The objective function J ;comprises of robust performance, stability criteria, restriction in gain crossover frequency and high frequency gain which is required to minimize the use of control effort; is to be optimized. The objective function is given by: N

J = β1 J hfg + ∑ ( β 2i J sta + β 3i J bi ) + β 4 J gcf i =1

(5) Where , J bi are the robust stability indices , J sta is the stability index , J hfg is the high frequency gain and J gcf is index for upper limit in gain crossover frequency for β 2i , β 3i and β 4 are compensated loop transmission. β 1 , weighting factors. In general β 1 should be reasonably large. In this paper β 1 is simply chosen as 1.83e3. Penalties associated with the respective indices for stability, robust stability index and gain crossover frequency bound are β 2i , β 3i , β 4 and N indicates the number of discrete points taken as designed frequencies. For an automated design, the stability is analyzed by checking the roots of the characteristics equation of compensated nominal loop(L(s)).A simple cost function to penalize unstable design is : ⎧0 if stable J sta = ⎨ ⎩1 if unstable To obtain the cost function J bi for robust stability indices at all designed frequencies, it is required to generate robust stability bound for the given robust stability specifications and amount of plant uncertainties by using QFT tool box. Then, with the capability and flexibility of an evolutionary algorithm, these numerical bounds can be used directly in an automated design. At each frequency point, the gain and phase of the open loop transmission L( jω i ) is calculated and then checked to see whether or not the QFT bound at all designed frequencies are satisfied. A simple robust stability bound index is given by if QFT bound at ϖ i satisfied ⎧0 J bi = ⎨ ⎩d bi otherwise Where d bi is the distance to the QFT bound at i th frequency point. Since the nominal plant is fixed the cost function for high frequency gain is calculated from the high b frequency gain of the controller given by: J hfg = r am In case of a NMP system occurrence of the gain crossover frequency is restricted by the location of non-minimum phase zero and L( jω gcf ) is then checked to see whether it is unity or not. A simple performance index function for gain crossover frequency restriction is defined as follows:

Figure. 2. PSO algorithm flow chart

VI.

PROCESS MODEL

In article[1], Khodabakhshian et al illustrated a third order NMP model for analysis and design of PID controller concern with the change in load frequency in a power system that includes the turbine, governor, load & machine dynamics, given by [13-15]: (1 − Tw s ) Δω ( s ) = ΔPref ( s ) (1 + sTg )(1 + 0.5* Tw s )( D + 2* Hs )

(6) Figure.3. shows the overall control loop for hydro turbine with transient droop compensator. Block Diagram Representation of Governor, Turbine and Load Machine Dynamics is shown in Figure. 4.

Figure 3: Hydro power system with the transient droop compensator

Figure 4: Block Diagram Representation of Governor, Turbine and Load Machine Dynamics

Where D and H are the parameters representing the Load & Machine dynamics and subjected to vary with the

environmental changes. Tw and Tg are the parameters of hydro turbine and governor respectively where hydro turbine model contain a non-minimum phase zero. For a case study, a typical power system has been modeled following [1, 13-14]. Nominal values of all parameters and their corresponding uncertainty ranges are given in Table I. TABLE 1 DETAIL OF PARAMETERS Nominal value Range (%)Variation 0.2 -----

Parameter

Tw Tg

4.0

----

---

H D

3.0

2.5-7.5

150%,-17%

1.0

0.5-1.5

± 50%

VII. SYNTHESIS OF PID CONTROLLER USING PSO ENABLED QFT

Figure.5. Plot of plant templates for minimum phase plant

In the case study QFT design is carried out to meet the following control objectives. [8], [10] 1) Robust Stability Specification: The associated QFT robustness constraint in terms of the nominal loop transfer function, L0 ( s ) = P0 ( s )C ( s ) , is given by,[1]

Δω ( s) L( s ) (7) = ≤ μ = 1.2, ω > 0 ΔPref ( s) 1 + L( s) which implies an approximately 1.58dB gain margin and 43 degree phase margin for the closed loop system. 2) Plant input disturbance rejection specification is considered here as, [1], [8]

Δω ( s ) s 2 + 948s + 2400 ≤ 0.01* 2 ΔPL ( s ) s + 8.4s + 110

(8)

Design frequencies are selected as Ω={0.01,0.02,0.05,0.5,1,20 } rad/s. The nominal plant is taken as with the uncertain parameters D=1,H=3 given by (1 − 4 s ) (9) P (s) = 0

(1 + 0.2 s )(1 + 2 s )(1 + 6 s )

Again for Non-minimum phase plant

−−

P(s) =

N ( s) N (− s) D( s) 1

,

P / ( s)

Compensated minimum phase nominal loop L/ ( s) is given by a s 2 + a1s + a0 (1 + 4s) * 2 (1 + 0.2s)(1 + 2s)(1 + 6s ) s (11) To permit optimization, all the coefficient of the chosen controller is allowed to vary in the 3 dimensional search spaces. During the optimization process the upper and lower bound of the coefficients within which each parameter can vary are given in the Table 2. L/ ( s) = P / 0 ( s)C ( s) =

TABLE 2 SEARCH RANGE OF CONTROLLER PARAMETERS Parameter of the Lower Bound Upper Bound Controller to be Search 1e-1 1e2

a2 a1 a0

1e-2

1e1

1e-2

1e1

:

From the above design specifications (7), (8) and nonminimum phase nominal plant composite bounds are generated at all designed frequencies shown in the Figure.6.

−−

where N (− s ) represents the non-minimum phase zero. −− −− for this particular problem , N ( s ) N ( s ) N (− s ) / P( s ) =

D( s)

.

−−

= P ( s ). A( s )

N (s)

minimum phase plant P / ( s ) =

(1 + Tw s ) (1 + sTg )(1 + 0.5 * Tw s )( D + 2 * Hs )

, and all pass system A( s ) = 1 − Tw s . [1], [8, 10]. The plant 1 + Tw s templates in Nichols chart are shown in Figure. 5 at all design frequency points , where “x” denotes the nominal case. For PSO enabled automatic loop shaping a PID controller is tentatively chosen as C (s) =

a2 s 2 + a1 s + a0 s

Figure.6. Composite of all bounds

(10)

The objective of this bound is, when we synthesized minimum phase nominal open loop transfer function, it must lie on or just above the composite bound. The gain and phase

contribution of the minimum phase open loop transmission

L/ 0 ( jω ) = P0/ ( jω ) A( jω ) at all designed frequencies are obtained graphically to satisfy the robust stability and tracking performance at each frequencies are given in the Table.3. TABLE.3. GRAPHICALLY OBTAINED VALUES OF ROBUST STABILITY Design Frequencies 0.01 0.02 0.05 0.5 1.0 20

Phase (degree) -88.98 -87.62 -84.89 -35.41 -26.78 -77.63

Magnitude (dB) 23.02 17.97 12.53 -5.68 -6.44 -18.96

L’(jω) 7.46-j12.02 7.45+j2.67 -4.22+j.28 -3.42+j0.39 -0.036-j0.47 -0.069-j0.089

The objective function for Robust stability indices are defined in such a way, that if the numerical value obtained from algebraic representation of L/ ( jω ) is same as graphically obtained value at all frequencies respectively J bi is set to zero otherwise it represents the difference between the numerical and graphical obtained values. The cost function for restriction in gain crossover frequency is define in such a way, that J gcf is set to zero if

The convergence of coefficients representing the minimum value of objective function for population size of 20 candidates is given below. 1 . 351

0 . 9246

0 . 1375

1 . 352

0 . 9248

0 . 1375

1 . 352

0 . 9247

0 . 1375

1 . 351

0 . 9247

0 . 1375

1 . 352

0 . 9247

0 . 1375

1 . 352

0 . 9248

0 . 1375

1 . 352

0 . 9247

0 . 1375

1 . 352

0 . 9248

0 . 1375

1 . 352

0 . 9246

0 . 1375

1 . 352 1 . 352

0 . 9246 0 . 9247

0 . 1375 0 . 1375

1 . 352

0 . 9246

0 . 1375

1 . 351

0 . 9246

0 . 1375

1 . 352

0 . 9246

0 . 1375

1 . 352

0 . 9246

0 . 1375

1 . 352

0 . 9248

0 . 1375

1 . 352

0 . 9246

0 . 1375

1 . 352

0 . 9246

0 . 1375

1 . 352

0 . 9246

0 . 1375

1 . 352

0 . 9246

0 . 1375

L/ ( jω ) = 1 at w=0.25(i.e. location of non-minimum phase

VIII. DESIGN VALIDATION

zero). The PSO algorithm converged to give the following robust PID controller 1.3552 s 2 + 0.9248 s + 0.1375 C ( s) = s (12) Whose loop shaping result is verified in the Figure.7.The control parameter of the PSO algorithm chosen is given in the Table.4.

The robust stability margin requirement should be checked first, and the result is shown in Figure 8. The worst closedloop response magnitude (covering all uncertainty cases) is plotted in the solid line, together with the design specification plotted in the doted line. Fig shows that robust stability specification matches perfectly for desired stability value (µ=1.2=2.0dB).

TABLE.4. PSO PARAMETERS Parameters

values

Maximum Iteration

10000

Population Size

20

Dimension

4

Value of K1

2.0

Value of K2

2.0

Maximum Weight Minimum Weight

0.90 0.40

Figure.8. Validation of closed loop robust stability

Moreover using the Nicholas chart stability criterion, the nominal plant under the controller (12) is stable since the loop transmission does not intersect the stability line R = {(r , φ ) : r > 0dB, φ = −180 } .it can be shown that there is no right half zeros-pole cancellation in C(s)P(s) for all H and D within the prescribed ranges(Table1).So all requirements in the current formulation of QFT for robust stability are satisfied and the plant family under this controller should be robustly stable. The system is robustly stable, as shown in the Figure.9 the Nyquist plot of the closed loop system does not encircle the critical point (-1+j0). o

Figure.7.: Loop Shaping

error involved in the manual synthesis of nominal loop. This paper has described the application of PSO enabled QFT technique to the development of a load frequency controller for hydro power plant considering a simplified single machine infinite bus system. Design validation studies and the overall result imply that a fixed robust PID controller is successfully implemented. This ensures required robust stability as well as good disturbance rejection sensitivity. X. REFERENCES [1]

[2] Figure.9. Nyquist plots under the resulting controller for all uncertain nonminimum phase plants

Figure.10.shows that worst case response of plant input disturbance sensitivity function is bounded by the specified disturbance weighting function. Robustness of the controller for a small variation in load can be validated from Figure.11.

[3] [4]

[5] [6] [7] [8] [9]

[10] [11] [12]

Figure.10.Validation of disturbance rejection requirement

[13] [14] [15]

Figure. 11. Frequency variation for a small change in load

IX. CONCLUSION A particle swarm optimization procedure has been employed to automate the loop shaping process in QFT. The automatic loop shaping using PSO minimizes the trial and

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