An LMI Based PID Controller for Load Frequency Control in Power System Subashish Datta and Debraj Chakraborty Abstract— In this article a methodology for designing a minimum gain PID controller for load frequency control in power system is proposed. The resulting PID controller reduces the required control effort while guaranteeing that pre-defined design specifications like settling time and damping ratio are achieved in the response. For this purpose a subset of the complex plane corresponding to the design specifications is mapped onto a stability region in the coefficient space of the characteristic polynomial. This stability region in the polynomial coefficient space defines an LMI which is used as a constraint in minimizing the PID gains of the controller. The effectiveness of the proposed design methodology is demonstrated through case studies with steam and hydraulic units. Index Terms— Power systems, PID Controller, LMIs, Pole placement
TABLE I L IST OF N OTATIONS R[s] kd kp ki Tg Rp Tw Tt H D △ωr △PL △Pm △Pv
Set of all polynomials with real coefficients Derivative gain Proportional gain Integral gain Time constant of governor (Sec) Permanent droop Water starting time at rated load (Sec) Turbine time constant (Sec) Inertia constant of the generator (p.u.) Load damping constant Rotor speed deviation or frequency deviation (p.u.) Load power deviation (p.u.) Deviation in mechanical power output of turbine (p.u.) Real power command signal sent to theprime mover from governor (p.u.)
I. INTRODUCTION Load frequency control, where the objective is to maintain constant specified frequency within an area, is an important consideration in power system design. It is well known [1] that in a certain isolated area if the load changes from its nominal value, the frequency of that area will change and the area will operate with a new frequency. Load frequency control (LFC) arrangement prevents this undesirable frequency fluctuation by sensing the frequency deviation following the load change and increasing/decreasing the generated power to maintain nominal frequency. Conventionally, proportional integral (PI) controllers have been used in LFC to eliminate the steady state frequency deviation [1]. However, these controllers exhibits poor dynamical behavior [2], [3]. Furthermore, a wrong choice of integral gain may lead to the system instability. To improve the transient response of the closed loop system in LFC, several authors ([2], [3], [4], [5] and others) have proposed proportional integral derivative (PID) controllers, LQR controllers and variable structure controllers. However, because of the simplicity in implementation and its success in industrial applications, designing a PID controller is still an interesting area of research. The conventional approach for designing a PID controller is by Ziegler-Nichols method and Cohen-Coon method where the gains for the PID controller are obtained by trail and error [6], [7], [5]. To minimize the required control effort, researchers like [8], [9] have designed optimal PID controllers for LFC where algorithms like particle swarm optimization and genetic algorithms (see [6] and the references therein) have been considered. Similarly, [6] has proposed multiobjective particle swarm optimization where the optimization is formulated to minimize the damping ratio and settling time S. Datta and D. Chakraborty are with the Department of Electrical Engineering, Indian Institute of Technology Bombay, India.
[email protected],
[email protected]
of the system. However, the above methods cannot guarantee that the resulting design meets pre-defined specifications like damping ratio and settling time requirements. In this work we propose a novel method for obtaining a minimum gain PID controller for LFC, which will simultaneously guarantee that the closed loop system response achieves pre-defined design specifications. This is achieved by ensuring that all the closed loop poles are placed within some pre-defined stability region in the complex plane. Additionally the control effort associated with the PID controller is minimized by optimizing the norm of the gain vector, consisting of PID gains, through a linear matrix inequality (LMI) optimization. The requirements on the closed loop poles are transfered to the coefficients of closed loop characteristic polynomial. Using results from [10], it is shown that the regional constraint on closed loop poles defines a linear matrix inequality (LMI) in the polynomial coefficient space. The resulting LMI is used as a constraint in the optimization which in turn can be solved using standard LMI solvers [11]. Rest of the paper is organized as follows. In Section II we show that the coefficients of the closed loop characteristic polynomial are linear in PID gains. A brief descriptions on power system components is presented in Section II-B and the method of designing a minimum norm PID controller is presented in Section III. Case studies with steam as well as hydraulic units are demonstrated in Section IV. II. P RELIMINARIES AND S YSTEM D ESCRIPTIONS A. Preliminaries Consider a continuous time, linear time invariant singleinput single-output (SISO) system represented by the following transfer function. P(s) =
b(s) bn−1 sn−1 + . . . + b1 s + b0 = a(s) an sn + an−1 sn−1 + . . . + a1 s + a0
(1)
where the polynomials a(s) and b(s) are co-prime and an 6= 0. Let us consider a PID controller kd s2 + k p s + ki (2) s Then, the transfer function of the closed loop system (P(s) in forward path and C(s) in reverse path with negative feedback) would be C(s) =
P(s)C(s) G(s) = 1 + P(s)C(s) Using (1) and (2) it can be shown that the closed loop characteristic polynomial α (s) = (bn−1 kd + an )sn+1 + (an−1 + bn−2 kd + bn−1 k p )sn + . . . + (a2 + b1 kd + b2 k p + b3 ki )s3 + (a1 + b0 kd + b1 k p + b2 ki )s2 + (a0 + b0 k p + b1 ki )s + b0 ki
4) Controller: A minimum norm PID controller is proposed in the frequency control loop to eliminate the steady state frequency error and provide required damping. The transfer function (from △Pre f to △ωr ) of the closed loop, which is depicted in Fig. 1, is G(s) =
where
0 0 b0 .. .
0 b0 b1 .. .
b0 b1 b2 .. .
(3)
F = ∈ R(n+2)×3 ; bn−3 bn−2 bn−1 bn−2 bn−1 0 bn−1 0 0 T k = kd k p ki ∈ R3 ; T g = 0 a0 · · · an−1 an ∈ Rn+2
In the following section we will represent the closed loop characteristic polynomial associated with hydraulic as well as steam turbines in the form of (3) following the brief description on power system components required for LFC. B. System Descriptions The various components of a load frequency controller are reviewed briefly for completeness. 1) Speed Governor: The transfer function which describe 1 . the function of a speed governor is: Gg (s) = 1 + sTg 2) Turbine: Transfer functions of the hydraulic and steam 1 − sTw turbines are given by Gt (s) = and Gt (s) = 1 + sTt 1 respectively. For hydraulic unit, turbine time 1 + sTt constant Tt is Tw /2. 3) Load: The Load and generator can be represented by the following first order transfer function: Gl (s) = 1 where M = 2H where H is the inertia constant Ms + D of generator.
(4)
Note that the turbine transfer function Gt is different for hydraulic and steam unit. In (4) the notation Re = 1/R p where R p and C are the permanent droop and the PID controller respectively. △PL △Pre f
Let us define the coefficient vector α := [α0 α1 · · · αn αn+1 ]T ∈ Rn+2 corresponding to the polynomial α (s) where αi ’s for i = 1, . . . , n are the coefficients of α (s). Notice that the coefficient vector α can be compactly written as
α = Fk + g
Gg Gt Gl 1 + Gg Gt Gl (Re +C)
+
Gg
Gt
Governor
Primary LFC loop
+ +
Turbine
Gl
△ωr
Load
Re
Droop
PID Controller C(s) Secondary LFC loop Fig. 1. Block diagram of LFC with primary as well as secondary control loop. Primary LFC loop consists of governor, turbine, load and droop. Secondary LFC loop is comprised of primary LFC loop and PID controller.
The closed loop characteristic equations (the denominator polynomials of (4)) from △Pre f to △ωr and △PL to △ωr are equal in the case of hydraulic as well as steam unit. Hence the stability of the system subjected to change in reference signal as well as disturbance can be guaranteed if the characteristic polynomial in (4) is stable. Furthermore, the closed loop characteristic equations, shown in (4) can be written in the form of (3). After some calculations, it can be shown that the characteristic polynomial, denoted as α (s), is a 4th degree polynomial for steam as well as hydraulic unit. The associated matrices F ∈ R5×3 and g ∈ R5 are
0 0 F = 1 −T w 0
0 1 −Tw 0 0
0 1 e −Tw D+R 0 ; g = M + DTt + DTg − Tw Re 0 DTg Tt + MTt + MTg 0 MTg Tt
(5)
for hydraulic unit and 0 0 F = 1 0 0
0 1 0 0 0
0 1 0 D + Re 0 ; g = M + DTt + DTg 0 DTg Tt + MTt + MTg 0 MTg Tt
(6)
for steam unit. In the next section, following the problem formulation, a methodology to obtain PID controllers for steam as well as hydraulic unit is presented. III. PID CONTROLLER FOR LOAD FREQUENCY CONTROL It is well known that the desired transient behavior of the closed loop system can be achieved by designing a controller which will place the closed loop poles within some stability
region in the complex plane C. Several regions, like open left half of complex plane, left of a vertical line in C, conic sections and discs have been considered as stability regions to achieve various objectives [10], [12]. Here we will define a general stability region S in the complex plane as follows: s11 s12 1 ∗ s ] 0. A feasible solution to the above problem will give the gains of PID controller and with this PID controller the requirements on closed loop poles can be achieved. Furthermore, the resulting PID gain vector would be the minimum norm gain vector with respect to the LMI stability region SLMI . Note that Problem 3 is an LMI constrained
IV. C ASE S TUDIES
To demonstrate the effectiveness of the proposed approach for designing a PID controller, case studies are carried out with following steam and hydraulic units. A. Steam Unit Let us consider an isolated power system consisting of steam turbines which are running coherently, with Tg = 0.2s, Tt = 0.5s, H = 5 p.u., D = 0.8, R p = 0.05 p.u.. According to the above data, the matrices F and g would be 0 0 0 1 20.8 0 1 0 1 0 0 F = ; g = 10.56 7.08 0 0 0 1 0 0 0 The turbine rated output is 1000MW (considered as base power i.e. 1 p.u.) at nominal frequency of 50Hz. Assume that a sudden load change (∆PL ) of 200MW (0.2 p.u.) occurs in the system. Then, at the steady state the frequency deviation would be [1] ∆ωrss =
0.2 −∆PL =− = −0.0096p.u. = −0.576Hz 1/R p + D 20.8
and hence the new frequency of the system would be 49.4240Hz. The steady state frequency error is shown in Fig. 2. Hence a minimum norm PID controller, as discussed in Section III-B, is designed to eliminate the steady state frequency error and improve the transient behavior of the overall system. For this purpose a stability region S in C is chosen which will guarantee that the closed loop response is satisfying the following properties: i) the settling time ts ≤ 8 second and ii) the damping ratio ζ ≥ 0.55. Note that these objectives can be achieved if all the closed loop poles lie within the shaded region as shown in Fig. 3. However, it is difficult to represent such regions according to the
0
p.u. frequency error
optimization with variables γ , k and P and can be solved by LMI solvers like SeDuMi [11] in MATLAB. It should be noted, however, that this formulation is sensitive to the choice of the stability region SLMI , which in turns b (s). depend on the selection of the central polynomial α b (s) ∈ It might happen that corresponding to a chosen α Cs , Problem 3 may turn out to be infeasible even though a PID controller satisfying the constraints might actually exist. However, at the current state of research, the central polynomial needs to be chosen heuristically. This leads to the following three steps design procedure to obtain the minimum norm PID controller. Design Steps: 1) Compute matrices F, g as defined in (5) for hydraulic and (6) for steam unit. 2) Define a stability region S in the complex plane for the closed loop poles according to the requirement. b (s). 3) Choose poles from S to form a central polynomial α Some trial and error adjustment may be required here b (s). Solve problem 3. for the choice of α
Steady State Frequency Deviation
−0.005
−0.01
−0.015 0
5
10
15
20
25
Time in Second
Fig. 2. A step change load of 0.2 p.u. is occurred at 5 second. The steady state frequency error is −0.0096 p.u.. TABLE II C LOSED LOOP POLES AND GAINS OF PID CONTROLLER Closed loop poles −1.8019 ± 0.4312i −2.4763, −1.0009
Gain Values k p = 0.0566, ki = 8.5002 kd = 7.8765
Remarks All closed loop poles are within S
formulation given in (7). Hence, a disc having center at (−3, 0) and radius 2 is considered inside the shaded region as stability region S. Corresponding to the above disc, the elements of matrix S defined in (7) would be as follows: s11 = 5, s12 = 3 and s22 = 1. Next, the central polynomial b (s) needs to be fixed and hence the complex numbers −1.5, α −2 ± 1i and −3.5 are chosen arbitrarily inside S to form b (s) := (s + 1.5)(s + 3.5)(s + 2 − i)(s + 2 + i). The gains of α the PID controller and the associated closed loop poles are obtained by solving Problem 3 and are shown in Table II. Im
Damping Ratio E
Z
Stability Region P
O
G
X
Re
Settling Time
F
Fig. 3. Stability region in the complex plane. The cone EGF corresponds to the damping ratio ζ and the vertical line XZ corresponds to the settling time ts . The defined stability region S is a disc having center at O and radius OP.
The resulting PID controller has been implemented in the LFC loop and simulated in MATLAB Simulink. The simulation results are shown in Fig. 4. Notice that the frequency error has converged to near zero within the specified time limit (i.e. 8 second). In addition to this, the real power command signal △Pv , which is sent to the prime mover from governor, and the change in mechanical power output △Pm of the turbine, has increased and settled at a value corresponding to 0.2 p.u. extra power requirement. To verify the effectiveness of the designed controller corresponding to increase/decrease the load, a step input with different magnitudes, varying from 0.1 p.u to −0.1 p.u., is used as load signal △PL . The simulated results are shown in
0.4
0.3
∆Pv
0.25
Step change load Frequency error Output of turbine Output of governor
∆Pm
p.u. Amplitude
p.u. Amplitude
0.15 0.1 0.05
0.25 0.2 0.15 0.1 0.05 0
0
−0.05 4 4
6
8
10
12 Time in Second
14
16
18
0.2 Step change load Frequency error Output of turbine Output of governor
0.15
p.u. Amplitude
0.1 0.05 0 −0.05 −0.1 −0.15 −0.2 0
10
20
30 Time in Second
40
50
60
Fig. 5. The output △Pv , △Pm and the frequency error in p.u. corresponding to the PID controller having gains k p = 0.0566, ki = 8.5002 and kd = 7.8765 respectively for steam unit. The closed loop system is excited with different magnitudes step input corresponding to the signal △PL .
Fig. 5. It is clear from the response that the controller can stabilize the system as well as provide appropriate transient for different values of load change. To compare the results of proposed approach, a PID controller is designed in MATLAB sisotool. The gains of the PID controller are tuned according to Ziegler-Nichols open loop tuning algorithm [14]. The simulation results corresponding to p.u. frequency error are shown in Fig. 6. The gains of the PID controller are as follows: k p = 11.3366, ki = 11.5680, kd = 1.9301. From Fig. 6 it is observed that comparatively better response in the frequency error is achieved with proposed approach. In addition, there is around 29% reduction in norm of the PID controller with proposed approach. The output of the governor (△Pv ) and the turbine (△Pm ) corresponding to PID control by conventional method is −3
x 10
0
p.u. Amplitude
−2 −4 −6 −8 −10 −12
Frequency error by conventional approach Frequency error by proposed approach 5
10
15 Time in Second
20
6
20
Fig. 4. Output of governor △Pv , turbine △Pm and frequency error in p.u. corresponding to the PID controller having gains k p = 0.0566, ki = 8.5002 and kd = 7.8765 for steam unit. The peak overshoot of △Pv is 0.26 p.u..
2
∆Pm
0.3
0.2
Step change load Frequency error Output of turbine Output of governor
∆Pv
0.35
25
Fig. 6. The frequency error comparison corresponding to the conventional and proposed PID control approach for a single area load frequency control of steam unit.
8
10
12 Time in Second
14
16
18
20
Fig. 7. The output of the governor △Pv , the turbine △Pm and the frequency error in p.u. corresponding to the conventional PID controller for steam unit. The peak overshoot of △Pv is 0.38 p.u..
depicted in Fig. 7. It is observed that △Pv and △Pm are oscillatory (damped) in nature. These oscillations, in general, are undesired from practical point of view because they create more vibrations in the turbine shaft. Furthermore, note that the power rating of gate/valve servomotor, which is required to move the position of gate/valve, depends on peak overshoot of the real power command signal △Pv . Higher peak overshoot leads to costlier servomotors. It is observed that around 31.58% reduction in p.u. peak overshoot is achieved with the proposed approach in comparison to the conventional (Ziegler-Nichols) approach. In the next section we will design a PID controller for hydraulic unit to address the same situation, that is, to eliminate the steady state frequency error (0.2 p.u.) and improve the transient response of the closed loop system. B. Hydraulic Unit Let us find out a minimum norm PID controller for hydraulic unit having the following parameters: Tg = 0.2s, Tt = 0.5s, H = 3 p.u., D = 1 and R p = 0.05 p.u.. The matrices F and g are as follows: 0 0 0 1 21 0 1 −1 F = 1 −1 0 ; g = −13.3 4.3 −1 0 0 0.6 0 0 0 In this case a disc having center at (−3.5, 0) and radius 2.8 is chosen as stability region inside the shaded region shown in Fig. 3. The elements of matrix S corresponding to this stability region are as follows: s11 = 4.41, s12 = 3.5 b (s), the poles and s22 = 1. To form the central polynomial α −0.8, −1.5, −1.5 ± 0.2i are used. The gain values, obtained by solving Problem 3, and the associated closed loop poles are shown in Table III. With the resulting PID controller the closed loop system is simulated in MATLAB Simulink. The simulation results are shown in Fig. 8. It is observed that there is no oscillatory response in the output of governor and turbine. Furthermore, the frequency error has converged to zero within 5 second. To compare the results of proposed approach, a PID controller is designed in MATLAB sisotool. The gains of the PID controller are tuned according to Singular Frequency
TABLE III C LOSED LOOP POLES AND GAINS OF PID CONTROLLER Gain Values k p = −15.5403 ki = 1.5330, kd = 2.0159
Frequency error by conventional approach Frequency error by proposed approach
0.01
Remarks All closed loop poles are within S
0 p.u. Amplitude
Closed loop poles −0.9760 ± 1.2092i −0.9274 ± 0.4448i
0.02
−0.01 −0.02 −0.03 −0.04
0.35 Step change load Frequency error Output of turbine Output of governor
∆P m ∆Pv
0.3 0.25
−0.05 −0.06
p.u. Amplitude
0.2
5
10
15 Time in Second
20
25
30
0.15
Fig. 10. The frequency error comparison corresponding to the conventional and proposed PID control approach for a single area load frequency control of hydraulic unit.
0.1 0.05 0 −0.05 −0.1 −0.15 0
5
10
15
20 Time in Second
25
30
35
40
Fig. 8. The output of the governor △Pv , the turbine △Pm and the frequency error in p.u. corresponding to the PID controller having gains k p = −15.5403, ki = 1.5330 and kd = 2.0159 for hydraulic unit.
Based Tuning algorithm (Ziegler-Nichols method will not work since the primary LFC loop is unstable). The gains of the PID controller are as follows: k p = −15.6902, ki = 2.2512, kd = 2.6023. With the resulting PID controller, the closed loop system is simulated in MATLAB Simulink and results are shown in Fig. 9. Notice that there is a damped oscillation in the signal △Pv , which is sent to the prime mover from governor, and in mechanical power output △Pm of the turbine. Furthermore, with the proposed approach around 5.71% reduction in p.u. peak overshoot is achieved in comparison to conventional approach. The p.u. frequency error simulation results corresponding to the conventional and proposed PID controller are depicted in Fig. 10. It shows that unlike the conventional PID controller the frequency error corresponding to the PID controller obtained by proposed approach has converged to zero within 5 second. V. CONCLUSION A methodology to obtain minimum norm PID controller for load frequency control is presented here. The gains of the PID controller are chosen in such a way that all the closed loop poles are placed within a pre-defined stability
0.4
0.2 p.u. Amplitude
Step change load Frequency error Output of turbine Output of governor
∆P ∆Pv m
0.3
0.1 0 −0.1 −0.2 5
10
15 Time in Second
20
25
30
Fig. 9. The output of the governor △Pv , the turbine △Pm and the frequency error in p.u. corresponding to the conventional PID controller for hydraulic unit.
region in the complex plane. Furthermore, if there are multiple number of PID controllers which are achieving the required objectives then the one having minimum norm is chosen through a semi-definite program. For this purpose the requirements on the closed loop poles are transfered to the coefficients of the closed loop characteristic polynomial. Finally, the problem is formulated as a norm minimization LMI problem. Case studies have been carried out with steam as well as hydraulic units. In both cases it is observed that the resulting PID controller has achieved the pre-defined objectives like settling time and damping ratio. R EFERENCES [1] H. Saadat, Power system analysis. McGraw-Hill, 1999. [2] W. Tan, “Tuning of PID load frequency controller for power systems,” Energy Conversion and Management, vol. 50, pp. 1465–1472, 2009. [3] W. Tan, “Unified tuning of PID load frequency controller for power systems via imc,” IEEE Transactions on Power Systems, vol. 25, no. 1, pp. 341–350, 2010. [4] E. Yesil, M. Guzelkaya, and I. Eksin, “Self tuning fuzzy PID type load and frequency controller,” Energy Conversion and Management, vol. 45, pp. 377–390, 2004. [5] A. Khodabakhshian and M. Edrisi, “A new robust PID load frequency control,” Control Engineering Practice, vol. 16, pp. 1069–1080, 2008. [6] K. Sahabi, A. Sharifi, M. A. Sh., M. Teshnehlab, and M. Aliasghary, “Load frequency control in interconnected power system using multiobjective PID controller,” Journal of Applied Sciences, vol. 8, no. 20, pp. 3676–3682, 2008. [7] E. Poulin and A. Pomerleau, “Unified PID design method based on a maximum peak resonance specification,” Proceedings of the Control Theory Applications, vol. 144, pp. 566–574, 1997. [8] S. Ghoshal, “Optimizations of PID gains by particle swarm optimizations in fuzzy based automatic generation control,” Electric Power Systems Research, vol. 72, pp. 203–212, 2004. [9] J. Talaq and F. Al-Basri, “Adaptive fuzzy gain scheduling for load frequency control,” IEEE Transactions on Power Systems, vol. 14, no. 1, pp. 145–150, 1999. [10] D. Henrion, M. Sebek, and V. Kucera, “Positive polynomials and robust stabilization with fixed-order controllers,” IEEE Transaction on Automatic Control, vol. 48, no. 7, pp. 1178–1186, 2003. [11] J. F. Sturm, “Using SeDuMi 1.02, a matlab toolbox for optimization over symmetric cones,” Optim. Meth. Software, vol. cs.SC, pp. 625– 653, 2005. [12] S. Datta, D. Chakraborty, and B. Chaudhuri, “Partial pole placement with controller optimization,” IEEE Transactions on Automatic Control, vol. 57, no. 4, pp. 1051–1056, 2012. [13] F. Yang, M. Gani, and D. Henrion, “Fixed-order robust hin f ty controller design with regional pole assignment,” IEEE Transaction on Automatic Control, vol. 52, no. 10, pp. 1959–1963, 2007. [14] J. Ziegler and N. Nichols, “Optimum settings for automatic controllers,” Transactions of the ASME, vol. 65, pp. 759–766, 1942.
