State of art fractional order controller for power system stabilizer

Abstract

Small perturbations quite often lead to power system instability. Power system stabilizer damp the electro mechanical oscillations in generator. Design of power system stabilizer has changed over past few decades. Artificial intelligence based controllers are found to be more effective to handle complex control situations during perturbations in power system. Fuzzy logic supported power system stabilizers were proposed to control such perturbations in more efficient way than conventional one. Fractional order controllers perform better than their counterparts in various engineering applications. This paper suggests use of such state of art fractional order controller in conjunction with existing power system stabilizer to enhance performance in terms of damping of electromechanical oscillations. From the results obtained from simulation models, it is observed that judicious design of fractional order controller in power system gives better handling of control operations. Comparison of conventional, fuzzy based and Fractional Order based power system stabilizer reveals usefulness of proposed controller.

Keywords

Power system stabilizer fractional order power system stabilizer fuzzy power system stabilizer

1 Introduction

Power system stability issues are closely associated with design of power system stabilizer (PSS). Role of power system stabilizer is to damp out power oscillations. Internal architecture of conventional power system stabilizer (CPSS) consist of wash out filter and cascaded phase lead networks. Various control system techniques helps in designing conventional PSS based on rigorous mathematical analysis and characteristic equation of overall transfer function. Root locus design is related to setting different values of constant K and location of open loop and closed loop poles. Using pole placement, desired location of dominant poles can be set. Eigen values helps in identifying different modes in which system equation responds. Frequency domain technique will highlight the frequency related behavior of PSS [7, 10].

As an advancement, numerous techniques based on neural networks, genetic algorithm, rule based fuzzy logic for power system stabilizer design have been proposed [1 , 16]. Fuzzy logic based designs related to single alternator connected to grid or infinite bus is expected to work efficiently due to inclusion of expert rules involved in design philosophy. Generator speed is used as an input to the FPSS. The rule base of fuzzy part of controller, consist of few rules for fuzzification and defuzzification.

New type of controller were introduced to control dynamics of power system based on fractional order calculus. Several engineering applications used them for better performance. These controllers gave flexibility and freedom to control system designer for deciding order of integration and differentiation in fractional way [13 , 17].

This paper investigates the performance of new state of art controller which uses fractional order of derivative and integration terms in controller architecture, for controlling the power system oscillations in a better way than presently used controllers. Further sections discusses the design of such controller. Single machine based power system models are used frequently to prove efficacy of controller as benchmark models.

The paper starts from analyzing the conventional design and its merits and demerits. Next generation of fuzzy logic based design of PSS is also analyzed on the same benchmark model as used in conventional one. Both of these designs have used finite order of derivative and integration. Latest controllers play with the order of these derivatives and integration terms in fractional sense. It opens up a huge space of freedom for control system designer and thus realization of fractional order based power system stabilizer (FoPSS) which performs well and proves to be the next generation of controller useful for power system related issues. Simple performance indices can be used to judge the performance of such controllers in terms of deviation from expected behavior.

Modern state of art controllers are equipped with Internet of Things (IoT) and artificial intelligence. Fractional order controller design can be made adaptive if it is backed by artificial intelligence. Parameters of controller can very well be monitored at remote end with the help of IoT. This gives extra dimension to controller and in turn improves its performance.

2 Mathematical model of conventional design of PSS

Different types of PSS realization for various power system applications as referenced in various CIGRE committees and IEEE papers can be summarized by a tree structure as shown in Fig. 1. As per this figure, conventional designs were based on linear approximations and simple transfer function. Challenges were posed due to complexity of power system models and this gave rise to need of nonlinear and adaptive realization of PSS design.

Fig.1

Different Types of PSS.

Rotor angle stability and power swings can be handled by the suitable design of PSS. While voltage stability is ensured by proper design of automatic voltage regulator (AVR) control at the terminals of alternator. Oscillations in rotor angle are dangerous to system and leads to loss of synchronism between machines in power system and ultimately leads to loss of active power. Stability can be improved by proper design of controller gains. constants and parameters involved in the process. Adverse effects of power swing can be handled by better damping ratio in output response of PSS. Different possible modes of oscillations can be avoided by knowing eigen values of system characteristic equation.

Sungle machine connected to infinite bus has several components realized by different transfer function involving various machine parameters such as inertia constants, time constants of electrical and mechanical parts including exciter model as well. This is standard model as approved by IEEE and called as Type-I exciter model. It neglects saturation of exciter. Also upper and lower limits of amplifier are also not included for simplified analysis.

Heffron-Phillip model has two basic loops. One involving mechanical rotational system involving inertia constant along with speed and rotor angle variation. Another control loop involves electrical model related to field winding and voltage signals. Electrical and mechanical loops have different time constants. But this complexity of interconnection between two different types of control loops give freedom to play with parameters for desired response. PSS acts as connecting link since it gives voltage as output while its input is variation in speed due to some disturbance or perturbation. Various constants as specified by benchmark model are K₁, K₂, K₄, K₅ and K₆ [7 , 11]. Following are the mathematical equations relating different parameters of generator used in SMIB model of power system [5, 11].

$Δ \dot{ω} = \frac{1}{2 H} (K_{1} Δ δ - K_{2} Δ E_{q}^{s} quo) + (\frac{1}{2 H}) Δ T_{m}$ (1) $Δ \dot{δ} = 2 π f Δ ω$ (2) $Δ \dot{E_{q}^{s} quo} = \frac{1}{T_{do}^{s} {quoK}_{3}} (- Δ E_{q}^{s} quo - K_{3} K_{4} Δ δ + K_{3} Δ E_{f} d)$ (3) $Δ \dot{E_{fd}} = \frac{1}{T_{E}} (K_{E} Δ E_{f} d + Δ V_{R})$ (4) $Δ \dot{V_{E}} = (- \frac{Δ V_{E}}{T_{F}}) + \frac{K_{F}}{T_{F} T_{E}} (- K_{E} Δ E_{fd} + Δ V_{R})$ (5)

All above variables are included in one state vector x and state equation is written as $\dot{x} = Ax + B Δ T_{m}$ where $x = [Δ ω, Δ δ, Δ E_{q}^{s} quo, Δ E_{fd}, Δ V_{E}, . . .]^{t}$ A is system matrix, ΔT_m is variation in mechanical torque and B is the disturbance matrix.

Fig. 3, represents simulation model of Fig. 2 scheme as implemented in MATLAB software tool. The simulation model is implemented with the help of several toolbox available in MATLAB. ΔT_m is shown as step change and acts as disturbance parameter which will result in variation of power swing i.e. rotor angle δ. The trajectory of rotor angle needs to be improved in case of sudden change in mechanical torque. For this PSS should give an output voltage, which along with exciter will help in damping the oscillations in rotor angle effectively if designed properly. Rule based design will ensure better performance and hence next generation of controller used Artificial Intelligence to some extent.

Fig.2

Heffron-Phillip suggested model of single alternator connected to infinite bus.

Fig.3

Conventional PSS model.

3 Design of fuzzy logic based power system stabilizer

Next generation of PSS used fuzzy set theory for better performance of controller action. Trend of speed deviation during perturbation in torque of machine was found to be a fuzzy variable which means speed deviation was either too slow, too fast or moderate. Controller action needs to care about the degree of deviation. Fuzzy mathematics overcame the problem of degree of speed deviation hence it was possible to sense too slow or too fast deviations clearly. Hence fuzzy theory based controller gave better response in understanding the implicit nature of speed deviation [1 , 9]. The input variable i.e. speed deviation is discretized resulting in three discrete levels. The inclusion of fuzzy logic block in power system model model is as shown in Fig. 4.

Fig.4

Fuzzy based PSS.

First derivative and second derivative of speed deviation has mathematical understanding in terms of dynamics of machine hence input to fuzzy logic based controller was divided in three parts. One will be actual speed deviation and other being first and second derivative of same. Wash out filter of conventional PSS is connected to output of fuzzy logic block as shown in Fig. 4. Three inputs to Fuzzy Logic block are U₁, U₂ and U₃ respectively. In this first input comes directly from speed deviation. To get first derivative of speed deviation as second input U₂, memory block is used in MATLAB software. Similarly second derivative of speed deviation as third input U₃ is obtained by another memory block connected to U₂ Second. Sampling time is denoted by kT. Where k is sample number. The generalized equations of input to FPSS at K sample are $U 1_{kT} = Δ ω$ (6) $U 2_{kT} = Δ ω_{kT} - Δ ω_{(k - 1) T}$ (7) $U 3_{kT} = U 2_{kT} - U 2_{(k - 1) T}$ (8)

Fuzzy design also needs judicious selection of membership function type. Most popular choice of membership functions is either Gaussian or trapezoidal respectively.

All three input variables of fuzzy based design passes through membership functions and according to degree of input variable they are processed by designed rules. The rule viewer in MATLAB software after designing certain finite number of rules is as shown in Figs. 5 and 6. This output needs to be verified by expert before its use in actual system.

Fig.5

FIS editor.

Fig.6

Rule viewer.

After passing through rules and output of fuzzy design is defuzzified so as to be fed to actual system in acceptable physical nature to further stage. This way detailed fuzzy design gave better performance over conventional PSS as will be clear in Section V.

4 Design of state of art controller based on fractional order

Controller design for a given plant, system or model becomes difficult due to cascading of various blocks with different time response. Hence overall impact on output quantity should possess following characteristics.

to increase the speed of the response;

better noise rejection;

robust and sensitive;

better signal to noise ratio;

better damping;

to reduce or eliminate the steady-state error.

Proportional Integral Derivative (PID) controllers have three fold attack to achieve most of the above specifications. Variation in derivative constant K_D will improve relative stability but signal to high frequency noise ratio is poor. Variation in integral constant K_I will improve or eliminate steady state error but it deteriorates the relative stability.

So it is difficult to adjust to given specifications hence optimal design is chosen for maximum accommodation of above specified expectations. Next generation designs based on fuzzy logic also worked on the same three constants i.e. K_P, K_I and K_D. Only difference in new designs was that, the values of K_P, K_I and K_D are adjusted according to human rules set by expert based on knowledge inference of system under consideration. Though rule based design is quite intuitive, it gave better output response when system was subjected to sudden disturbance.

Paradigm shift in mathematical theory of derivative and integration was introduced and it opened stringent boundaries of three constants i.e. K_P, K_I and K_D and increased degree of freedom from three parameters to five parameters i.e. λ and μ. Thus extra space and freedom to control system designer. Fractional orders help in exploring the areas beyond hard limits of derivative i.e. 1. Range of derivative order became variable which means one can find half or fractional derivative as well. So fractional calculus came into existence and it was birth of new generation of controllers. Several engineering applications started using fractional calculus and were successful enough to realize the power of freedom to choose order of derivative and integration.

PI^λ and D^μ controller proves to better when compared to conventional PID controllers [8 , 15]. Figure 7 shows use of fractional order controller block in PSS design. Conventional PID controller is special case of fractional order where order of integration and differentiation is full i.e. 1. $C (s) = \frac{U (s)}{E (s)} = K_{P} + K_{D} s^{- λ} + K_{I} s^{μ} (λ, μ > 0)$ (9) where λ is fractional order of integration ranging from 0 to 1, μ is fractional order of differentiation ranging from 0 to 1, while K _p , K _I and K _D are regular constants associated with proportional, derivative and integral processing of error.

Fig.7

Fractional order PSS.

Engineering systems are basically governed by integro-differential equations. Integral and derivative term in such equations is of full order. Similarly fractional order systems are governed by fractional orders of derivative and integral terms. Thus the form of fractional order equation is simplified to few terms on L.H.S. involving fractional derivatives of output variable y (t). R.H.S. of same equation involve fractional derivatives of input term v (t) [2 , 19]. $a_{1} D_{t}^{β_{1}} y (t) + a_{2} D_{t}^{β_{2}} y (t) + . . . . . + a_{n} D_{t}^{β_{n}} y (t)$ $= b_{1} D_{t}^{γ_{1}} v (t) + b_{2} D_{t}^{γ_{2}} v (t) + . . . . . . + b_{m} D_{t}^{γ_{m}} v (t)$ (10)

Right hand side of the above equation can be replaced by u (t) for getting simplified equation for output. $u (t) = b_{1} D^{γ_{1}} v (t) + . . . . + b_{m} D^{γ_{m}} v (t)$ (11) $y (t) = \frac{1}{\sum_{i = 1}^{n} \frac{a_{i}}{h^{β_{i}}}}$ $[u (t) - \sum_{i = 1}^{n} \frac{a_{i}}{h^{β_{i}}} \sum_{j = 1}^{[(t - a) / h]} w_{j}^{(β_{i})} y (t - jh)]$ (12)

Analytical solution of the above equation and associated MATLAB code is as referenced in [14].

$\begin{matrix} a_{n} D_{t}^{β_{n}} y (t) + a_{n - 1} D_{t}^{β_{n - 1}} y (t) \\ + . . . . . + a_{0} D_{t}^{β_{0}} y (t) = u (t) \end{matrix}$ (13)

$(m; k_{0}, k_{1}, . . . k_{n - 2}) = \frac{m!}{k_{0}! k_{1}! . . . ., k_{n - 2}!}$ (14) With the help of above mentioned equation, fractional order system related differential equation can be solved efficiently.

5 Simulation results

Speed of alternator and rotor angle are two crucial parameters related to stability of system under study. Extensive simulations were carried out with conventional, fuzzy logic based and fractional order controller in conjunction with conventional power system stabilizer. Models were subjected to extensive simulation under different conditions. Fractional order range was varied from 0 to 1 and it can always be optimized to a single specific value of fractional order for both integration as well as differentiation.

Optimality exist inherently in fractional order design. Performance indices based on deviation of output response are calculated for different values of fractional order of integration and differentiation ranging from 0 to 1 in discrete way. Out of which a single value of fractional order which gives better response is finalized, by algorithm implemented for selection of fractional order.

5.1 Work flow

Simulink model of CPSS, Fuzzy based PSS and fractional order PSS were subjected to unit step change in input i.e. mechanical torque of synchronous machine and output response in terms of rotor angle deviation and speed variation for different values of K_P, K_I and K_D, λ and μ was recorded. All models were subjected to step by step increase in one parameter say λ as fractional order of integration from zero to unity value, while keeping other parameters constant. Results were noted and visual inspection of rotor angle deviation shows improvement for some specific value of fractional order of integration. Performance indices such as Sum Square Error, Absolute Error, Integral Absolute Error were computed for each and every condition. Tables 1, and 2 shows the performance under different values of λ and μ.

Table 1
Integral Absolute Error for different values of λ for μ = 0.7

λ IAE in δ

0.1 0.269

0.2 0.352

0.3 0.372

0.4 0.399

0.5 0.452

0.6 0.172

0.7 0.209

0.8 0.152

0.9 0.272

1.0 0.569

λ	IAE in δ
0.1	0.269
0.2	0.352
0.3	0.372
0.4	0.399
0.5	0.452
0.6	0.172
0.7	0.209
0.8	0.152
0.9	0.272
1.0	0.569

Table 2

Integral Absolute Error for different values of μ for λ = 0.8

μ	IAE in δ
0.1	0.671
0.2	0.582
0.3	0.912
0.4	0.712
0.5	0.852
0.6	0.711
0.7	0.309
0.8	0.453
0.9	0.772
1.0	0.585

Table 1 shows that optimum value of fractional order of integration for better performance index is λ = 0.8, while Table 2 decides the fractional order of derivative term μ = 0.7. This way huge database for complete range of fractional order was analyzed and optimized values of K_P, K_I and K_D, λ and μ are finalized based on best performance reflected by reduced errors. It was seen power swings (in terms of rotor angle deviation) can be damped effectively with the use of all controllers but from performance indices fractional order was best which was already evident from rotor angle waveforms as well.

5.2 Graphical results

Fig. 8 represents comparison of Fuzzy based PSS and conventional PSS. It is clear from speed response to sudden variation in torque of machine; settling time of Fuzzy PSS is lower and it has less oscillations and better damping as compared to conventional PSS.

Fig.8

Speed response to step change in mechanical torque.

Fig. 9 represents comparison of Fuzzy based PSS and Fractional Order based PSS. This figure shows that speed response to sudden variation in torque of machine; settling time of fractional order based PSS is much lower and it has very few oscillations, better and efficient damping as compared to conventional PSS.

Fig.9

Speed response to step change in mechanical torque.

Fig. 10 represents comparison of Fuzzy based PSS and Fractional Order based PSS related to rotor angle swing. This figure shows that angle response to sudden variation in torque of machine; rotor angle in case of fractional order based PSS is less which is safe for power system from the point of view of transient stability as compared to fuzzy logic based PSS.

Fig.10

Rotor angle response to step change in mechanical torque.

6 Conclusion

Rotor angle response can be controlled in a better way with the use of new generation of fractional order based controllers. Stringent requirement on rotor angle response to curb instability is a challenge to controller designers. Fractional order opened infinite space of freedom by adding two more additional dimensions related to choice of order of differentiation and integration. Different power system model can have different values of fractional order chosen for control purpose. This makes the design of controller to be unique to that particular system. Analysis of response of various controller designs in previous sections, shows better performance in terms of rotor angle response of system with fractional order based controller over fuzzy and conventional designs in terms of peak overshoot and settling time. Performance indices such as Integral Absolute Error for different fractional order gives optimum selection of order of integration and differentiation. Full order integration and differentiation possess high IAE error as demonstrated in tables. This ensures better stability when power system is subjected to perturbations in real electrical power system.

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