Instability,chaos and bifurcation control in nonlinear dynamical system behavior using perturb-boost fuzzy logic controller

Abstract

This paper presents a novel Perturb-Boost Fuzzy Logic Controller for controlling the instability of nonlinear dynamical system behavior. Several applications can make use of the small perturbation technique discussed in the paper related to industrial control, mechanical nonlinear systems, electrical systems and other systems governed by nonlinear differential equations. This paper presents the power system as an application for use of novel controller to control voltage instability problem. The power system is an electro-mechanical nonlinear dynamical system, and is described by a combination of electrical and mechanical parameter based differential equations together. The power system faces problems related to voltage instability and chaos. Voltage instability exists in almost every power system for a specific set of mechanical power, electrical loading and initial conditions. Voltage instability can be controlled by injecting a small amount of reactive power using a power electronic device called a Static Volt Ampere Reactive Compensator. The amount of reactive power to be injected is trivial for different types and sizes of the power system. To control voltage instability, reactive power in a power system needs to be boosted. Proposed controller output decides amount of reactive power which perturbs the system equation to the stable operating point. The proposed Perturb-Boost Fuzzy Logic Controller differs from conventional controllers due to its single shot boost action, which perturbs system dynamics in such a way as to push it to safe zones of voltage stability. This paper analyzes the performance of the proposed controller to control the voltage instability for the generalized three node power system benchmark model. Reactive power to be injected is momentary due to single shot boost action. Time to reach instability gets delayed by approximately fifteen seconds using proposed controller for the benchmark model. Mitigation of voltage collapse is discussed in view of simulation results using proposed novel controller.

Keywords

Fuzzy logic controller chaos nonlinear dynamical systems nonlinear systems power system stability

1 Introduction

Nonlinear dynamical system behavior depends on several parameters and initial conditions. These parameters are nonlinear functions of other parameters. For simplification, several assumptions have been made in order to understand the behavior of vital parameters of the dynamic system. Under dynamic conditions, nonlinear systems are very much sensitive to initial conditions and parameter variation. One of the possible routes to the instability of nonlinear systems, is through bifurcation and chaos. Bifurcation and chaos can be evident visually, from close observation of phase plane trajectories or trajectories due to parameter variation. The pattern of these trajectories reveals the behavior of parameters of the system. These patterns are signatures of the nonlinear dynamic state of the system. The power system is one such application where dynamics play a very vital role and it affects millions if there is instability in the system due to parameter variation. The use of an intelligent system based on an artificial intelligence tool such as the fuzzy logic controller has been tried by several authors [4 , 15] for different objectives, but their need is felt in the effective control of chaos due to the highly sensitive nature of parameters involved in the power system.

This paper analyzes the power system model and control of its dynamic behavior using the proposed technique. Bifurcation and chaos are analyzed on Power-Voltage (P-V) nose curves in a previous study of different power system models [24]. Dynamical behavior of the power system can be well understood by state equations involving noise and stochastic variables [6, 14]. State of the power system at any instant of time is highly dependent on previous conditions [23]. These conditions, if slightly changed, may lead to exponential growth of the trajectory of power flows, angle or voltage which may lead to chaos or instability.

Apart from knowing the effect of initial conditions, the effect of parameter variation on the dynamic trend of vital parameters such as voltage, rotor angle and load angle needs to be understood by the power system operators.

Analytical solution of nonlinear equations with time varying realistic parameters is complicated and sometimes not possible to solve. Numerical solution and trajectories are the closest solution to such systems. Close observation of non linear power system dynamic equations reveals parameters such as mechanical power, electrical power, speed, rotor angle, voltage, active power demand, reactive power demand and load angle [18], which affect phase plane trajectories or instability trajectories.

Mechanical power is assumed to be constant for a given electrical power demand. But in real time the turbine output power is not smooth or constant as assumed theoretically. Hence the classical mathematical models need modification to include low frequency ripples present in turbine output power. During transition state, the mechanical power output of a turbine possesses a high magnitude of the ripples, which settles to lower values under static conditions. It can be attributed to transfer function between change in position of steam valve and associated changes in mechanical power output. Theoretical assumptions of mathematical models of the power system need modification by adding an extra term in existing differential equations [7, 21].

Analysis of voltage instability in the higher order model of the power system is complicated. It is due to the nonlinear and indirect relation between turbine ripple and voltage instability. The rule based fuzzy logic state feedback controller helps in perturbing the system to a nearby stable operating point. Voltage instability is curbed by injecting reactive power at load bus. The amount of reactive power to be injected is decided by fuzzy logic rules and membership functions. The inclusion of the real nature of mechanical quantities in proposed models of the power systems leads to different behavior from other normal known behavior. Noise is one such inclusion in mathematical models [6], which leads to bifurcation and chaos, hence the modification in the on the existing controller is required. This modification is essential to avoid unpleasant situations from the point of view of avoiding instability [24]. This paper proposes an artificial intelligence technique based novel perturb-boost fuzzy logic controller to curb the ill effects of turbine torque ripple on voltage instability. Conventional control of chaos and instability is reported in several case studies in the power system [2].

Response of nonlinear dynamical system is very much sensitive to small variations in parameters of state equations related to the dynamics of the generator. Computation of the Lyapunov Exponent (LE) of nonlinear equations of the dynamical power system model indicates bifurcation and chaotic behavior. Chaotic and instability behavior is very much related to the reactive power term in analytical equations. If we perturb this parameter by a small amount as a booster dose of injecting reactive power, it will shift the operating point of existing dynamics of the power system. Range of reactive power required for safe zones of stability are already known a priori for such models by performing simulations with different values of reactive power. Hence perturbation of the existing system shifts the operating point with the help of a single shot booster dose of injecting reactive power using a novel controller.

Section 2 explains the voltage instability scenario for a benchmark model of the power system. This section discusses the three node power system model as an application to understand the dynamics of systems subjected to parameter variation. This section highlights need for inclusion of a mechanical system response in conjunction with the electrical system for investigating potential threats of voltage and/or angle instabilities and chaos. Section 3 explains schematic diagram of Fuzzy Logic Controller (FLC) in conjunction with the power system model, to mitigate the chaos and instability trajectories by injecting reactive power through rule based design based on voltage collapse magnitude. Section 4 describes design details of software based MATLAB platform based Simulink model of proposed Perturb-Boost Fuzzy Logic Controller (FLC) along with numerical solution of differential equations of the fourth order model of the power system. This section discusses the design and implementation of proposed controller. Section 5 shows the results obtained from a most suitable benchmark model of the power system subjected to mechanical power ripple in turbine output causing instability and discusses the aspect of controlling voltage stability of systems with and without FLC using phase plane trajectory and time domain waveforms. Finally the paper concludes with specific attention to voltage instability due to turbine power ripples owing to the response of a governor control system and its mitigation through FLC.

2 Voltage instability in nonlinear dynamical power system

Benchmark three node model of the power system [8] is used to study chaos, bifurcation and voltage instability studies subjected to different initial conditions. Potential possibility of bifurcation, chaos or instability arising due to initial condition is already reported by several researchers [8, 23] by in depth study of the benchmark model. The model is versatile due to the fact that any power system can be shown equivalent to three interconnected. Node 1 connects to group of generators represented by single equivalent generator. Node number 2 connects to load bus. Node number 3 connects to the grid. So, the model is an equivalent representation of small scale to large scale power system as illustrated in Fig. 1.

The model is governed by a set of four first order nonlinear differential equation associated with parameters such as rotor angle, speed, voltage and load angle respectively. This model consists of a generator connected to bus 1, bus 3 is connected to an infinite bus-bar with voltage V _o and bus 2 is connected to a set of static and dynamic load. Variation in reactive power demand at bus 2 gives rise to different types of bifurcation and chaos corresponding to different values of reactive power demand Q.

In three node model of power system, δ is rotor angle, ω _m is angular speed, δ _L is load angle, V _L is load voltage and Q is reactive power load Demand as illustrated in Fig. 1. Following equations (with different coefficient values) based on 4D model are widely used by the power system researchers for bifurcation and chaos studies [15]. $x = [δ ω_{m} δ_{L} V_{L}]^{T}$ (1) where x is a state variable vector. An associated set of state equation for the 4-D model of the power system are $\dot{δ} = ω_{m}$ (2)

${\dot{ω}}_{m} = K_{1} \sin (δ_{L} - δ + θ_{1}) V_{L} - K_{2} ω_{m} + P_{m}$ (3) $\begin{matrix} {\dot{δ}}_{L} & = & K_{3} V_{L}^{2} - K_{4} \cos (δ_{L} - δ + θ_{1}) V_{L} - K_{5} V_{L} \\ - K_{6} \cos (δ_{L} - θ_{2}) V_{L} + K_{7} Q + K_{8} \end{matrix}$ (4) $\begin{matrix} {\dot{V}}_{L} & = & - K_{9} V_{L}^{2} + K_{10} \cos (δ_{L} - δ + θ_{3}) V_{L} \\ + K_{11} V_{L} + K_{12} \cos (δ_{L} - θ_{4}) V_{L} \\ - K_{13} Q - K_{14} \end{matrix}$ (5)

Where P _m is turbine power output and K ₁ , K ₂ ,... K ₁₄ are constants with values mentioned in Appendix A, for a case study of the 4-D power system [23]. The load model at bus 3 is a composite of constant power, constant current, and constant impedance as follows. It includes dynamic model of induction motor as well. Thus load model envisages effect of all types of load. Equation (6) and (7) illustrates a detailed load model. $P = P_{o} + P_{A} * V + P_{B} * V^{2}$ (6) $Q = Q_{o} + Q_{A} * V + Q_{B} * V^{2}$ (7)

Detail explanation of terms and coefficients in (8) and (9) is cited in [11]. $\begin{matrix} P & = & - V_{0}^{'} {VY}_{0}^{'} \sin (δ + θ_{0}^{'}) \\ - V_{m} {VY}_{m} \sin (δ - δ_{m} + θ_{m}) \\ + [Y_{0}^{'} \sin (θ_{0}^{'}) + Y_{m} \sin (θ_{m})] V^{2} \end{matrix}$ (8) $\begin{matrix} Q & = & V_{0}^{'} {VY}_{0}^{'} \cos (δ + θ_{0}^{'}) \\ + V_{m} {VY}_{m} \cos (δ - δ_{m} + θ_{m}) \\ + [Y_{0}^{'} \cos (θ_{0}^{'}) + Y_{m} \cos (θ_{m})] V^{2} \end{matrix}$ (9)

The model with specific values of constants shows different types of bifurcation behavior at following values [3] of reactive power Q.

Sub critical Hopf bifurcation at Q = 10.9461.

Period doubling bifurcation at Q = 10.8859.

Period doubling bifurcation at Q = 11.3776.

Sub critical Hopf bifurcation at Q = 11.4066.

Saddle-node bifurcation at Q = 11.4106.

Apart from reactive power loading this model is also sensitive to mechanical power term P_m in (3). Turbine power output term P_m in (3) needs to be modified due to its ripple content in real systems. The mechanical power output of the turbine will not be smooth under dynamic conditions of the power system. Hence the mechanical power output of the turbine is time dependent quantity. So the constant mechanical power term is augmented with an additional ripple term as in (10). $P_{m} (t) = P_{m} + P_{osc} \sin (φ) where φ = ω_{T} t$ (10) where P _m is average power, P _osc is turbine power ripple amplitude, ω _T is the angular frequency of turbine power ripple and t is time in seconds. The 4D model of the power system, was subjected to variation in turbine power ripple amplitude. Its impact on phase plane trajectories of voltage is as illustrated in Fig. 2. If the ripple content in turbine power exceeds 5%, then the voltage magnitude collapses near towards a value below 0.8 pu. Figure 2 represents, how ripple in turbine power affects voltage trajectory and oscillating part of power aggravates the situation and push the system near to voltage instability. By considering the effect of turbine ripple, (3) gets additive term related to an oscillatory component of turbine power and (12) is modified form of (3) due to (10). $\dot{δ} = ω_{m}$ (11) $\begin{matrix} {\dot{ω}}_{m} & = & K_{1} \sin (δ_{L} - δ + θ_{1}) V_{L} - K_{2} ω_{m} \\ + P_{m} + P_{osc} \sin (φ) \end{matrix}$ (12) $\begin{matrix} {\dot{δ}}_{L} & = & K_{3} V_{L}^{2} - K_{4} \cos (δ_{L} - δ + θ_{1}) V_{L} - K_{5} V_{L} \\ - K_{6} \cos (δ_{L} - θ_{2}) V_{L} + K_{7} Q + K_{8} \end{matrix}$ (13) $\begin{matrix} {\dot{V}}_{L} & = & - K_{9} V_{L}^{2} + K_{10} \cos (δ_{L} - δ + θ_{3}) V_{L} \\ + K_{11} V_{L} + K_{12} \cos (δ_{L} - θ_{4}) V_{L} \\ - K_{13} Q - K_{14} \end{matrix}$ (14)

Voltage instability for such model needs control using artificial intelligent control. Equations (11)–(14) are highly nonlinear and there is no linear relation between Pm and Q as evident from observed bifurcation and chaotic behavior of the power system models. Owing to different sizes and scales of the power system models, control law becomes complicated. Hence human rule based controller design using Fuzzy Logic is a better choice. State feedback controller will keep watch on voltage instability issue as explained in the next section.

3 Fuzzy logic controller for mitigation of voltage instability

Ripple content in turbine power due to governor response for variation in active power demand, usually range from 1% to 10%. Figure 2 illustrates phase plane trajectories of the model discussed in the previous section. It is a clear observation that as ripple content increases, voltage instability peeps into the system. This voltage collapse can be controlled intelligently using novel formulated Fuzzy Logic Controller. State variable based feedback will help in controlling the system to a desired stable point of operation. Since we are targeting voltage instability, voltage is chosen as the obvious choice for the state variable feedback parameter. If we need to control chaos, then windowed Lyapunov Exponent can be used as input to fuzzy logic based state feedback controller. Since Lyapunov Exponent computation is based on “average”, it may not be a proper choice as control input which needs to be under check for every instant. There is a possibility of instantaneous upsurge or downfall in voltage, which will give a lower average value of LE. Power system control can never be risked based on average concept. But at the same time LE is a good parameter for forecasting of futuristic disaster in observed parameter, but not a parameter for input to the control system. Voltage and its rate of change is sufficient for perturbing system to a stable state and boost action ensures minimized control efforts.

Role of fuzzy logic controller will be to inject a suitable amount of reactive power to improve voltage instability situation. By writing suitable MATLAB based script or embedded code in con-junction with Simulink Based Fuzzy Logic Controller with RuleViewer, one can decide the time of injection and quantity of reactive power to be injected based on desired safe zone stable phase plane trajectory of voltage. This will ensure control of voltage instability.

State equations of the 4D power system are sensitive to reactive power demand Q at load bus. If we can add a certain reactive power dQ judiciously, then trajectories associated with voltage-angle or voltage-time will change accordingly. Following equations will hold good when Q gets modified by an amount dQ. $\dot{δ} = ω_{m}$ (15) $\begin{matrix} {\dot{ω}}_{m} & = & K_{1} \sin (δ_{L} - δ + θ_{1}) V_{L} - K_{2} ω_{m} \\ + P_{m} + P_{osc} \sin (φ) \end{matrix}$ (16) $\begin{matrix} {\dot{δ}}_{L} & = & K_{3} V_{L}^{2} - K_{4} \cos (δ_{L} - δ + θ_{1}) V_{L} \\ - K_{5} V_{L} - K_{6} \cos (δ_{L} - θ_{2}) V_{L} \\ + K_{7} (Q + dQ) + K_{8} \end{matrix}$ (17) $\begin{matrix} {\dot{V}}_{L} & = & - K_{9} V_{L}^{2} + K_{10} \cos (δ_{L} - δ + θ_{3}) V_{L} \\ + K_{11} V_{L} + K_{12} \cos (δ_{L} - θ_{4}) V_{L} \\ - K_{13} (Q + dQ) - K_{14} \end{matrix}$ (18)

Equation (13) and (14) are modified to (17) and (18) due to the addition of reactive power dQ. Amount of dQ to be injected depends on FLC design. FLC design can be formulated for desired stability. For example, if the membership function for the amount of dQ is defined over a range of –0.1 to 0.1 (1% of the reactive power loading value of 11.3165), it will give voltage stability up to 34-36 seconds as discussed in the results section. Hence a slight shift in Q will give relief from the ill effects of turbine ripple leading to voltage instability. So mitigation using FLC is effective. Figure 3 gives brief idea about close loop FLC for mitigation of undesirable trajectories. As per this schematic voltage and its rate of change are selected as two input state variables fed from power system. Output of FLC is used to decide the amount of reactive power to be injected or withdrawn. To get voltage and its rate of change from power system equation, solution of differential equation is required. The solution of differential equations of power system is realized using MATLAB software based Simulink blocks. Simulink in MATLAB software, is huge collection of different mathematical blocks. Simulink based blocks along with the use of Integrator block will solve four sets of interconnected differential equations of the power system. Alternatively, one can use MATHEMATICA software to solve nonlinear differential equations. But programming flexibility is easy in Simulink based models as compared to directly write scripts in MATHEMATICA software.

The dQ amount of power is injected or extracted based on voltage sag and swell and its dynamics due to bifurcation, chaos or unstable divergent fall in voltage. This reactive power is supplied by power electronic based Flexible Alternating Current Transmission System (FACTS) devices such as static VAR compensators (SVC).

4 Simulink model with novel fuzzy logic controller

Fourth order model of the power system was modeled in Simulink with parameters as mentioned in Appendix [A]. MATLAB-Simulink software based model of Perturb-boost Fuzzy Logic Controller is illustrated in Fig. 4. The Perturb-boost Fuzzy Logic Controller is built using the integration of Fuzzy Logic Controller with Rule-Viewer block available in Simulink, clock, injector time, IF block and IF-ACTION subsystem in Simulink model realization. MATLAB script file controls the injector-time and runs the Simulink mathematical model of four state Equations (15)–(18), with four integrator blocks, one each for every state variable. The complete Simulink model uses four differential equations interconnected with each other due to state variable associations between them. Voltage solution and its rate of change available at discrete interval of time are fed to FLC as illustrated in Fig. 4. Figure 5, illustrates fuzzy inference system built using MATLAB software.

Hardware realization of injection of dQ is possible with the help of Flexible Alternating Current Transmission System (FACTS) devices such as static VAR compensators (SVC). Control of chaos using the FACTS device in a power system is already in use [5, 16].

Figure 6 illustrates solution of only one differential Equation (18) with injection of reactive power as an output of perturb-boost fuzzy logic controller. The complete simulation model consist of four such differential equation with integrator blocks and other coefficient blocks. Every term on the right hand side of (18) is modeled. Each term gets input from other state variables. Derivative of variable such as voltage is computed with the help of right hand side terms of (18). Integrator block is kept in front of the voltage derivative term, thus voltage is available as output. It will act as feedback signal, since right hand side terms consist of voltage variable also. Reactive power term in (18) is the addition of an output of FLC block and reactive power demand at load bus. FLC block is connected to injector time and other conditional blocks, whose purpose is to inject reactive power for a very short duration to perturb the system slightly to push the system out of bifurcation or chaotic state. Perturb-boost type FLC injects a single shot of required amount of reactive power and system needs to be perturbed slightly from its earlier stable zone operation of stability. Hence it is momentary action and reduces control efforts as compared to continuous proportional-integral (PI) controller. Figure 5 represents Mamdani type fuzzy inference system with voltage and its rate of change as two inputs and output variable as reactive power dQ to be injected. Membership functions associated with voltage were given linguistic names as COLLAPSE, LOW and NORMAL. Reactive power dQ to be injected/withdrwan is in the range of –0.1 to 0.1 (1% of the actual Q loading value of 11.3165). Nine rules as per Table 1, were framed so as to inject reactive power dQ. Linguistic terms related to membership functions are, N for Negative, Z is Zero and P is Positive for rate of change of voltage input and output dQ respectively. It is evident from fuzzy based rule design of controller that for the LOW range of voltage and a negative rate of change of voltage (which represents a falling trend of voltage and may lead to further reduction in voltage situation), the output of FLC i.e. dQ is positive which means the injection of reactive power as illustrated in Fig. 7.

The reactive power dQ needs to be extracted if voltage is NORMAL (when its rate of change is positive, which represents a rising trend of voltage and may lead to over-voltage situation) as shown in Fig. 8.

Amount of dQ injected (or extracted) could be anything between –0.1 to 0.1 (1% of nominal Q ). This injection causes voltage to boost up in presence of turbine power ripple. This will simply perturb voltage slightly to cause the local death of bifurcation and chaos. From the above design of FLC, we can see suggested controller successfully injects (or withdraws) required amount of dQ based on voltage approaching COLLAPSE, LOW and NORMAL range with second input as rate of change of voltage.

5 Results and discussions

MATHEMATICA software and MATLAB software based computing platform was used for the numerical solution of a set of differential equations. Numerical solution of all above mentioned equations was obtained using Runge-Kutta Method (RKM-45) method and corresponding trajectories of voltage are plotted. Ripple of magnitude 1 to 10 percent was applied to the model and it was observed that for similar initial conditions of parameters, trajectories of voltage started deviating very fast for higher levels of ripple amplitude and led to instability within few seconds.

A numerical solution to a set of simple first-order differential equations offers the closest approximation to understanding the problem and formulating a solution. For a constant magnitude of mechanical power P _m dynamic trajectory of voltage and load angle are within permissible limits of stability as shown in Fig. 9 (solid curve), while leading to instability (dashed curve) for mechanical power with ripple.

With approximately 5.5% ripple in mechanical power P _m , voltage trajectory enters an unstable zone (<0.8 p.u.) after 20 seconds in the absence of any controller, as illustrated in Fig. 10.

The dynamics of a power system can be understood visually by inspecting the phase plane trajectories as illustrated in Fig. 11. They seem to be in a certain order compared to their time-domain related waveforms. Time-domain related waveforms do not reveal much about the dynamics of system variables under multiple parameters except time to reachinstability.

Figure 11(a) shows the effect of FLC design with only one input as voltage. The “Without FLC” condition is shown by a dashed curve, which indicates voltage collapse below 0.9. With FLC, the phase plane trajectory does not lead to instability for 20 seconds. So in a given time, the presence of FLC with a single or two inputs ensures that the voltage trajectory remains safely within 0.95 to 1.05 per unit for a long period of time. Figure 11(b) shows the effect of a two parameter FLC. Figure 11 (c) shows a comparison between FLC with one input and an FLC with, two inputs. One can observe from Fig. 11 (c) that the fluctuation in angle is greater in the case of two-input FLC design.

Figure 12 shows the time-domain situation of the instability; one can clearly see that instability is delayed by 14 to 15 seconds with a FLC design with perturb-boost feature, and hence that it is very effective in giving power system operators more time to shift the loading point accordingly, and while load curtailment or a change of operating point will mitigate instability completely.

The dynamics of a system under consideration can very well be better understood by a closer look at the pattern of phase plane trajectory plots than through analytical solutions.Various analytical approaches based on stability and Lyapunov function are difficult to solve for higher order power system models [10] hence the numerical solution of a set of state equations is preferred for getting a simplified solution for voltage trajectory [9, 20]. To prevent voltage instability issues several intelligent methods need to be tried out. Additionally, terms associated with the dynamics of system stability should be clearly delineated for a better design of the proposed perturb-boost FLC based controller [19].

Use of the conventional FLC to prevent voltage collapse is proposed by researchers due to the simplicity of the rule-based design [1, 17]. Our results and design reduce the control effort and keeps the system stable for a long period of time. The role of turbine governor response can further be analyzed in better stochastic terms. Further, stability can be correlated to mechanical parameters that are usually ignored or assumed to be constant for theoretical simplifications [12, 22].

To summarize, we can say proposed FLC helps in perturbing the system to a more stable zone for a longer period of time as evident from the results obtained. This gives sufficient time for the power system operational experts to act. FACTS devices such as SVC can inject a small amount of reactive power with variation in firing angle of power electronic switches. The perturbed point of operation will be away from unwanted bifurcation, chaos and instability and can be confirmed by detailed Lyapunov Characteristic Exponent computation. The modified value of reactive power will change the trajectory of nonlinear dynamical system and push the system back to a stable state.

6 Conclusion

From results obtained and discussions in earlier sections, we can see that a nonlinear dynamical system possess the potential for bifurcation, chaos, strange trajectories and instability on account of the specific numerical values of different parameters. This paper highlights the role of a proposed new perturb-boost fuzzy logic controller in curbing unexpected behaviors in power systems as such voltage instability arising from the ripple content of mechanical power.

A power system was chosen as an application for the proposed controller. Using the proposed controller for the benchmark model, the onset of instability is delayed by approximately 15 seconds. Similar control action can be applied in other systems because the chaotic trajectories are generally very sensitive.

A small perturbation is sufficient to end the chaos and bring the system to order. The inclusion of real observed parameters in the mathematical equation of a nonlinear dynamical system will help to analyze correctly the behavior of vital power system stability in a better way. Power system operators will therefore experts will benefit from this research.

Higher order models of power systems, along with the actual nature of turbine power ripple, needs to be analyzed and simulated accordingly. The inclusion of oscillatory components of turbine power and their impact on phase plane trajectories was also discussed. Controlling the adverse effect of torque ripple on voltage stability, perturb-boost fuzzy logic controller based reactive power injection is proposed and found to be effective for normal ripple content in turbine power. The amount of reactive power required is only 1%, which is sufficient to stabilize voltage for an extended period of 14 to 15 seconds compared to the case without the controller.

Reactive power to is injected momentarily in a single-shot boost action. Injector time varies from a minimum of half a cycle to a maximum of one cycle. Further work is required on the inclusion of more rules and input parameters for 6D and 7D, i.e. higher-order models of power system. Steam, hydro and wind turbines have different responses in the actual power system, so any potential instability and controller design using perturb-boost fuzzy fuzzy-logic based state feedback needs to be investigated a priori.

Footnotes

Appendix

Following are the values of constants used for the 4-D model of power system [3]. Initial conditions are δ(0)=0.3, ω_m(0)=0, δ_L(0)=0.2 and V_L(0)=0.97. $\begin{matrix} K_{1} & = & 16.667, K_{2} = 1.881, K_{3} = 496.872 \\ K_{4} & = & 166.667, K_{5} = 93.333, K_{6} = 666.667 \end{matrix}$

$\begin{matrix} K_{7} & = & 33.333, K_{8} = 43.333, K_{9} = 78.764 \\ K_{10} & = & 26.217, K_{11} = 14.523, K_{12} = 104.869 \\ K_{13} & = & 5.229, K_{14} = 7.033, θ_{1} = 0.087 \\ θ_{2} & = & 0.209, θ_{3} = 0.012, θ_{4} = 0.135 \\ Q_{1} & = & 11.3165 \end{matrix}$

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