Keywords

1 Introduction

The management of modern power systems became most difficult task to ensure the system security and reliability with good quality. India is aiming at forming a single national grid with fine tuning of frequency all over the system, constituted with many of the power system components and distinct loads. The load perturbation on all over the grid is dynamically varying, leads to frequency deviations over each of the control areas. The focus of this article is to maintain the constant or tolerable range of frequency in all control areas of power grid. The single area power system is modelled as a single transfer function block diagram, comprised of governor, turbine and generator including load transfer function models. The two-area model is composed of two single areas with different individual block models of different parameters connected together with a tie line model. The Simulink models have been developed for both single area and two area power systems to carryout simulation study and analysis. These simulation model of three area power system with three different case studies viz. case I: with 5% load disturbance with all three sub cases of (a), (b) and (c) without any controller, with PI controller and with Fuzzy controller respectively. Case II: with 5% load disturbance with all three sub cases of (a), (b) and (c) without any controller, with PI controller and with Fuzzy controller respectively. Case III: with 5% load disturbance with all three sub cases of (a), (b) and (c) without any controller, with PI controller and with Fuzzy controller respectively [1,2,3,4,5,6].

2 Single Area Power System

The power system network comprehends of many distinct loads with huge transmission and distribution networks. The complete network is categorized into different control areas based on frequency deviation over the part of power network. The small signal model is derived with individual power system components viz. Speed Governor, Turbine and Generator including load models.

2.1 Speed Governor Model

The load on power system is continuously varying may lead to speed deviation subsequently it leads to frequency fluctuations. The speed governing system is used to control these fluctuations. The speed governing system composed of fly ball speed governor, hydraulic amplifier and speed changer with linkage mechanism. The governor has two basic inputs, one is changes in reference power and the second is frequency variations are modelled into an equation, which describes both of these inputs as portray in the below equation and Fig. 1. Time constant (Tg) (as expressed in terms of a constant, depends on orifice, cylindrical geometries and fluid pressure of Hydraulic Amplifier and its typical value is in the range of 0.1 to 0.6 s [3,4,5,6,7].

$$ P_{v} \left( s \right) = ~[P_{{ref}} - ~{}_{R}^{1} F\left( s \right)]~\frac{{K_{g} }}{{1 + T_{g} }} $$
(1)

2.2 Turbine Model

Turbine model has been derived with a single stage turbine with a single time constant of Tt and typical value is in the range of 0.3 to 0.7 s. The transfer function model of a single time constant turbine model is depicted in below equations (Fig. 2).

$${P}_{m}\left(s\right)= {P}_{t}\frac{{K}_{t}}{1+{T}_{t}}$$
(2)
$${T}_{t}=\frac{1}{{K}_{h}}$$
(3)
Fig. 1.
figure 1

Governing system model

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Fig. 2.
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Small signal model of turbine

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2.3 Generator Load Model

The incremental power input to the system can fractionated towards stored kinetic energy in the generator being proportional to square of the frequency and load changes of sensitive loads due to frequency deviations. These two events have been modelled and derived the transfer function of generator load model as depicted in the following Eq. (4) and (5) below. The inertia constant is designated as H and its typical value is about 2 to 10 s, hence the power system time constant i.e. Tps lies in the range of 4 to 20 s. The following equations illustrate expressions for Tps and Kps in terms of inertia constant (H) and load damping factor (D) [5,6,7,8,9,10,11,12] (Fig. 3).

$${T}_{PS}= \frac{2H}{{DF}^{O}}$$
(4)
$${K}_{PS}= \frac{1}{D}$$
(5)
$$ \left[ { P_{m} \left( s \right) - } \right[ P_{m} \left( s \right) = \frac{2HP}{F}\frac{dF}{{dt}} $$
(6)
$$F(s)=[ {P}_{m}\left(s\right)- {P}_{e}\left(s\right)\frac{{K}_{ps}}{1+{T}_{ps}}]$$
(7)
Fig. 3.
figure 3

Generator load model

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The model of LFC as illustrated in Fig. 4, LFC with PI Controller is illustrated by the Figs. 5 and 6 shows the Single Area Load Frequency Control with Fuzzy Controller [4,5,6,7,8,9,10,11,12].

Fig. 4.
figure 4

LFC without Controller

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Fig. 5.
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LFC with PI Controller

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Fig. 6.
figure 6

LFC with Fuzzy Controller

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3 Single Area Power System

Power Systems can be sub divided into number of areas through tie lines, without out loss of generality, consider a simple two area power system with a single tie line as depicted by Fig. 7 as shown below. The power transfer of the tie line is P12 is given in Eq. (10), the synchronizing torque coefficient T12 as illustrated in Eq. (11), subsequently expressions have been derived for Area Control Error (ACE1 and ACE2) in terms of P12 and P21, bias factors B1 and B2 respectively as shown in equations from (12) to (15). Figure 8 illustrates the Block diagram model of Fuzzy controller based two area power system, with fuzzy based controller in each of the areas respectively [1,2,3,4,5,6,7,8] (Figs. 9 and 10).

Fig. 7.
figure 7

Two area power system with tie line

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$${\mathrm{P}}_{12}=\frac{{\mathrm{V}}_{1}{\mathrm{V}}_{2}}{{\mathrm{X}}_{12}}\mathrm{sin}\left(\mathrm{\Delta \delta }\right)$$
(8)
$${\mathrm{T}}_{12}=\frac{{\mathrm{V}}_{1}{\mathrm{V}}_{2}}{{\mathrm{X}}_{12}}\mathrm{cos}\left(\mathrm{\Delta \delta }\right)$$
(9)
$${\mathrm{P}}_{12}=2\uppi \int \mathrm{\Delta F}\frac{{\mathrm{V}}_{1}{\mathrm{V}}_{2}}{{\mathrm{X}}_{12}}\mathrm{cos}\left(\mathrm{\Delta \delta }\right)$$
(10)
$${\mathrm{P}}_{12}=2\uppi \int \mathrm{\Delta F}{\mathrm{T}}_{12}$$
(11)
$${\mathrm{P}}_{12}\left(\mathrm{s}\right)=\frac{2\uppi {\mathrm{T}}_{12}\mathrm{\Delta F}\left(\mathrm{s}\right)}{\mathrm{s}}$$
(12)
$${\mathrm{P}}_{21}\left(\mathrm{s}\right)=-\frac{2\uppi {\mathrm{T}}_{12}\mathrm{\Delta F}\left(\mathrm{s}\right)}{\mathrm{s}}=- {\mathrm{P}}_{12}\left(\mathrm{s}\right)$$
(13)
$${ACE}_{1}={\mathrm{\Delta P}}_{12}+ {B}_{1}{\mathrm{\Delta F}}_{1}$$
(14)
$${ACE}_{2}={\mathrm{\Delta P}}_{21}+ {B}_{2}{\mathrm{\Delta F}}_{2}$$
(15)
Fig. 8.
figure 8

Block diagram model of Fuzzy controller based two area power system

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Fig. 9.
figure 9

Three Area Power System with tie lines

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Fig. 10.
figure 10

Block diagram model of Fuzzy controller based multi area power system showing only one area signals

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3.1 Tie Line Bias Control of Multi Area Power Systems

Each of the control area is connected with as many areas as possible in interconnected systems, with tie lines. Consider one area in the interconnected system, the net interchange with other control areas equal to the sum of tie line powers through outgoing tie lines and is given by the following equation below Eq. (16) similarly other area controllers can be obtained.

$${ACE}_{i}=\sum \Delta {P}_{ij}+{B}_{i}\Delta {F}_{i}$$
(16)

4 Power System Stability

Power system stability is defined as a system ability to regain its initial equilibrium state after being subjected to a disturbance and its classification are given in Fig. 11.

Fig. 11.
figure 11

Power system stability stratification

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Fig. 12.
figure 12

Power angle curve

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4.1 Rotor Angle Stability

The system should remain in synchronism even after being subjected to the disturbance, which involves output power oscillates reflected in rotor oscillations.

4.2 Rotor Angle Stability

Relation between the power and angular position of a rotor in synchronous machine is nonlinear relation, when the synchronous generator is feeding a synchronous motor through transmission line. The power transferred to the motor from the generator is depends on the function of angular displacement between rotors of the generator and motor this is because of Motor internal angle, generator internal angle, and the angular displacement between motor and generator terminal voltage and power flow is given by the Eq. 17.

$$\mathrm{P}=\frac{{\mathrm{V}}_{1}{V}_{2}}{{\mathrm{X}}_{\mathrm{T}}}\mathrm{sin\delta }$$
(17)

This equation say that the power transferred to a motor from generator is maximum when the angle is 90°, if the angle is further increased beyond 90°, power transferred starts decreasing is shown in Fig. 12. The maximum power transferred is directly proportional to machine internal voltage [8,9,10,11,12].

$${\mathrm{P}}_{\mathrm{max}}=\frac{{\mathrm{V}}_{1}{V}_{2}}{{\mathrm{X}}_{\mathrm{T}}}$$
(18)

4.3 Dynamic Stability

Whenever the synchronous machine is subjected to small and medium load perturbations then the machine is subjected to rotor oscillations. These oscillations will die out very soon due to damping provided in the machine or external damping provided by the controller or compensator. The swing curves of the machine will illustrate the dynamic instability mechanism of the machine for such disturbances as illustrated in the below Fig. 13(a) and (b). The Fig. 13(a) depicts that the system is stable for small and medium disturbances with quiet effective damping and is unstable for medium disturbances with insufficient damping.

4.4 Transient Stability

Whenever the synchronous machine is subjected to medium and large load variations or contingencies then the machine is subjected to wide rotor oscillations. These oscillations will die out very soon due to damping provided in the machine or external damping provided by the controller or compensator. The swing curves of the machine will illustrate the transient instability mechanism of the machine for such disturbances as illustrated in the below Fig. 13(a) and (c). The Fig. 13(a) depicts that the system is stable for medium disturbances with suitable controller and is unstable for large disturbances without any controller [1,2,3,4,5,6,7,8,9,10,11,12].

Fig. 13.
figure 13

Power system stability swing curves (a) Stable System (b) Oscillatory Unstable System or dynamic instability and (c) Non-oscillatory Instability or Transient Instability

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5 Fuzzy Logic Control

Fuzzy control is used when vaguness in the decission making is present or when non linearities are involved in the system dynamics. Figure 14 shows Mamdani Rule based Fuzzy Logic Controller input and output functional relationship in simulink model and Fig. 15 3-d surface of the Fuzzy rules. The Table 1 shows the Fuzzy rules table for the single and two area power systems. There are two input membership functions of Fuzzy logic controller, one is error i.e. the reference voltage minus actual voltage of seven input triangular membership functions varying range of + or −16%. The second is derivative of error seven input triangular membership functions varying range of + or −16% as depicted in Fig. 14 and seven output triangular membership functions varying range of + or −18%. The Fuzzy rules [6,7,8,9,10,11,12].

Table 1 shows the 49 rules framed with input1 variables viz. nb1, nm1, ns1, zo1, ps1, pm1 and pb1, similarly input2 variables viz. nb2, nm2, ns2, zo2, ps2, pm2 and pb2 and output variables viz. nb, nm, ns, zo, ps, pm and pb [4,5,6,7,8,9,10,11].

Fig. 14.
figure 14

Triangular input and output membership functions of fuzzy controller

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Fig. 15.
figure 15

Fuzzy 3-D rule surface

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Table 1. Fuzzy rule table
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6 Case Study and Simulation Results

The single area and two area power system Simulink models have been derived from the small signal transfer function models of both areas. Both of these areas have been simulated for three cases, one is without controller, second is with PI controller and third is with Fuzzy logic based controller. In subsequent part of the article single area system has been presented and succeeding part describes the simulation results of two area system.

6.1 Simulation Results of Three Area Power System

The Simulink models have been developed for three area power system for all three cases, case I (a): without any controller as depicted by the Fig. 16, case I (b): Three area system with PI Controller as shown in Fig. 17 and case I (c): Single area system with Fuzzy Controller as illustrated in Fig. 18.

Fig. 16.
figure 16

Three area system Simulink model without Controller

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Fig. 17.
figure 17

Three area system Simulink model with PI Controller

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Fig. 18.
figure 18

Three area system with Fuzzy Controller

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Case I: 5% disturbance results of three area system

Area1 frequency deviations with 5% disturbance

In Case I results have been presented with 5% disturbance only with sub cases of Case I-1 for area1, Case I-2 for area2 and Case I-3 for area3 respectively. Case I-(a) Area 1 without controller error is −15% which is not acceptable and unstable, Case I-(b) with PI Controller the peak undershoot is −22% and settling time of 25 s and Case I-(c) with fuzzy controller the peak undershoot is −12.5% and settling time of 14 s with zero steady state error which is most accepted one among all systems as illustrated by the below Fig. 19.

Fig. 19.
figure 19

Area1 frequency deviations with 5% disturbance

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Case I-2 (a) Without controller error is −19% which is not acceptable and unstable, Case I-3 (a) with PI Controller the peak undershoot is −29% and settling time of 25 s and Case I-3 (a) with fuzzy controller the peak undershoot is −18% and settling time of 14 s with zero steady state error as depicted by Fig. 20 below.

Fig. 20.
figure 20

Area2 frequency deviations with 5% disturbance

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Case I-(a) without controller error is −22% which is not acceptable and unstable, Case I-3 (b) with PI Controller the peak undershoot is −43% and settling time of 25 s and Case I-3 (c) with fuzzy controller the peak undershoot is −23% and settling time of 14 s with zero steady state error and Fig. 21 shows the Area3 frequency deviations with 5% disturbance.

Fig. 21.
figure 21

Area3 frequency deviations with 5% disturbance

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Case I-4 the power and delta deviation without controller it is unstable and with controllers the system is stable for small load disturbances in Area1 and the deviation is very small for fuzzy controller as depicted in Figs. 22 and 23 below.

Case II: 10% disturbance results of three area system

Case II-1 (a) without controller error is −29% which is not acceptable and unstable, Case II-1 (b) with PI Controller the peak undershoot is −49% and settling time of 15 s and Case II-1 (c) with fuzzy controller the peak undershoot is −11.5% and settling time of 6 s with zero steady state error as illustrated in the below Fig. 24.

Fig. 22.
figure 22

Tie line Power in area1 without, with PI and with Fuzzy Controllers

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Fig. 23.
figure 23

Area1 deviation in delta1, delta2 and delta3 without, with PI and with Fuzzy Controllers

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Fig. 24.
figure 24

Area1 frequency deviations with 10% disturbance

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Case II-1 (a) Without controller error is −39% which is not acceptable and unstable, Case II-2 (b) with PI Controller the peak undershoot is −64% and settling time of 16 s and Case II-2 (c) with fuzzy controller the peak undershoot is −11.5% and settling time of 12 s with zero steady state error as Fig. 25 depicts Area2 frequency deviations with 10% disturbance.

Fig. 25.
figure 25

Area2 frequency deviations with 10% disturbance

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Case II-3 (a) Without controller error is −49% which is not acceptable and unstable, Case II-3 (b) with PI Controller the peak undershoot is −83% and settling time of 16 s and Case II-3 (c) with fuzzy controller the peak undershoot is −40% and settling time of 12 s with zero steady state error as Fig. 26 shows Area3 frequency deviations with 10% disturbance.

Fig. 26.
figure 26

Area3 frequency deviations with 10% disturbance

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Case II-(d) the power and delta deviation without controller it is unstable and with controllers the system is stable for small load disturbances in Area1 and the deviation is very small for fuzzy controller as depicted in Figs. 27 and 28 respectively.

Fig. 27.
figure 27

Tie line Power in area2 without, with PI and with Fuzzy Controllers

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Fig. 28.
figure 28

Area2 deviation in delta4, delta5 and delta6 without, with PI and with Fuzzy Controllers, without controller it is unstable and with controllers the system is stable for small load disturbances in Area2

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Case III: 30% disturbance results of three area system

Transient Stability Analysis

Case III-1 (a) Without controller error is −88% which is not acceptable and unstable, Case III-1 (b) with PI Controller the peak undershoot is −130% and settling time of 25 s and Case III-1 (c) with fuzzy controller the peak undershoot is −29% and settling time of 5 s with 15% steady state error as Fig. 29 depicts Area1 frequency deviations with 30% disturbance.

Fig. 29.
figure 29

Area1 frequency deviations with 30% disturbance

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Case III-2 (a) Without controller error is −148% which is not acceptable and unstable, Case III-3 (b) with PI Controller the peak undershoot is −252% and settling time of 23 s and Case III-3 (c) with fuzzy controller the peak undershoot is −15% and settling time of 5.7 s with a small acceptable steady state error as Figs. 30 and 31 illustrates Area2 and Area3 frequency deviations with 30% disturbance respectively.

Fig. 30.
figure 30

Area2 frequency deviations with 30% disturbance

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Fig. 31.
figure 31

Area3 frequency deviations with 30% disturbance

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Case III-4 the power and delta deviation without controller it is unstable and with controllers the system is stable for small load disturbances in Area1 and the deviation is very small for fuzzy controller as depicted in Figs. 32 and 33 respectively.

Fig. 32.
figure 32

Tie line Power in area3 without, with PI and with Fuzzy Controllers

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Fig. 33.
figure 33

Area3 deviation in delta4, delta5 and delta6 without, with PI and with Fuzzy Controllers

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7 Conclusions

The article entente the three area load frequency control and dynamic and transient stability analysis. It dispenses the simulation of three area systems with small, medium and large load disturbances, with three cases. Case I with 5% load disturbances, Case II with 10% and case III with 30% load disturbances. In each of the three cases, there are three sub cases (a), (b) and (c) sub case (a) without any controller, sub case (b) with PI controller and sub case (c) with Fuzzy Controllers. For all three cases.

The simulation is carried out for all three cases of three area power system with all three sub cases of (a), (b) and (c). In the first part of the case study, the simulation results of three area power system for all three cases have been presented.

The frequency deviation for the three area system for all three cases shows that the system is working well with less settling time and zero steady state error, when it concerned with PI controller, the response is showing that the steady state error is there but still it growing and is not completely accepted as far as performance is concerned and without any controller the system performance is not accepted and it is unstable.

In the second part, the simulation results of three area power system for last part of all three cases have been presented. The power and swing curves being illustrated that for small, medium and also for large disturbances the system is completely stable with fuzzy logic controller and where as it is not accepted with PI controller and completely unstable without controller.