**A Two-Layered Load and Frequency Controller of a Power System**

Mavungu Masiala, Mohsen Ghribi and Azeddine Kaddouri

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48404

## **1. Introduction**

502 Fuzzy Controllers – Recent Advances in Theory and Applications

*Fuzzy Systems*, Vol. 14, N° 3, pp.386–397, June 2006.

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Automatic generation control (AGC) or called load frequency control (LFC) has gained a lot of interests in the past 30 decade (Benjamin &. Chan, 1982; Pan & Liaw, 1989; Kothari et al., 1989; Y. Wang et al., 1994; Indulkar & Raj, 1995; Karnavas & Papadopoulos, 2002; Moon et al., 2002; Sherbiny et al., 2003). LFC insures a sufficient and reliable supply of power with good quality. To ensure the quality of the power supply, it is necessary to deal with the control of the generator loads depending on the frequency with a proper LFC design. Therefore, the design of the controller is faced with nonlinear effects due to the physical components of the system, such as governor dead zone and generation rate constraints (GRC) and its complexity and the inherent characteristics of changing loads and parameters. Most actuators used in practice contain static (dead zone) or dynamic (backlash) nonsmooth nonlinearities. These actuators are present in most mechanical and hydraulic systems such as servo valves. Their mathematical models are poorly known and limit the static and dynamic performance of feedback control system (Corradini & Orlando,2002). Conventional PI controller has been often used to achieve zero steady state frequency deviation. However, because of the load changing, the operating point of a power system may change very much during a daily cycle (Pan & Liaw, 1989). Therefore, a PI controller which is fixed and optimal when considering one operating point may no longer be suitable with various statuses. On the other hand, it is known that the classical LFC does not yield adequate control performance with consideration of the speed – governor non-smooth nonlinearities and GRC (Karnavas & Papadopoulos, 2002; Moon et al., 2002).

The problem of controlling systems with dead-zone nonlinearity has been addressed in the literature using various approaches some of which are dedicated to power systems. Reference (Tao & Kokotovic, 1994; X.-S. Wang et al., 2004) proposed adaptive schemes with and without dead zone inverse scheme, respectively, to track the error caused by the dead

© 2012 Kaddouri et al., licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

zone effect to zero. In the past decade, fuzzy logic controllers (FLC) have been developed successfully for analysis and control of nonlinear systems (Lee, 1990).

A Two-Layered Load and Frequency Controller of a Power System 505

(1)

*<sup>g</sup> B T* (4)

*p p F KT* (5)

∆PC = – Kx(t) (7)

y(t) = Cx(t) (2)

(3)

The investigated model consists of a tandem-compound single non reheat turbine. The state

() () () *<sup>L</sup> x t Ax t BU t F P*

1 00 0 11 0 1 01 1

001 0

 *C* = [1 0 0 0] (6) The time constant, Tg, in the governor model is quite small and often it is neglected, which means that the governor is assumed to act very fast compared to the change in speed or frequency. This leads to a second order dynamic power system model. But for accuracy and

Hence, the closed loop eigenvalues become insensitive to variations of the system parameters.

Consider the same isolated non reheat power system model reported in (Benjamin &. Chan, 1982; Pan & Liaw, 1989; Y. Wang et al., 1994) as shown in Figure 1, with the system

**Figure 1.** Block diagram of a isolated non reheat power system with supplementary control

*p pp*

*T KT*

*i*

In linear control system theory, it is required that a state feedback controller:

*K*

*A*

**2.1. Model 1: Single-area non reheat power system** 

0 00

000

*T*

*T*

*t t g g g*

*T T*

*RT T T*

space model can be expressed as following:

comparison purposes, Tg is considered.

parameters given in appendix A.

where:

However, it has been sown by (Kim et al., 1994) that usual "Fuzzy PD" controller suffers from poor transient performance and a large steady state error when applied to systems with dead zones. On the other hand, when dealing with complex systems, the single-loop controller may not achieve the control performances and a multilayered controller turns out to be very helpful. The main advantage of the multilevel control lies in the freedom of the design of each layer (Yeh & Li, 2003; Oh & Park, 1998). The layers are designed to target particular objectives, so that design is simpler and performance improved. Motivated by the success of FLC, (Koo, 2001; Rubai, 1991) proposed new adaptive fuzzy controllers with online gain – tuning algorithm.

However, lots of computations are needed to calculate the adaptive control law with and without fuzzy system. To simplify the controller design (Kim et al., 1994) proposed a Twolayered fuzzy logic controller in which a fuzzy pre-compensator and a "fuzzy PD" controller were introduced to control plants with dead zones. Stimulated by ((Kim et al., 1994; L. X. Wang, 1997) designed a 2 layered fuzzy LFC (FLC-FLC) with the dead zone and GRC effects. Reference (Rubai, 1991) proposed a 2 layers fuzzy controller for the transient stability enhancement of the electric power system.

Based on the alternative choices proposed by (L. X. Wang, 1997) and the previous works (Kim et al., 1994; Sherbiny et al., 2003; Rubai, 1991], in this paper we study the case of a twolayer control architecture (FLC-CC) where the pre-compensator layer is constructed from fuzzy systems as a control supervisor and the other layer from the conventional method. In addition, we demonstrate that the proposed scheme exhibits a good transient and steady state performance, and is robust to load variations and system nonlinearities.

This paper is organized as follows. In section 2 we briefly introduce the systems investigated. Section 3 describes the idea underlying the approach and the design procedure of the proposed controller. The simulation plots that illustrate the behaviour of our scheme, taking into account parameters variations, GRC and speed governor dead zone are provided in section 4. Finally, conclusions based on extensive simulation results, recommendations and further research are drawn in section V.

## **2. Plant model**

Power systems can be modeled by their power balance equations, linearized around the operating point. Since power systems are only exposed to small changes in load during their normal operation, a linear model can be used to design LFC. We consider the same single–area non reheat power system model as shown in Fig. 1. The investigated system consists of a speed–governor, a turbine that produces mechanical power, *Pg*, and the rotating mass (or power system). In steady state, *Pg* is balanced by the electrical power output, *Pe*, of the generator. Any imbalance between *Pg* and *Pe* produces accelerating power and thereby creates an incremental change in frequency, *∆f*. All parameters are given Appendix A.

The investigated model consists of a tandem-compound single non reheat turbine. The state space model can be expressed as following:

$$\stackrel{\bullet}{\max}(t) = A\boldsymbol{\omega}(t) + B\boldsymbol{U}(t) + F\boldsymbol{\Delta P}\_{L} \tag{1}$$

$$\mathbf{y}(\mathbf{t}) = \mathbf{C}\mathbf{x}(\mathbf{t})\tag{2}$$

where:

504 Fuzzy Controllers – Recent Advances in Theory and Applications

stability enhancement of the electric power system.

and further research are drawn in section V.

**2. Plant model** 

online gain – tuning algorithm.

zone effect to zero. In the past decade, fuzzy logic controllers (FLC) have been developed

However, it has been sown by (Kim et al., 1994) that usual "Fuzzy PD" controller suffers from poor transient performance and a large steady state error when applied to systems with dead zones. On the other hand, when dealing with complex systems, the single-loop controller may not achieve the control performances and a multilayered controller turns out to be very helpful. The main advantage of the multilevel control lies in the freedom of the design of each layer (Yeh & Li, 2003; Oh & Park, 1998). The layers are designed to target particular objectives, so that design is simpler and performance improved. Motivated by the success of FLC, (Koo, 2001; Rubai, 1991) proposed new adaptive fuzzy controllers with

However, lots of computations are needed to calculate the adaptive control law with and without fuzzy system. To simplify the controller design (Kim et al., 1994) proposed a Twolayered fuzzy logic controller in which a fuzzy pre-compensator and a "fuzzy PD" controller were introduced to control plants with dead zones. Stimulated by ((Kim et al., 1994; L. X. Wang, 1997) designed a 2 layered fuzzy LFC (FLC-FLC) with the dead zone and GRC effects. Reference (Rubai, 1991) proposed a 2 layers fuzzy controller for the transient

Based on the alternative choices proposed by (L. X. Wang, 1997) and the previous works (Kim et al., 1994; Sherbiny et al., 2003; Rubai, 1991], in this paper we study the case of a twolayer control architecture (FLC-CC) where the pre-compensator layer is constructed from fuzzy systems as a control supervisor and the other layer from the conventional method. In addition, we demonstrate that the proposed scheme exhibits a good transient and steady

This paper is organized as follows. In section 2 we briefly introduce the systems investigated. Section 3 describes the idea underlying the approach and the design procedure of the proposed controller. The simulation plots that illustrate the behaviour of our scheme, taking into account parameters variations, GRC and speed governor dead zone are provided in section 4. Finally, conclusions based on extensive simulation results, recommendations

Power systems can be modeled by their power balance equations, linearized around the operating point. Since power systems are only exposed to small changes in load during their normal operation, a linear model can be used to design LFC. We consider the same single–area non reheat power system model as shown in Fig. 1. The investigated system consists of a speed–governor, a turbine that produces mechanical power, *Pg*, and the rotating mass (or power system). In steady state, *Pg* is balanced by the electrical power output, *Pe*, of the generator. Any imbalance between *Pg* and *Pe* produces accelerating power and thereby creates an incremental change in frequency, *∆f*. All parameters are given Appendix A.

state performance, and is robust to load variations and system nonlinearities.

successfully for analysis and control of nonlinear systems (Lee, 1990).

$$A = \begin{bmatrix} -1/T\_p & K\_p/T\_p & 0 & 0 \\ 0 & -1/T\_t & 1/T\_t & 0 \\ -1/RT\_\mathcal{g} & 0 & -1/T\_\mathcal{g} & -1/T\_\mathcal{g} \\ K\_i & 0 & 0 & 0 \end{bmatrix} \tag{3}$$

$$B = \begin{bmatrix} 0 & 0 & 1/T\_{\mathcal{S}} & 0 \end{bmatrix}^T \tag{4}$$

$$F = \begin{bmatrix} -K\_p \int T\_p & 0 & 0 & 0 \end{bmatrix}^T \tag{5}$$

$$C = \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix} \tag{6}$$

The time constant, Tg, in the governor model is quite small and often it is neglected, which means that the governor is assumed to act very fast compared to the change in speed or frequency. This leads to a second order dynamic power system model. But for accuracy and comparison purposes, Tg is considered.

In linear control system theory, it is required that a state feedback controller:

$$
\Delta \mathbf{P} \mathbf{c} = -\mathbf{K} \mathbf{x}(\mathbf{t})\tag{7}
$$

Hence, the closed loop eigenvalues become insensitive to variations of the system parameters.

#### **2.1. Model 1: Single-area non reheat power system**

Consider the same isolated non reheat power system model reported in (Benjamin &. Chan, 1982; Pan & Liaw, 1989; Y. Wang et al., 1994) as shown in Figure 1, with the system parameters given in appendix A.

**Figure 1.** Block diagram of a isolated non reheat power system with supplementary control

Where u(t) denotes the existence of the proportional gain Kp, whereas its absence leads to an integral controller. The above described model has a tandem-compound single non reheat turbine and does not consider the speed-governor dead zone and GRC.

A Two-Layered Load and Frequency Controller of a Power System 507

u(k) = D(v(k)) (14)

yp(k) = P(s) F[e(k), ∆e(k)] (11)

yp(k) = Ks F[ess , 0] = ym – ess (12)

By assuming that CF is well tuned and that the load reference deviation ym = ∆Pc = 0, (12)

Ks F[ess , 0] = – ess (13)

Taking into account the feedback negative input sign into the controller (see Fig. 1), it can be verified from the description of Fuzzy PD controller (Indulkar & Raj, 1995; Karnavas & Papadopoulos, 2002). that the law F( . , 0) is an increasing odd function that can satisfy the following condition f(x) = -f(-x), with f(x) = F( . , 0). Therefore, it is clear that the solution to

with v(k) as input and u(k) as output. The operator D(v(k)) has been described in detail by (Corradini & Orlando,2002; Tao & Kokotovic, 1994; X.-S. Wang et al., 2004; Oh & Park, 1998; Koo, 2001).The parameters of D(v(k)) are specified by the width 2d of the dead zone and the slop m of the response outside the dead zone. In this case, equation (12) can be written as :

yp(k) = Ks D(v(k)) F[e(k), 0] = ym – e(k) (15)

Ks D(v(k)) F[ess , 0] - ym = – ess (16)

From (13) and (16) it can be seen that the steady state error ess in (16) is no longer zero. This

It has been demonstrated by (Yeh & Li, 2003) that the steady state error due to the dead zone in the actuator can be eliminated by adding some other constant η to the reference input ym. We deliberately avoided using explicit knowledge of the value D(v(k)) because its

(13) is ess = 0, i.e., the steady state error of the system output is zero, as expected.

The dead zone nonlinearity can be denoted as an operator is written as following:

The solution to the equation (15) results in the steady state error as follow:

is due to the presence of the dead zone in the speed governor actuator.

parameters are poorly known or uncertain. Therefore, (16) becomes:

Let ym(k) = ym. For steady state ∆e(k) = 0 since the system is supposed to have reached the stabilizing time. Therefore, CF becomes a function of e(k) alone and (11) can be written as

**3.1. Case 1: No actuator with dead zone** 

The plant output in this case can be written as

where Ks is the system static gain and is given by Ks = lims→0P(s)

**3.2. Case 2: Speed governor dead zone is present** 

following:

becomes:

## **2.2. Model 2: two-areas reheat hydrothermal power system**

The linearized mathematical model – 2 (see Fig.2), comprises an interconnection of two areas: single stages reheat thermal system (area 1) and a hydro system (area 2). The system parameters are given in appendix B. Figure 2 shows the small perturbation transfer function model of the hydro thermal system (Sherbiny et al., 2003). The speed governor dead zone and the GRC effects are also included in the model.

**Figure 2.** Bloc diagram of two-area reheat hydrothermal system with nonlinearities

## **3. Design of two layered controller for the system investigated**

Considering the system shown in Figures 1, let P(s) represent the plant and D the speed governor actuator with dead zone (not present in Fig. 1). Recall that the supplementary PI control law can be written as following:

$$\text{C.c.}\{\text{e(k)}, \text{ je(k)}\} = \text{Kpe(k)} + \text{K} \downarrow \text{e(k)}\tag{8}$$

And in the case of "Fuzzy PD" controller, neglecting the scale factors, we get

$$\operatorname{Cr}[\operatorname{e}(\mathbf{k}), \Delta \mathbf{e}(\mathbf{k})] = \operatorname{F}[\operatorname{e}(\mathbf{k}), \Delta \mathbf{e}(\mathbf{k})] \tag{9}$$

CC is a linear function of the error e(k) between the system output yp(frequency deviation ∆f) and the reference input Ym (load reference ∆Pc ) and the integral of the error whereas CF is a function of the error and the change of error. From the above, we get

$$\mathbf{e(k) = y\_m(k) - y\_{\mathbb{P}}(k)} \text{ or } \quad \mathbf{y\_{\mathbb{P}}(k) = y\_m(k) - e(k)} \tag{10}$$

Let us assume that the supplementary control of the system is ensured by a "fuzzy PD" controller of the same type as (Indulkar & Raj, 1995; Karnavas & Papadopoulos, 2002).

#### **3.1. Case 1: No actuator with dead zone**

506 Fuzzy Controllers – Recent Advances in Theory and Applications

and the GRC effects are also included in the model.

control law can be written as following:

Where u(t) denotes the existence of the proportional gain Kp, whereas its absence leads to an integral controller. The above described model has a tandem-compound single non reheat

The linearized mathematical model – 2 (see Fig.2), comprises an interconnection of two areas: single stages reheat thermal system (area 1) and a hydro system (area 2). The system parameters are given in appendix B. Figure 2 shows the small perturbation transfer function model of the hydro thermal system (Sherbiny et al., 2003). The speed governor dead zone

turbine and does not consider the speed-governor dead zone and GRC.

**Figure 2.** Bloc diagram of two-area reheat hydrothermal system with nonlinearities

**3. Design of two layered controller for the system investigated** 

And in the case of "Fuzzy PD" controller, neglecting the scale factors, we get

function of the error and the change of error. From the above, we get

Considering the system shown in Figures 1, let P(s) represent the plant and D the speed governor actuator with dead zone (not present in Fig. 1). Recall that the supplementary PI

CC[e(k), ∫e(k)] = Kpe(k) + Ki ∫e(k) (8)

CF[e(k), ∆e(k)] = F[e(k), ∆e(k)] (9)

CC is a linear function of the error e(k) between the system output yp(frequency deviation ∆f) and the reference input Ym (load reference ∆Pc ) and the integral of the error whereas CF is a

Let us assume that the supplementary control of the system is ensured by a "fuzzy PD" controller of the same type as (Indulkar & Raj, 1995; Karnavas & Papadopoulos, 2002).

e(k) = ym(k) – yp(k) or yp(k) = ym(k) – e(k) (10)

**2.2. Model 2: two-areas reheat hydrothermal power system** 

The plant output in this case can be written as

$$\mathbf{y}\_{\mathbb{P}}(\mathbf{k}) = \mathbf{P}(\mathbf{s}) \cdot \mathbf{F}[\mathbf{e}(\mathbf{k}), \Delta \mathbf{e}(\mathbf{k})] \tag{11}$$

Let ym(k) = ym. For steady state ∆e(k) = 0 since the system is supposed to have reached the stabilizing time. Therefore, CF becomes a function of e(k) alone and (11) can be written as following:

$$\mathbf{y}\_{\mathbb{P}}(\mathbf{k}) = \mathbf{K}\_{\mathbf{s}} \, \mathrm{F}[\mathbf{e}\_{\mathbf{s}\mathbf{s}} \, , \, \mathbf{0}] = \mathbf{y}\_{\mathbb{P}} - \mathbf{e}\_{\mathbf{s}\mathbf{s}} \tag{12}$$

where Ks is the system static gain and is given by Ks = lims→0P(s)

By assuming that CF is well tuned and that the load reference deviation ym = ∆Pc = 0, (12) becomes:

$$\mathbf{K} \circ \mathbf{F}[\mathbf{e}\_{\text{se}}, \mathbf{0}] = -\mathbf{e}\_{\text{se}} \tag{13}$$

Taking into account the feedback negative input sign into the controller (see Fig. 1), it can be verified from the description of Fuzzy PD controller (Indulkar & Raj, 1995; Karnavas & Papadopoulos, 2002). that the law F( . , 0) is an increasing odd function that can satisfy the following condition f(x) = -f(-x), with f(x) = F( . , 0). Therefore, it is clear that the solution to (13) is ess = 0, i.e., the steady state error of the system output is zero, as expected.

#### **3.2. Case 2: Speed governor dead zone is present**

The dead zone nonlinearity can be denoted as an operator is written as following:

$$\mathbf{u}(\mathbf{k}) = \mathbf{D}(\mathbf{v}(\mathbf{k})) \tag{14}$$

with v(k) as input and u(k) as output. The operator D(v(k)) has been described in detail by (Corradini & Orlando,2002; Tao & Kokotovic, 1994; X.-S. Wang et al., 2004; Oh & Park, 1998; Koo, 2001).The parameters of D(v(k)) are specified by the width 2d of the dead zone and the slop m of the response outside the dead zone. In this case, equation (12) can be written as :

$$\mathbf{y}\_{\mathbb{P}}(\mathbf{k}) = \mathbf{K} \circ \mathbf{D}(\mathbf{v}(\mathbf{k})) \text{ [F[e(k), 0] = y\_m - e(k)]} \tag{15}$$

The solution to the equation (15) results in the steady state error as follow:

$$\mathbf{K} \circ \mathbf{D}(\mathbf{v}(\mathbf{k})) \text{ F[e\$e\$\kappa\$, 0] - y\_m = -e\_{\kappa}} \tag{16}$$

From (13) and (16) it can be seen that the steady state error ess in (16) is no longer zero. This is due to the presence of the dead zone in the speed governor actuator.

It has been demonstrated by (Yeh & Li, 2003) that the steady state error due to the dead zone in the actuator can be eliminated by adding some other constant η to the reference input ym. We deliberately avoided using explicit knowledge of the value D(v(k)) because its parameters are poorly known or uncertain. Therefore, (16) becomes:

$$\mathbf{K} \circ \mathbf{D}(\mathbf{v}(\mathbf{k})) \text{ F}[\mathbf{e} + \mathbf{v}\_{\parallel} \text{ }, 0] \text{ } \cdot \text{ y}\_{\text{m}} = -\mathbf{e} \tag{17}$$

A Two-Layered Load and Frequency Controller of a Power System 509

(19)

*m*

*i i i m i i*

*w y*

*y*

**e(k)** NB NM ZE PM PB

NB NB NB NB NM PM NM NB NB NM ZE PM ZE NB NM ZE PM PB PM NM ZE PM PB PB PB NM PM PB PB PB

The performance of the FLC is affected by scaling factors of the inputs/output variables, MFs and the control rules. The selection of the optimum values of these factors is necessary in order to achieve satisfactory response [6]. But for the control system shown in Fig. 3, we can design the FLC without considering stability and use the stabilizer layer to deal with

The stabilizing layer consists of a conventional controller which is described in (8) and its design procedure is detailed in (Karnavas & Papadopoulos, 2002). The input to the present

The power systems under investigation are simulated and subjected to different load disturbances in order to validate the effectiveness of the proposed scheme. The nonlinear

Where nη is the output control gain, *wi* is the grade of the *i*th output MF, *yi* is the output label for the value contributed by the *i*th MF, and *m* is the number of contributions from the rules. The fuzzy output variable is determined by same MFs shown in Fig. 4 and labelled as

CF[e(k), ∆e(k)] = n<sup>η</sup>

**Figure 4.** Membership functions of control input/output variables

**Table 1.** Fuzzy logic rules for pre-compensator C1

The associated fuzzy matrices used in this work are given in table 1.

following L(η) = {NB, NM, ZE, PM, PB}

**∆e(k)**

stability related problems.

**6. Simulation results** 

**5. Second layer: Stabilizer** 

layer is the y'm as shown in Fig. 3.

Since PI controller is still the most used controller in power system (Pan & Liaw, 1989; Sherbiny et al., 2003) in our approach we use fuzzy logic rules to determine the appropriate value of η to be added to ym. In this case, FLC plays the rule of a pre-compensator (supervisor) ensuring that the appropriate value of η is added to e(k) in order to eliminate the steady state error due to the dead zone. The conventional controller, called the stabilizer, present in the system, ensures the stabilization of the system.

The price to pay for changing the existing conventional controller into a FLC, which is proposed by (Sherbiny et al., 2003), would be in the computation. FLC is driven by a set of control rules rather than by two constant proportional and integral gains. As we shall see, the proposed scheme exhibits good transient and steady state behaviour. The proposed control scheme is depicted in Fig. 3 where C1 is a FLC and C2 a conventional controller. The feed-forward gain K1 is normally set to the reciprocal of the Ks and constitutes an additional design parameter.

**Figure 3.** Proposed control structure

### **4. Design of the supervisor controller**

As previously discussed, the first layer of the proposed control structure consists of the fuzzy logic based pre-compensator. The FLC law is based on standard fuzzy logic rules. It is well known that FLC consists of 3 stages, namely fuzzification, control rules inference engine and defuzzification. The reader is refereed to (Lee, 1990; L. X. Wang, 1997) for details on FLC. For LFC the process operator is assumed to respond to error e and change of error ∆e (Indulkar & Raj, 1995); defined in (9). Considering the scale factors, (9) becomes as

CF[e(k), ∆e(k)] = F[ne e(k), n∆<sup>e</sup> ∆e(k)] (18)

where ne and n∆e are the error and change of error scale factors respectively.

A label set corresponding to the linguistic variables control input e(k) and ∆e(k) with a sampling time of 0.1 sec is as follows: L(e, ∆e) = {NB, NM, ZE,PM, PB} where, NB - Negative Big, NM - Negative Medium, ZE – Zero, PM - Positive Medium and PB - Positive Big. The membership functions (MFs) for the control input variables are shown in Fig. 4. The universe of discourse of each control variable is normalised from -1 to 1. The proposed control structure uses the center of gravity defuzzification method to determine the output control as following:

A Two-Layered Load and Frequency Controller of a Power System 509

$$\text{Cr}[\text{e(k), }\Delta\text{e(k)}] = \text{n}\_{\mathbb{T}} \frac{\sum\_{i}^{m} w\_{i} y\_{i}}{\sum\_{i}^{m} y\_{i}} \tag{19}$$

Where nη is the output control gain, *wi* is the grade of the *i*th output MF, *yi* is the output label for the value contributed by the *i*th MF, and *m* is the number of contributions from the rules. The fuzzy output variable is determined by same MFs shown in Fig. 4 and labelled as following L(η) = {NB, NM, ZE, PM, PB}

**Figure 4.** Membership functions of control input/output variables


The associated fuzzy matrices used in this work are given in table 1.

**Table 1.** Fuzzy logic rules for pre-compensator C1

The performance of the FLC is affected by scaling factors of the inputs/output variables, MFs and the control rules. The selection of the optimum values of these factors is necessary in order to achieve satisfactory response [6]. But for the control system shown in Fig. 3, we can design the FLC without considering stability and use the stabilizer layer to deal with stability related problems.

## **5. Second layer: Stabilizer**

508 Fuzzy Controllers – Recent Advances in Theory and Applications

design parameter.

**Figure 3.** Proposed control structure

control as following:

**4. Design of the supervisor controller** 

present in the system, ensures the stabilization of the system.

Ks D(v(k)) F[e + η , 0] - ym = – e (17)

Since PI controller is still the most used controller in power system (Pan & Liaw, 1989; Sherbiny et al., 2003) in our approach we use fuzzy logic rules to determine the appropriate value of η to be added to ym. In this case, FLC plays the rule of a pre-compensator (supervisor) ensuring that the appropriate value of η is added to e(k) in order to eliminate the steady state error due to the dead zone. The conventional controller, called the stabilizer,

The price to pay for changing the existing conventional controller into a FLC, which is proposed by (Sherbiny et al., 2003), would be in the computation. FLC is driven by a set of control rules rather than by two constant proportional and integral gains. As we shall see, the proposed scheme exhibits good transient and steady state behaviour. The proposed control scheme is depicted in Fig. 3 where C1 is a FLC and C2 a conventional controller. The feed-forward gain K1 is normally set to the reciprocal of the Ks and constitutes an additional

As previously discussed, the first layer of the proposed control structure consists of the fuzzy logic based pre-compensator. The FLC law is based on standard fuzzy logic rules. It is well known that FLC consists of 3 stages, namely fuzzification, control rules inference engine and defuzzification. The reader is refereed to (Lee, 1990; L. X. Wang, 1997) for details on FLC. For LFC the process operator is assumed to respond to error e and change of error

CF[e(k), ∆e(k)] = F[ne e(k), n∆<sup>e</sup> ∆e(k)] (18)

A label set corresponding to the linguistic variables control input e(k) and ∆e(k) with a sampling time of 0.1 sec is as follows: L(e, ∆e) = {NB, NM, ZE,PM, PB} where, NB - Negative Big, NM - Negative Medium, ZE – Zero, PM - Positive Medium and PB - Positive Big. The membership functions (MFs) for the control input variables are shown in Fig. 4. The universe of discourse of each control variable is normalised from -1 to 1. The proposed control structure uses the center of gravity defuzzification method to determine the output

∆e (Indulkar & Raj, 1995); defined in (9). Considering the scale factors, (9) becomes as

where ne and n∆e are the error and change of error scale factors respectively.

The stabilizing layer consists of a conventional controller which is described in (8) and its design procedure is detailed in (Karnavas & Papadopoulos, 2002). The input to the present layer is the y'm as shown in Fig. 3.

## **6. Simulation results**

The power systems under investigation are simulated and subjected to different load disturbances in order to validate the effectiveness of the proposed scheme. The nonlinear effects, such as GRC and speed governor dead-zone, are also included in the simulations. The proposed controller (FLC-CC) will be evaluated qualitatively and quantitatively with

A Two-Layered Load and Frequency Controller of a Power System 511

FLC-CC FLC CC

FLC-CC FLC CC

When GRC is applied to the system, its dynamic responses experience longer transient setting time ts and larger overshoots OS compare to cases without the GRC. GRC of 3% p.u. MW/min and 10% p.u. MW/min are usually applied to reheat and non reheat turbines, respectively. In addition to GRC, the dead zone effect is also added to the system investigated. A dead zone width of 0.05 p.u. is considered. GRC and dead zone are taken into account by adding limiters to the turbines and an actuator to the speed governor input, respectively. Fig. 7 plots the responses of the system under nonlinear effects and a step load

<sup>0</sup> 0.5 <sup>1</sup> 1.5 <sup>2</sup> 2.5 <sup>3</sup> 3.5 <sup>4</sup> 4.5 <sup>5</sup> -0.25

Time (sec)

0 1 2 3 4 5 6 7 8 9 10

Time (sec)

perturbation ∆Pd = 0.05 p.u.

**Figure 6.** Frequency deviation at parameter changes of +30%





df (p.u.)

0

0.05

0.1

**Figure 7.** Frequency deviation due to ∆Pd = 0.05 p.u with nonlinear effects





df (p.u.)


0

0.02

A single layer "PD FLC" (FLC) proposed in the first part of the work of (Karnavas & Papadopoulos, 2002) and a single layer conventional control (CC). The reader is referred to (Sherbiny et al., 2003) in order to compare our controller responses with the two-layered fuzzy controller proposed by (Sherbiny et al., 2003). The parameters of our controller for the two investigated models are given in table 2.


**Table 2.** Proposed controller parameters

## **6.1. Model 1 (Appendix A)**

A step load perturbation of 10% (∆Pd = 0.1 p.u.) of the nominal loading is considered. Fig 5 shows the simulation results of the frequency deviation ∆f response to the step load change. The responses, obtained by the PD Fuzzy and the conventional controller (Karnavas & Papadopoulos, 2002) are also shown for comparison purpose.

Assume that the parameters R, Tg, Tt, Tp, Kp are subjected to a simultaneous changes of +30% from their nominal values. The frequency deviation response of the system is plotted in Fig. 6.

**Figure 5.** Frequency deviation for a load change of 0.1 p.u.

When GRC is applied to the system, its dynamic responses experience longer transient setting time ts and larger overshoots OS compare to cases without the GRC. GRC of 3% p.u. MW/min and 10% p.u. MW/min are usually applied to reheat and non reheat turbines, respectively. In addition to GRC, the dead zone effect is also added to the system investigated. A dead zone width of 0.05 p.u. is considered. GRC and dead zone are taken into account by adding limiters to the turbines and an actuator to the speed governor input, respectively. Fig. 7 plots the responses of the system under nonlinear effects and a step load perturbation ∆Pd = 0.05 p.u.

**Figure 6.** Frequency deviation at parameter changes of +30%

510 Fuzzy Controllers – Recent Advances in Theory and Applications

two investigated models are given in table 2.

Papadopoulos, 2002) are also shown for comparison purpose.

**Figure 5.** Frequency deviation for a load change of 0.1 p.u.





df (p.u.)

0

0.05

**Table 2.** Proposed controller parameters

**6.1. Model 1 (Appendix A)** 

in Fig. 6.

effects, such as GRC and speed governor dead-zone, are also included in the simulations. The proposed controller (FLC-CC) will be evaluated qualitatively and quantitatively with

A single layer "PD FLC" (FLC) proposed in the first part of the work of (Karnavas & Papadopoulos, 2002) and a single layer conventional control (CC). The reader is referred to (Sherbiny et al., 2003) in order to compare our controller responses with the two-layered fuzzy controller proposed by (Sherbiny et al., 2003). The parameters of our controller for the

Model 2 area 1 10.0 20.0 0.15 4.00 6.00

A step load perturbation of 10% (∆Pd = 0.1 p.u.) of the nominal loading is considered. Fig 5 shows the simulation results of the frequency deviation ∆f response to the step load change. The responses, obtained by the PD Fuzzy and the conventional controller (Karnavas &

Assume that the parameters R, Tg, Tt, Tp, Kp are subjected to a simultaneous changes of +30% from their nominal values. The frequency deviation response of the system is plotted

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

FLC-CC CC FLC

Time (sec)

 ne n∆e nη Kp Ki Model 1 0.25 10.0 0.14 0.425 0.212

area 2 10.0 10.0 0.05 30.0 20.0

**Figure 7.** Frequency deviation due to ∆Pd = 0.05 p.u with nonlinear effects

From Figs. 5-7 it can be observed that the proposed controller acts as fast as the FLC-FLC controller with less oscillatory, less undershoot and setting time. In addition the proposed scheme is also robust to load and parameters changes with and without nonlinearities. Fig. 7 shows the system steady state error for the PD FLC as previously predicted.

A Two-Layered Load and Frequency Controller of a Power System 513

FLC-CC CC

FLC-CC CC

**Figure 10.** Tie line power deviation response due to ∆Pd2




df1, p.u.

0.02




dPtie, p.u. MW



0

0.5 x 10-3

reader is referred to (Sherbiny et al., 2003) for comparison.

It can be observed from the results obtained in Figs 8-10 that the proposed controller exhibits less oscillations and setting time compare to the conventional controller. Our controller responses are faster with smaller overshoot than the FLC-FLC controller. The

0 10 20 30 40 50 60 70 80

Time (sec)

Now assume that a dead zone width of 0.5 p.u. and GRC effects are considered in both area 1 and 2 simultaneously. Figs 11-13 plot the responses of the system under nonlinear effects.

**Figure 11.** Frequency deviation of area 1 due to ∆Pd1 = 0.05 p.u. with nonlinear effects

0 5 10 15 20 25 30 35 40

Time (sec)

#### **6.2. Model 2: Appendix (B)**

Fig 8-10 show the simulation results of the frequency deviations and the tie line power responses of the two area power system due to a step load perturbation ∆Pd = 0.05 p.u. without non-linear effects. The responses obtained by the conventional controller (CC) are also shown for comparison purpose.

**Figure 8.** Frequency deviation of area 1 due to ∆Pd1 = 0.05 p.u.

**Figure 9.** Frequency deviation of area 2 due to ∆Pd2 = 0.05 p.u

**Figure 10.** Tie line power deviation response due to ∆Pd2

**6.2. Model 2: Appendix (B)** 

also shown for comparison purpose.

0.02





df2, p.u.



0

0.01



df1 (p.u.)


0

**Figure 8.** Frequency deviation of area 1 due to ∆Pd1 = 0.05 p.u.

**Figure 9.** Frequency deviation of area 2 due to ∆Pd2 = 0.05 p.u

From Figs. 5-7 it can be observed that the proposed controller acts as fast as the FLC-FLC controller with less oscillatory, less undershoot and setting time. In addition the proposed scheme is also robust to load and parameters changes with and without nonlinearities. Fig. 7

Fig 8-10 show the simulation results of the frequency deviations and the tie line power responses of the two area power system due to a step load perturbation ∆Pd = 0.05 p.u. without non-linear effects. The responses obtained by the conventional controller (CC) are

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> -0.1

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>25</sup> <sup>30</sup> <sup>35</sup> <sup>40</sup> -0.06

Time (sec)

Time (sec)

FLC-CC CC

FLC-CC CC

shows the system steady state error for the PD FLC as previously predicted.

It can be observed from the results obtained in Figs 8-10 that the proposed controller exhibits less oscillations and setting time compare to the conventional controller. Our controller responses are faster with smaller overshoot than the FLC-FLC controller. The reader is referred to (Sherbiny et al., 2003) for comparison.

Now assume that a dead zone width of 0.5 p.u. and GRC effects are considered in both area 1 and 2 simultaneously. Figs 11-13 plot the responses of the system under nonlinear effects.

**Figure 11.** Frequency deviation of area 1 due to ∆Pd1 = 0.05 p.u. with nonlinear effects

A Two-Layered Load and Frequency Controller of a Power System 515

The performance Evaluation of The proposed controller (FLC-CC) is given by table 3.

**Table 3.** Performance Evaluation Of The Proposed Controller (FLC-CC)

the optimum tuning method for the conventional controller parameters.

*Greater Research Group, University of Moncton, New-Brunswick, Canada* 

∆Ptie (∆f1)

∆f1

∆f2

**7. Conclusion** 

**Author details** 

**Appendix A** 

Corresponding Author

 \*

Mavungu Masiala, Mohsen Ghribi\*

 ts [sec] OS [p.u.] ·10־³ US [p.u.] ·10־³ w/NE NE w/NE NE w/NE NE

In this paper a two layered controller with a fuzzy pre-compensator is used to damp the power system frequencies and tie line power error oscillation and track their errors to zero. The price to pay for changing a conventional controller into a FLC in order to obtain a two layered controller, where both layers are FLC, would be in the computation. FLC is driven by a set of control rules rather than by two constant proportional and/or integral gains. In our approach, simple tuning of the conventional controller parameters enables the easy and cheap implementation of the proposed controller. Extensive simulations for a single area and an interconnected systems with no reheat, reheat and hydro turbines, taking into account a number of practical aspects such as the loads and parameters disturbances and the nonlinear effects, have verified the validity of our scheme over the conventional controller. Therefore, the proposed controller should be preferred. Further research is based on finding

and Azeddine Kaddouri

Nominal Parameters Of A Typical Single-Area Nonreheat Power System (Model – 1):

FLC-CC 10 17 0 0 1.2 3 FLC-FLC 25 25 2.5 2.5 3.5 3.7

FLC-CC 5 5 0 0.1 45 52 FLC-FLC 30 32 4.0 5 17 18

FLC-CC 12 10 0 0.1 45 45 FLC-FLC 30 30 4 4 20 20

**Figure 12.** Frequency deviation of area 2 due to ∆Pd2 = 0.05 p.u. with nonlinear effects

Figures 11-13 demonstrate the robustness of the proposed controller under a large dead zone width and GRC. In addition, it has been possible to reduce the steady state error in the tie line power flow deviations (Fig 10, 13) using the proposed controller while it has been difficult with a single layered FLC proposed by (Indulkar & Raj, 1995).

**Figure 13.** Tie line power deviation response due to ∆Pd2 = 0.05 p.u.


The performance Evaluation of The proposed controller (FLC-CC) is given by table 3.

**Table 3.** Performance Evaluation Of The Proposed Controller (FLC-CC)

## **7. Conclusion**

514 Fuzzy Controllers – Recent Advances in Theory and Applications






df2, p.u.



0

0.01

**Figure 12.** Frequency deviation of area 2 due to ∆Pd2 = 0.05 p.u. with nonlinear effects

difficult with a single layered FLC proposed by (Indulkar & Raj, 1995).

**Figure 13.** Tie line power deviation response due to ∆Pd2 = 0.05 p.u.


dPtie, p.u. MW

Figures 11-13 demonstrate the robustness of the proposed controller under a large dead zone width and GRC. In addition, it has been possible to reduce the steady state error in the tie line power flow deviations (Fig 10, 13) using the proposed controller while it has been

0 5 10 15 20 25 30 35 40

Time (sec)

0 5 10 15 20 25 30 35 40

FLC-CC CC

FLC-CC CC

Time (sec)

In this paper a two layered controller with a fuzzy pre-compensator is used to damp the power system frequencies and tie line power error oscillation and track their errors to zero. The price to pay for changing a conventional controller into a FLC in order to obtain a two layered controller, where both layers are FLC, would be in the computation. FLC is driven by a set of control rules rather than by two constant proportional and/or integral gains. In our approach, simple tuning of the conventional controller parameters enables the easy and cheap implementation of the proposed controller. Extensive simulations for a single area and an interconnected systems with no reheat, reheat and hydro turbines, taking into account a number of practical aspects such as the loads and parameters disturbances and the nonlinear effects, have verified the validity of our scheme over the conventional controller. Therefore, the proposed controller should be preferred. Further research is based on finding the optimum tuning method for the conventional controller parameters.

## **Author details**

Mavungu Masiala, Mohsen Ghribi\* and Azeddine Kaddouri *Greater Research Group, University of Moncton, New-Brunswick, Canada* 

## **Appendix A**

Nominal Parameters Of A Typical Single-Area Nonreheat Power System (Model – 1):


\* Corresponding Author

## **Appendix B**

Nominal Parameters Of ATwo-Area Reheat Hydrothermal Power System (Model – 2) :

A Two-Layered Load and Frequency Controller of a Power System 517

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Mamoh, J. A. and K. Tomsovic, K. (1995). *Overview and literature survey of fuzzy set theory in power systems*, IEEE Trans. Power Systems, (August 1995), Vol. 10, no. 3, pp. 1676-1690 Das, D. ; Kothari, M.L.; Kothari, D.P. and Nanda, J., (1991). *Variable structure control strategy to automatic generation control of interconnected reheat thermal system,* IEEE Proceedings on

Control Theory and Applications, (November 1991), Vol. 138, no. 6, pp. 579-585 Djukanovic, M. (1995). *Conceptual development of optimal load frequency control using artificial neural networks and fuzzy set theory*, International Journal of Engineering Intelligent Systems for Electrical Engineering & Communication, vol. 3, no. 2. pp. 95-108, 1995. Ha, Q. P. and Negnevitsky, M., (1997). *Fuzzy tuning in electric power system generation control*, Proceedings of the 4th Conference on APSCOM, (November 1997), vol. 2, no. 450,

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Yesil, E.; Guzelkaya, M. and Ekisin, I. (2004). *Self-tuning fuzzy PID type load and frequency* 

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## **8. References**


Corradini, M. L. & Orlando, G. (2002). *Robust stabilization of nonlinear uncertain plants with backlash or dead–zone in the actuator*, IEEE Trans. Control Systems Technology, vol. 10, No. 1, pp. 158 – 166

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Nominal Parameters Of ATwo-Area Reheat Hydrothermal Power System (Model – 2) :

Benjamin, N.N & Chan, W.C. (1982). *Variable structure control of electric power generation*, IEEE

Pan C. T., & Liaw, C. M. (1989). *An Adaptive controller for power system load-frequency control*,

Kothari, M. L.; Nanda, J.; Kothari, D. P. & Das, D. (1989). *Discrete-mode automatic generation control of a two-area reheat thermal system with new area control error*, IEEE Trans. Power

Wang, Y.; Zhou, Y. R. & Wen C. (1994). *New robust adaptive load-frequency control with system parametric uncertainties*, IEEE Trans. Generation, Transmission and Distribution, (May

Indulkar, C. S. & Raj B. (1995). *Application of fuzzy controller to automatic generation control*,

Karnavas, Y.L. & Papadopoulos, D.P. (2002). *AGC for autonomous power system using combined* 

Moon, Y-H.; Ryu, H-S.; Lee, J-G.; Song, K-B. & Shin, M-C. (2002). *Extended integral control for load frequency control with the consideration of generation-rate constraints*, Electrical Power

Sherbiny, M.K.; El-Saady, G. & Yousef, A.M. (2003). *Efficient fuzzy logic load-frequency* 

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**Appendix B** 

**8. References** 

	- Chown, G.A. and Hartman, R.C. (1998). *Design and experiment with a fuzzy controller for automatic generation control (AGC)*, *IEEE Trans. Power Systems,* (Aug. 1998), Vol. 13, pp. 965-970

**Chapter 22** 

© 2012 Tsengenes and Adamidis, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is

© 2012 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

properly cited.

**Performance Evaluation of PI** 

Georgios A. Tsengenes and Georgios A. Adamidis

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48394

required by the non-linear load.

**1. Introduction** 

**and Fuzzy Controlled Power Electronic** 

**Inverters for Power Quality Improvement** 

In recent years, the increasing use of power electronics in the commercial and industry processes results in harmonics injection and lower power factor to the electric power system [Emanuel A E. (2004)]. Conventionally, in order to overcome these problems, passive R-L-C filters have been used. The use of this kind of filters has several disadvantages. Recently, due to the evolution in modern power electronics, new device called "shunt active power filter (SAPF)" was investigated and recognized as a viable alternative to the passive filters. The principle operation of the SAPF is the generation of the appropriate current harmonics

For the reference currents generation, one of the best known and effective technique is the 'instantaneous reactive power theory' or '*p-q* theory' [Czarnecki L S. (2006)]. In the literature, various modifications of *p-q* theory for the reference currents generation have been proposed [Salmeron P, Herrera R S, Vazquez J R. (2007)], [Kilic T, Milun S, et al. (2007)]. It is a common phenomenon in an electric power system, the grid voltages to be non-ideal [Segui-Chilet S S, Gimeno-Sales F J, et al. (2007)]. In such condition, the *p-q* theory is ineffective. To improve the efficiency of the *p-q* theory various reference currents generation techniques [Kale M, Ozdemir E. (2005)], [Tsengenes G, Adamidis G. (2011)] have been proposed. Except from the reference currents generation method, the current control method plays an important role to the overall system's performance. Plenty of methods have been used in the current control loop [Buso S, Malesani L, et al. (1998)] (e.g. ramp comparison, space vector modulation), one of which is the hysteresis current control. Hysteresis current controller compared to other current control methods has a lot of

advantages such as robustness and simplicity [Tsengenes G, Adamidis G. (2010)].

and reproduction in any medium, provided the original work is properly cited.


**Chapter 22** 
