**Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems**

Xiaorong Xie

26 Will-be-set-by-IN-TECH

[21] R. J. Duffin, Bittner, L. [1967]. Geometric programming - theory and application, *ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und*

[24] Reyes-Sierra, M. & Coello, C. A. C. [2006]. Multi-objective particle swarm optimizers: A survey of the state-of-the-art, *International Journal of Computational Intelligence Research*

[25] Sáenz Noval, J. J., Roa Fuentes, E. F., Ayala Pabón, A. & Van Noije, W. [2010]. A methodology to improve yield in analog circuits by using geometric programming, *Proceedings of the 23rd symposium on Integrated circuits and system design*, SBCCI '10, ACM,

[26] Tiwary, S. K., Tiwary, P. K. & Rutenbar, R. A. [2006]. Generation of yield-aware pareto surfaces for hierarchical circuit design space exploration, *Design Automation Conference*,

[27] Weber, T. O. & Noije, W. A. M. V. [2011]. Analog design synthesis method using simulated annealing and particle swarm optimization, *Proceedings of the 24th symposium*

[28] Wilson, R. B. [1963]. *A Simplicial Algorithm for Concave Programming*, PhD thesis,

[29] Yu, G. & Li, P. [2011]. Hierarchical analog/mixed-signal circuit optimization under process variations and tuning, *IEEE Transactions on Computer-Aided Design of Integrated*

*on Integrated circuits and systems design*, SBCCI '11, ACM, pp. 85–90.

[22] Rao, S. S. [1996]. *Engineering Optimization: Theory and Practice*, New Age Publishers. [23] Razavi, B. [2001]. *Design of Analog CMOS Integrated Circuits*, 1 edn, McGraw-Hill, Inc.,

*Mechanik* 47(8): 561–561.

New York, NY, USA.

New York, NY, USA, pp. 140–145.

*Circuits and Systems* 30: 313.

2: 287–308.

pp. 31–36.

Cambridge.

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/50371

## **1. Introduction**

Series compensations (SCs) have been widely used in electrical power systems to enhance transmission capability through partial compensation of line reactors. However, they will trigger oscillatory modes inherent in the mass-spring system of turbine-generators, resulting in subsynchronous resonance (SSR), which, if not properly handled, could cause shaft failure. The SSR phenomenon was first discussed in 1937 [1] and until 1971, shaft torsional oscillations were ignored. Two shaft failures at the Mohave power plant in Southern Nevada, USA [2] led to the understanding and development of the theory of interaction between series-capacitor compensated lines and the torsional modes of steam turbinegenerators. After the second shaft failure at Mohave, the utility industry devoted considerable effort to the analysis and suppression of SSR. Much has been written on the subject as evidenced by the bibliography [3] and three supplements [4, 5, 6]. Generally, when preliminary analysis shows the system to be at risk of unstable SSR, control measures must be applied [7]. In this chapter, two types of controllers, namely, supplementary excitation damping control (SEDC) and static var compensator based subsynchronous damping control (SVC-SSDC), are used for the mitigation of SSR problem. The emphasis of our work is to achieve an optimal design of the controller parameters with the proposed genetic algorithm and simulated annealing (GA-SA) algorithm.

The SSR issue is generally addressed under three different types [8]: induction generator effect (IGE), torsional interaction (TI), and torque amplification (TA). In all cases, SSR is due to the interaction of a series capacitor compensated system with a turbine-generator.

© 2012 Xie, 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.

However, this chapter will focus on the TI type SSR, because it is the main problem we encountered in practice.

TI-SSR is the interplay between the mechanical shaft system (turbine-generator) and the series- capacitor compensated electrical network. Small-signal disturbances in a power system result in simultaneous excitation of all natural modes of the electrical and mechanical systems. The turbine-generator shaft system responds to disturbances with oscillations at its torsional natural frequencies. Therefore, TI-SSR can be viewed as a stability phenomenon of the linearized system model [9-10]. In other words, their stability and design of control parameter can be investigated with the eigenvalues of the small-signal model obtained at a certain working point [10]. In our study, for each of the concerned operation conditions, a detailed linearized system model of the studied system is developed. It can be expressed as

$$\begin{cases} \Delta \dot{\mathbf{X}} = \mathbf{A} \Delta \mathbf{X} + \mathbf{B} \Delta \mathbf{U} \\ \Delta \mathbf{Y} = \mathbf{C} \Delta \mathbf{X} \end{cases} \tag{1}$$

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 247

> **A BH <sup>A</sup> GC F**

**Γα Γα Γα** (4)

(5)

*<sup>i</sup>* means the calculation

are the

is the least

and Im( ) *ij*

is the extended

where <sup>T</sup> <sup>ˆ</sup> **X XZ** is the extended state vector and ˆ

Obviously, the eigenvalues of the matrices **A** and **A**ˆ that correspond to the torsional modes represent the stability or damping of these TI modes for the open-loop and closedloop systems respectively. Hence, by the eigen-analysis method, the relationship between the torsional damping with system variables and control parameters can be traced out under different operating conditions. Generally, for an uncontrolled practical series-compensated power system, several of the eigenvalues related to the TI-SSR problem would be negative or unstable under certain operating conditions, i.e., the SSR problem is a multimodal one. Thus, the key issue is to design an effective control strategy which can meet the following requirements: First, the multimodal SSR must be damped simultaneously and the controller parameters determined in a coordinated way. Second, the controller must be robust enough to stabilize SSR under all possible operating conditions. In addition, the controller should be practical, i.e., easy to implement and operate. Finally, its size and operational loss should be

In this chapter, the aimed power system and its SSR problem is represented with detailed electromagnetic models and the task of control-design is formulated into the constrained

subject to: ( ) ( ) ( )

Where *f* is the fitness function to be optimized, **Γ α**( ) is the function of the control

For our concerned multimodal SSR problem, the fitness function is generally defined as [13]

1 123 1 1

*f ww w w*

 

*ii N i i*

 

**A***j* is the closed-loop coefficient matrix for the *j*-th operating condition and is jointly

min , Re( ) / Im( ) ,

*<sup>i</sup> ij ij ij ij <sup>j</sup>*

*Ai j M*

damping among all operating conditions; *wi* are the positive weight coefficients; *N* is the number of concerned SSR modes; *M* is the number of small-signal system model with each representing an evaluation condition; the subscripts *i*, *j* denote the index of the SSR mode

1

 

is the close-loop damping; *<sup>i</sup>*

min{ , , }, 1, 0

parameter **α** , which has its lower and upper limits being **Γα Γα** ( ), ( ) respectively.

*N N*

*i i*

 

coefficient matrix with appropriate dimensions.

reasonable for reduction of cost, room and maintenance.

nonlinear optimization problem, which is of the following form:

max

*f*

 

ˆ ( ) ; 1,2,3; 1,...,

**α**

of the *i-*th SSR mode of the included close-loop coefficient matrix; Re( ) *ij*

1

*ij i j*

determined by the system variables and the control parameters;

 

real and imaginary parts of the SSR mode; *ij*

and the operating condition respectively.

where ˆ

*j M*

 

Where **X** is the state vector, **U** is the input vector and **Y** is the output or feedback vector.

In model (1), the following various sub-models are included:


The controller, either the SEDC or the SVC-SSDC to be discussed, can be also be linearized as

$$\begin{cases} \Delta \dot{\mathbf{Z}} = \mathbf{F}(\mathbf{a}) \Delta \mathbf{Z} + \mathbf{G}(\mathbf{a}) \, \Delta \mathbf{Y} \\ \Delta \mathbf{U} = \mathbf{H}(\mathbf{a}) \Delta \mathbf{Z} \end{cases} \tag{2}$$

where **Z** is the state vector, and **FGH** , , are the coefficient matrices with appropriate dimensions and their elements are determined by the control parameter vector **α** , which including the gains and time constants of the controller to be designed.

Therefore the close-loop system model under a certain operation situation can be obtained by combining the open-loop system model (1) and the controller (2), i.e.

$$
\Delta \dot{\hat{\mathbf{X}}} = \hat{\mathbf{A}} \Delta \hat{\mathbf{X}} \tag{3}
$$

where <sup>T</sup> <sup>ˆ</sup> **X XZ** is the extended state vector and ˆ **A BH <sup>A</sup> GC F** is the extended coefficient matrix with appropriate dimensions.

246 Simulated Annealing – Advances, Applications and Hybridizations

In model (1), the following various sub-models are included:

model with three damper windings [11, 12];

system stabilizer (PSS);

capacitors.

as

encountered in practice.

It can be expressed as

vector.

set;

However, this chapter will focus on the TI type SSR, because it is the main problem we

TI-SSR is the interplay between the mechanical shaft system (turbine-generator) and the series- capacitor compensated electrical network. Small-signal disturbances in a power system result in simultaneous excitation of all natural modes of the electrical and mechanical systems. The turbine-generator shaft system responds to disturbances with oscillations at its torsional natural frequencies. Therefore, TI-SSR can be viewed as a stability phenomenon of the linearized system model [9-10]. In other words, their stability and design of control parameter can be investigated with the eigenvalues of the small-signal model obtained at a certain working point [10]. In our study, for each of the concerned operation conditions, a detailed linearized system model of the studied system is developed.

**Y CX**

**X AX BU**

Where **X** is the state vector, **U** is the input vector and **Y** is the output or feedback

the lumped mass-spring model of the mechanical shaft system of the turbine-generator

the electromagnetic model of the generator, which is generally represented with the dq0

 the excitation system, for instance, the static self-parallel excitation system is modeled with the IEEE ST4B-type automatic voltage regulator (AVR) and PSS2B-type power

the model of the electrical network, incorporating the lines, transformers and the series

The controller, either the SEDC or the SVC-SSDC to be discussed, can be also be linearized

( ) 

**U H α Z**

including the gains and time constants of the controller to be designed.

by combining the open-loop system model (1) and the controller (2), i.e.

( ) ( )

**Z F α Z G α Y**

where **Z** is the state vector, and **FGH** , , are the coefficient matrices with appropriate dimensions and their elements are determined by the control parameter vector **α** , which

Therefore the close-loop system model under a certain operation situation can be obtained

(1)

(2)

ˆ ˆ <sup>ˆ</sup> **X AX** (3)

Obviously, the eigenvalues of the matrices **A** and **A**ˆ that correspond to the torsional modes represent the stability or damping of these TI modes for the open-loop and closedloop systems respectively. Hence, by the eigen-analysis method, the relationship between the torsional damping with system variables and control parameters can be traced out under different operating conditions. Generally, for an uncontrolled practical series-compensated power system, several of the eigenvalues related to the TI-SSR problem would be negative or unstable under certain operating conditions, i.e., the SSR problem is a multimodal one. Thus, the key issue is to design an effective control strategy which can meet the following requirements: First, the multimodal SSR must be damped simultaneously and the controller parameters determined in a coordinated way. Second, the controller must be robust enough to stabilize SSR under all possible operating conditions. In addition, the controller should be practical, i.e., easy to implement and operate. Finally, its size and operational loss should be reasonable for reduction of cost, room and maintenance.

In this chapter, the aimed power system and its SSR problem is represented with detailed electromagnetic models and the task of control-design is formulated into the constrained nonlinear optimization problem, which is of the following form:

$$\begin{aligned} \max f \\ \text{subject to: } \underline{\Gamma}(\mathfrak{a}) \le \Gamma(\mathfrak{a}) \le \overline{\Gamma}(\mathfrak{a}) \end{aligned} \tag{4}$$

Where *f* is the fitness function to be optimized, **Γ α**( ) is the function of the control parameter **α** , which has its lower and upper limits being **Γα Γα** ( ), ( ) respectively.

For our concerned multimodal SSR problem, the fitness function is generally defined as [13]

$$\begin{aligned} f &= \sum\_{i=1}^{N} w\_i \eta\_i + w\_{N+1} \min\{\eta\_1, \eta\_2, \eta\_3\} \sum\_{i=1}^{N+1} w\_i = 1, w\_i > 0\\ \eta\_i &= \min\_{j=1}^{j=M} \{\sigma\_{ij}\} \ \ \sigma\_{ij} = -\text{Re}\{\mathcal{A}\_{ij}\} / \left|\text{Im}(\mathcal{A}\_{ij})\right| \tag{5} \\ \mathcal{A}\_{ij} &= \mathcal{A}\_i \left\{\hat{\mathcal{A}}\_j(\mathbf{x})\right\}; i = 1, 2, 3; j = 1, \dots, M \end{aligned} \tag{6}$$

where ˆ **A***j* is the closed-loop coefficient matrix for the *j*-th operating condition and is jointly determined by the system variables and the control parameters; *<sup>i</sup>* means the calculation of the *i-*th SSR mode of the included close-loop coefficient matrix; Re( ) *ij* and Im( ) *ij* are the real and imaginary parts of the SSR mode; *ij* is the close-loop damping; *<sup>i</sup>* is the least damping among all operating conditions; *wi* are the positive weight coefficients; *N* is the number of concerned SSR modes; *M* is the number of small-signal system model with each representing an evaluation condition; the subscripts *i*, *j* denote the index of the SSR mode and the operating condition respectively.

The control-design problem can be summarized as follows:

Firstly, for a SSR-threatened system, representative conditions should be selected as the "evaluation conditions", which cover the full range of generator output levels as well as the status of transmission lines. While these evaluation conditions by no means limit the situations under which the power system operates, together they form the edge of practical operating conditions and the most unfavorable situations relevant to the controller to be designed. Therefore, the derived controller can be robust enough to stabilize the system under all possible conditions.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 249

particular procedures employed to design the SSR control (i.e., SEDC and SVC-SSDC) are

i. Define algorithm parameters, including: boundaries of control parameters, GA parameters (population size *n*, number of generations *k*GA, number of preferential chromosomes *n*b, crossover probability *p*c, mutation probability *p*m, initial perturbation

 , initial acceptance rate *pr*). ii. Set an initial population by randomly generating *n* feasible vectors or chromosomes in a

iii. Define the initial temperature as 0 max / ln *<sup>r</sup> t p* , where max max min *f f* and max min *f* , *f* are fitness values of the best and the worst chromosomes of the initial

**Step 3.** Apply GA operations to generate a new generation. This is fulfilled with three sub-

i. Selection: Select *<sup>b</sup> n* chromosomes as potential parents from the old population. The selection is simulated by spinning a weighted roulette wheel [18] in such a way that the fittest chromosomes have a higher probability of being selected, so that their genes have

)and SA parameters (number of state transition *k*SA, cooling rate *r*,

elaborated in the following.

The flow chart of the proposed GA-SA is shown in Fig. 1.

**Figure 1.** Flow chart of the implemented GA-SA

**Step 1.** GA-SA initialization.

amplitude GA0 

population.

operators:

perturbation amplitude SA0

The GA-SA is executed with the following steps.

**Step 2.** Evaluate each chromosome by using the fitness function.

a greater chance to be passed on to the next generation.

reasonable range of the control parameters;

Secondly, for each evaluation condition, a linearized model like (1)-(3) is derived for the nonlinear system as well as the controller via small-signal approximation, and the damping of torsional modes can be calculated by eigen-analysis. Consequently, the parameter sensitivity of the control on the stability of TI-type SSR is quantified.

Thirdly, the control-design task, i.e., simultaneously tuning of the control parameters, was formulated into a standard nonlinear optimization problem as described in (4)-(5), by appropriately determining the fitness function and the constraints on the parameters of the controller. As seen in (5), the ability of the controller to depress SSR is determined by the weighted sum of the damping of concerned torsional modes under all listed conditions. Furthermore, the least damped one is emphasized with additional weight to achieve better damping.

Next, by solving the optimization problem (4)-(5), the control parameters can be tuned simultaneously and an optimal control strategy to mitigate the SSR issue is obtained.

Finally, the designed controller should be verified with simulation study on the nonlinear system model, or even be implemented as hardware equipment and tested in practical applications.

## **2. The combined intelligent optimization method: Genetic algorithm and simulated annealing (GA-SA)**

The control-design problem (4)-(5) is a complex nonlinear optimization problem with many models, each corresponding to an evaluation condition. However, it is difficult for conventional methods due to its nonlinear nature and high dimension of the solution space. Here, we provide the solution by combining genetic algorithm (GA) and simulated annealing (SA), termed GA-SA. As a hybrid and global optimization strategy, GA-SA takes advantage of both GA's parallel-searching capability and SA's probabilistic jumping property [14-17]: The fast and global parallel searching ability of GA is retained, and the diversity is improved by SA state transition. Thus, premature convergence in GA can be avoided. Furthermore, SA renders GA mutation more controllable by its temperature. Thanks to the powerful global searching ability, the criteria for the selection of algorithm parameters are very much relaxed, resulting in improved performance and robust optimization. Since the basic principle of GA-SA has been well documented [14-17], only the particular procedures employed to design the SSR control (i.e., SEDC and SVC-SSDC) are elaborated in the following.

The flow chart of the proposed GA-SA is shown in Fig. 1.

248 Simulated Annealing – Advances, Applications and Hybridizations

under all possible conditions.

damping.

applications.

**simulated annealing (GA-SA)** 

The control-design problem can be summarized as follows:

sensitivity of the control on the stability of TI-type SSR is quantified.

Firstly, for a SSR-threatened system, representative conditions should be selected as the "evaluation conditions", which cover the full range of generator output levels as well as the status of transmission lines. While these evaluation conditions by no means limit the situations under which the power system operates, together they form the edge of practical operating conditions and the most unfavorable situations relevant to the controller to be designed. Therefore, the derived controller can be robust enough to stabilize the system

Secondly, for each evaluation condition, a linearized model like (1)-(3) is derived for the nonlinear system as well as the controller via small-signal approximation, and the damping of torsional modes can be calculated by eigen-analysis. Consequently, the parameter

Thirdly, the control-design task, i.e., simultaneously tuning of the control parameters, was formulated into a standard nonlinear optimization problem as described in (4)-(5), by appropriately determining the fitness function and the constraints on the parameters of the controller. As seen in (5), the ability of the controller to depress SSR is determined by the weighted sum of the damping of concerned torsional modes under all listed conditions. Furthermore, the least damped one is emphasized with additional weight to achieve better

Next, by solving the optimization problem (4)-(5), the control parameters can be tuned

Finally, the designed controller should be verified with simulation study on the nonlinear system model, or even be implemented as hardware equipment and tested in practical

**2. The combined intelligent optimization method: Genetic algorithm and** 

The control-design problem (4)-(5) is a complex nonlinear optimization problem with many models, each corresponding to an evaluation condition. However, it is difficult for conventional methods due to its nonlinear nature and high dimension of the solution space. Here, we provide the solution by combining genetic algorithm (GA) and simulated annealing (SA), termed GA-SA. As a hybrid and global optimization strategy, GA-SA takes advantage of both GA's parallel-searching capability and SA's probabilistic jumping property [14-17]: The fast and global parallel searching ability of GA is retained, and the diversity is improved by SA state transition. Thus, premature convergence in GA can be avoided. Furthermore, SA renders GA mutation more controllable by its temperature. Thanks to the powerful global searching ability, the criteria for the selection of algorithm parameters are very much relaxed, resulting in improved performance and robust optimization. Since the basic principle of GA-SA has been well documented [14-17], only the

simultaneously and an optimal control strategy to mitigate the SSR issue is obtained.

**Figure 1.** Flow chart of the implemented GA-SA

The GA-SA is executed with the following steps.

**Step 1.** GA-SA initialization.

	- ii. Crossover: This is carried out between each parent member (*pi*, *i*=1,…,*n*b) and the fittest chromosome (*p*0) of the current generation. The convex crossover method is adopted to produce the offspring *<sup>i</sup>*<sup>1</sup> *x* , *<sup>i</sup>*<sup>2</sup> *x* , where 1 0 (1 ) , *i i xp p* 2 0 (1 ) , *i i x pp* 01 [17]. All offspring are added to the original population and then *n* chromosomes of higher fitness are chosen to form a new generation.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 251

almost the same parameters. Each turbine-generator consists of four rotors, i.e., a high-andintermediate-pressure (HIP) turbine rotor, two low-pressure (LPA and LPB) turbine rotors, and the generator rotor, thus resulting in three subsynchronous torsional modes. The characteristic frequencies (in Hz) are 15.19~15.33 (mode 1), 26.01~26.12 (mode 2) and 30.25~30.54 (mode 3). To solve the SSR problem, torsional stress relays (TSRs) and supplementary excitation damping controllers (SEDCs) are applied to the SPP generators

A thorough evaluation of the severity of the SSR problem was conducted under all possible system conditions [19, 20], which comprised frequency scanning, eigenvalue analyses and

With eigenvalue analysis, the modal damping (i.e., the negative real part of the torsional eigenvalue, which is a combined outcome of mechanical and electrical damping) can be obtained with respect to different operating conditions. To illustrate, 24 representative conditions are selected as the "evaluation conditions" (listed in Table 1), which cover the full range of generator output levels as well as the status of the Shangdu-Chengde lines. Table 1 also lists the calculated modal damping. Thus, the severity of SSR is quantified and the most

Through the evaluation study, characteristics of the SSR problem are summarized as

i. Mode 1 is well-damped in all operating conditions; however, modes 2 and 3 may be under-damped or even unstable under some conditions. Thus, the SSR problem is a

ii. Mode 2 is the worst damped and tends to be unstable for numbers of operating conditions, especially when only one Shangdu-Chengde line is in service or the generator output is relatively low (corresponding to a lower mechanical damping).

**Figure 2.** The one-line diagram of the equivalent transmission system

(as shown in fig. 2).

**3.2. The SSR problem** 

electromagnetic transient (EMT).

risky situations are identified.

multimodal one.

follows:


## **3. Application of GA-SA for SEDC optimization**

## **3.1. The target system**

The Shangdu power plant (SPP), located in the Inner Mongolia Autonomous Region, is about 300 kilometers north of Beijing city. It has four 600 MW steam turbine-generators connected to the North China Power Grid through 500 kV transmission lines, including the 243 km double lines connecting SPP with the Chengde substation and the 130km double lines connecting the Chengde substation with the Jiangjiaying substation. Fig.2 illustrates the one-line diagram of the equivalent transmission system. To improve the transferring capability as well as system stability, fixed series capacitors (FSCs) are applied to the parallel transmission lines between SPP and the Chengde substation with 45% compensation degree. The four turbine-generators are subcritical air-cooled machines with almost the same parameters. Each turbine-generator consists of four rotors, i.e., a high-andintermediate-pressure (HIP) turbine rotor, two low-pressure (LPA and LPB) turbine rotors, and the generator rotor, thus resulting in three subsynchronous torsional modes. The characteristic frequencies (in Hz) are 15.19~15.33 (mode 1), 26.01~26.12 (mode 2) and 30.25~30.54 (mode 3). To solve the SSR problem, torsional stress relays (TSRs) and supplementary excitation damping controllers (SEDCs) are applied to the SPP generators (as shown in fig. 2).

**Figure 2.** The one-line diagram of the equivalent transmission system

## **3.2. The SSR problem**

250 Simulated Annealing – Advances, Applications and Hybridizations

perturbation, i.e., *<sup>n</sup>* o GA *xxm*

parameter of perturbation amplitude SA

generations. Otherwise, go back to Step 2.

**3. Application of GA-SA for SEDC optimization** 

distribution, and GA

ii. State acceptance: Let *<sup>i</sup>*

searches.

**3.1. The target system** 

<sup>1</sup> exp[( ) / ] *ii p p ff T* .

produce the offspring *<sup>i</sup>*<sup>1</sup> *x* , *<sup>i</sup>*<sup>2</sup> *x* , where 1 0 (1 ) , *i i xp p*

transition and acceptance are repeatedly used during the SA.

higher fitness are chosen to form a new generation.

ii. Crossover: This is carried out between each parent member (*pi*, *i*=1,…,*n*b) and the fittest chromosome (*p*0) of the current generation. The convex crossover method is adopted to

iii. Mutation with one elite reservation strategy: Keep the best chromosome and apply mutation to other chromosomes with a probability of *mp* . For each mutation, a new chromosome n*x* is generated from an old chromosome <sup>o</sup> *x* by applying a random

**Step 4.** Apply the parallel SA algorithm to enhance the quality of the new population. State

i. State transition: First, SA selects candidate chromosomes from the GA-generated population according to the roulette strategy [18]. State transition is then performed SA *k* times for each of these chromosomes. The operation of state transition is identical to that of GA mutation, the only difference being that the former has a much smaller

*xi*+1 obtained by state transition, respectively. If *i i* <sup>1</sup> *f f* , *xi*+1 is accepted as the starting point for the next state transition; otherwise, *xi*+1 is accepted with probability

**Step 5.** Cooling: As the optimization proceeds, the temperature *Tp* is dynamically updated according to the rule *k k* <sup>1</sup> *Tp r Tp* . Thus, at higher temperatures, SA will accept poorer status with certain probability to avoid plugging into local minimum; while at lower temperatures, it basically turns out to be an optimization function in small random

**Step 6.** Termination: GA-SA terminates if the maximum number of generations allowed ( GA *k* ) is reached or the best result has not seen improvement for a certain number of

The Shangdu power plant (SPP), located in the Inner Mongolia Autonomous Region, is about 300 kilometers north of Beijing city. It has four 600 MW steam turbine-generators connected to the North China Power Grid through 500 kV transmission lines, including the 243 km double lines connecting SPP with the Chengde substation and the 130km double lines connecting the Chengde substation with the Jiangjiaying substation. Fig.2 illustrates the one-line diagram of the equivalent transmission system. To improve the transferring capability as well as system stability, fixed series capacitors (FSCs) are applied to the parallel transmission lines between SPP and the Chengde substation with 45% compensation degree. The four turbine-generators are subcritical air-cooled machines with

is the perturbation amplitude.

. [17]. All offspring are added to the original population and then *n* chromosomes of

 

 2 0 (1 ) , *i i x pp* 

, where *m*( 5,5) is a random variable of Cauchy

*f* and *<sup>i</sup>* <sup>1</sup> *f* be the fitness values of the original state *xi* and the state

   01 

> A thorough evaluation of the severity of the SSR problem was conducted under all possible system conditions [19, 20], which comprised frequency scanning, eigenvalue analyses and electromagnetic transient (EMT).

> With eigenvalue analysis, the modal damping (i.e., the negative real part of the torsional eigenvalue, which is a combined outcome of mechanical and electrical damping) can be obtained with respect to different operating conditions. To illustrate, 24 representative conditions are selected as the "evaluation conditions" (listed in Table 1), which cover the full range of generator output levels as well as the status of the Shangdu-Chengde lines. Table 1 also lists the calculated modal damping. Thus, the severity of SSR is quantified and the most risky situations are identified.

> Through the evaluation study, characteristics of the SSR problem are summarized as follows:


iii. Modal damping is affected by several factors and each torsional mode has a most undesirable (or least damped) condition unique to its own, making it a challenge to design a controller adaptable to all operating conditions.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 253

> )

drives the excitation circuit to yield the field voltage ( *<sup>f</sup> E* ). There is a time delay (

**Figure 3.** The signal relationship of the SEDC and the excitation regulators

(AVR: automatic voltage regulator, PSS: power system stabilizer)

SEDC-controlled system.

deviation signal

SSR.

**Figure 4.** Configuration of SEDC

between *<sup>f</sup> u* and *<sup>f</sup> E* , mainly due to signal sampling, data processing and thyristor transport lag. This time delay, generally several to a dozen milliseconds, is important to the practicality of SEDC's design, since it is comparable with the period of torsional modes, which, if not taken into consideration, would deteriorate or even destabilize the

Fig.4 shows the block diagram of the proposed SEDC. The mechanical speed of the HIP turbine provides the input, which after proper filtering and conditioning becomes the

tuned to a specific mode, comprises a band-pass filter, an amplifier *Gk* and a unity-gain phase-shifter 2 2 (1 ) (1 ) *k k Ts Ts* to generate the control signal for the corresponding mode. The control signals of all torsional modes are summarized, clipped and finally added to the AVR output to modulate the field voltage. Consequently, three subsynchronous components are generated in the excitation current, which in turn produce subsynchronous torque upon the generator shafts. If the gains and phase-shifts are appropriately set, this torque provided by SEDCs will play a role in damping SSR. So proper determination of the gains and time constants, i.e., , ( 1,2,3) *GTk k k* in Fig.4, is crucial to mitigate the multimodal

<sup>1</sup> . It then passes through three separate control paths. Each control path,



## **3.3. Supplementary Excitation Damping Controller**

SEDC is a real-time control system that works through the excitation system by modulating the field voltage at the torsional frequencies. Fig.3 illustrates the signal relationship of the SEDC, the excitation regulators, the generators and the grid. As a supplementary control loop, SEDC uses the mechanical speed of the HIP turbine (<sup>1</sup> ) as the only feedback signal to generate the subsynchronous control output ( *SEDC u* ). *SEDC u* is then added to the output of AVR ( *Cu* ) to form a modulated control signal ( *<sup>f</sup> u* ), which drives the excitation circuit to yield the field voltage ( *<sup>f</sup> E* ). There is a time delay ( ) between *<sup>f</sup> u* and *<sup>f</sup> E* , mainly due to signal sampling, data processing and thyristor transport lag. This time delay, generally several to a dozen milliseconds, is important to the practicality of SEDC's design, since it is comparable with the period of torsional modes, which, if not taken into consideration, would deteriorate or even destabilize the SEDC-controlled system.

**Figure 3.** The signal relationship of the SEDC and the excitation regulators (AVR: automatic voltage regulator, PSS: power system stabilizer)

Fig.4 shows the block diagram of the proposed SEDC. The mechanical speed of the HIP turbine provides the input, which after proper filtering and conditioning becomes the deviation signal <sup>1</sup> . It then passes through three separate control paths. Each control path, tuned to a specific mode, comprises a band-pass filter, an amplifier *Gk* and a unity-gain phase-shifter 2 2 (1 ) (1 ) *k k Ts Ts* to generate the control signal for the corresponding mode. The control signals of all torsional modes are summarized, clipped and finally added to the AVR output to modulate the field voltage. Consequently, three subsynchronous components are generated in the excitation current, which in turn produce subsynchronous torque upon the generator shafts. If the gains and phase-shifts are appropriately set, this torque provided by SEDCs will play a role in damping SSR. So proper determination of the gains and time constants, i.e., , ( 1,2,3) *GTk k k* in Fig.4, is crucial to mitigate the multimodal SSR.

**Figure 4.** Configuration of SEDC

252 Simulated Annealing – Advances, Applications and Hybridizations

Oper. Cond. #

design a controller adaptable to all operating conditions.

Generator output (%) Status of

Gen 1 Gen 2 Gen 3 Gen 4 Line 1 Line 2 No

**Table 1.** Torsional damping of the open-loop system under the evaluation conditions

SEDC is a real-time control system that works through the excitation system by modulating the field voltage at the torsional frequencies. Fig.3 illustrates the signal relationship of the SEDC, the excitation regulators, the generators and the grid. As a

the only feedback signal to generate the subsynchronous control output ( *SEDC u* ). *SEDC u* is then added to the output of AVR ( *Cu* ) to form a modulated control signal ( *<sup>f</sup> u* ), which

supplementary control loop, SEDC uses the mechanical speed of the HIP turbine (

**3.3. Supplementary Excitation Damping Controller** 

iii. Modal damping is affected by several factors and each torsional mode has a most undesirable (or least damped) condition unique to its own, making it a challenge to

1 0 0 0 0 online offline 0.0312 **-0.5603** 0.0211 2 0 0 0 0 online online 0.0367 **-0.0429 -0.0053**  3 0 0 0 offline online offline 0.0338 **-0.1992 -0.0764**  4 0 0 0 offline online online 0.0380 0.0449 0.0236 5 0 0 offline offline online offline 0.0402 **-0.0235 -0.0921**  6 0 0 offline offline online online 0.0442 0.0628 0.0290 7 0 offline offline offline online offline 0.0506 0.0594 0.0267 8 0 offline offline offline online online 0.0533 0.0796 0.0325 9 40 40 40 40 online offline 0.0598 **-0.4887** 0.0511 10 40 40 40 40 online online 0.0654 0.0383 0.0242 11 40 40 40 offline online offline 0.0590 **-0.1882** 0.0229 12 40 40 40 offline online online 0.0641 0.0569 0.0460 13 40 40 offline offline online offline 0.0599 **-0.0123 -0.0864**  14 40 40 offline offline online online 0.0646 0.0748 0.0567 15 40 offline offline offline online offline 0.0664 0.0684 0.0529 16 40 offline offline offline online online 0.0717 0.0916 0.0620 17 100 100 100 100 online offline 0.0674 **-0.5455** 0.1344 18 100 100 100 100 online online 0.0716 0.0787 0.1026 19 100 100 100 offline online offline 0.0615 **-0.1835** 0.1035 20 100 100 100 offline online online 0.0648 0.1015 0.1284 21 100 100 offline offline online offline 0.0555 0.0248 **-0.0192**  22 100 100 offline offline online online 0.0580 0.1226 0.1409 23 100 offline offline offline online offline 0.0511 0.1163 0.1370 24 100 offline offline offline online online 0.0540 0.1414 0.1467

SPP-Chengde lines

Mode 1 ( <sup>1</sup> *s* )

SEDC

Mode 2 ( <sup>1</sup> *s* )

No SEDC Mode 3 ( <sup>1</sup> *s* )

No SEDC

> <sup>1</sup> ) as

#### **3.4. System modeling**

#### *3.4.1. The linearized open-loop system model*

The linearized state equations of the four generators and network around a specific operating point can be expressed by (1). Here the state vector **X** , input vector **U** and output vector **Y** are given respectively by <sup>T</sup> 41 41 61 C, , *d q* **<sup>X</sup> δ ω i u** T T , , *<sup>T</sup> <sup>f</sup>* **U YX** *E* where 4 1 **δ** are the mass angles; **ω**4 1 are the mass speeds; 6 1 **i** are the winding currents; C , *d q* **u** are the *d*-axis and *q*-axis voltages of the series capacitors; *<sup>f</sup> E* is the field excitation voltage and **ABC** , , are the coefficient matrices with proper dimensions.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 255

In our case, it is discovered that the performance of GA-SA is not very sensitive to these parameters. In fact, if the population size and the number of generations are chosen properly, GA-SA will generally converge to satisfactory results after running 8 to 12 iterations, which lasts only several minutes on a modern computer. The typical convergence characteristic of the GA-SA is displayed in Fig.5. The most time-consuming step during the optimization process is the calculation of the fitness function, which is essentially the close-loop damping of the torsional modes. Therefore, a highly efficient and reliable algorithm should be used to calculate the fitness function. In this aspect, the implemented GA-SA works very well. Table 2

> *G*<sup>1</sup> <sup>1</sup> *T G*<sup>2</sup> <sup>2</sup> *T G*<sup>3</sup> <sup>3</sup> *T* -378.99 0. 0535 311.45 0.0174 239.92 0.0018

To verify the tuned SEDC, both eigenvalue analysis and EMT simulation were fulfilled on

Eigenvalue analysis was conducted under the evaluation conditions, first in the absence and then in the presence of SEDC. The result is presented in Table 3. It shows that, without SEDC, modes 2 and 3 will incur negative damping, resulting in an unstable system. When SEDC is included into the control loop, the damping of torsional modes under all the specified conditions has been increased considerably and all unstable cases have been well

lists the optimized SEDC parameters obtained by the proposed GA-SA.

**Table 2.** The optimized SEDC parameters

**Figure 5.** Convergence of the GA-SA algorithm

*3.6.1. Model study* 

the close-loop system.

stabilized.

**3.6. Performance verification of the designed SEDC** 

#### *3.4.2. The combined system model*

From a combination of the models of the grid, the generators and the excitation control system, the close-loop system model can be derived (shown in Fig.3). For the convenience of further analysis and synthesis, the time delay *<sup>s</sup> e* is replaced with a rational approximation in the frequency domain, i.e. <sup>2</sup> (1 / 2) *<sup>s</sup> e s* . For the target system, the time delay is measured to be about 7 milliseconds. Since our SEDC scheme is actually a linear dynamic stabilizer, whose transfer function can be expressed with the general form as in (2). Therefore the close-loop system model (3) can be obtained by combining the open-loop system model and the SEDC controller.

#### **3.5. Optimal design of SEDC based on GA-SA**

For the Shangdu system, the evaluation conditions listed in Table 1 are used again for the purpose of parameters-tuning because these selected conditions together not only bound the range of practical operating conditions but also incorporate the most unfavorable situations. An SEDC based on these conditions will be robust enough to stabilize the system under all normal conditions. Then the nonlinear optimization problem can be formulated as in (4)-(5), in which the control parameter set **α** *GG G TTT* 1 2 3123 , , ,,, , *N*=3, *M*=24. During the parameter-tuning process, the gains and time constants of SEDC should be restricted within a reasonable range due to the control energy and hardware implementation limits. Thus the constraint can be written as

$$G\_{\rm lb,k} \le G\_k \le G\_{\rm ub,i'} \quad T\_{\rm lb,k} \le T\_k \le T\_{\rm ub,k} \tag{6}$$

where lb,k ub,k lb,k ub,k *GG TT* , , , are the lower and upper bounds of the gains and time constants.

Then the GA-SA is adopted to solve the problem. The following algorithm parameters have been chosen after running a number of trials: The weights in the fitness function: 123 4 *www w* 0.2, 0.4 ; population size = 20; crossover rate = 0.9; mutation rate = 0.1; number of generations = 30; cooling rate=0.95; and initial temperature = <sup>10</sup> 1\* / log (0.1) *f* , where *f* is the maximum margin of the fitness value of the initial population.

In our case, it is discovered that the performance of GA-SA is not very sensitive to these parameters. In fact, if the population size and the number of generations are chosen properly, GA-SA will generally converge to satisfactory results after running 8 to 12 iterations, which lasts only several minutes on a modern computer. The typical convergence characteristic of the GA-SA is displayed in Fig.5. The most time-consuming step during the optimization process is the calculation of the fitness function, which is essentially the close-loop damping of the torsional modes. Therefore, a highly efficient and reliable algorithm should be used to calculate the fitness function. In this aspect, the implemented GA-SA works very well. Table 2 lists the optimized SEDC parameters obtained by the proposed GA-SA.


**Table 2.** The optimized SEDC parameters

254 Simulated Annealing – Advances, Applications and Hybridizations

output vector **Y** are given respectively by <sup>T</sup>

The linearized state equations of the four generators and network around a specific operating point can be expressed by (1). Here the state vector **X** , input vector **U** and

*<sup>f</sup>* **U YX** *E* where 4 1 **δ** are the mass angles; **ω**4 1 are the mass speeds; 6 1 **i** are the winding currents; C , *d q* **u** are the *d*-axis and *q*-axis voltages of the series capacitors; *<sup>f</sup> E* is the field excitation voltage and **ABC** , , are the coefficient matrices with proper dimensions.

From a combination of the models of the grid, the generators and the excitation control system, the close-loop system model can be derived (shown in Fig.3). For the convenience of

measured to be about 7 milliseconds. Since our SEDC scheme is actually a linear dynamic stabilizer, whose transfer function can be expressed with the general form as in (2). Therefore the close-loop system model (3) can be obtained by combining the open-loop

For the Shangdu system, the evaluation conditions listed in Table 1 are used again for the purpose of parameters-tuning because these selected conditions together not only bound the range of practical operating conditions but also incorporate the most unfavorable situations. An SEDC based on these conditions will be robust enough to stabilize the system under all normal conditions. Then the nonlinear optimization problem can be formulated as in (4)-(5), in which the control parameter set **α** *GG G TTT* 1 2 3123 , , ,,, , *N*=3, *M*=24. During the parameter-tuning process, the gains and time constants of SEDC should be restricted within a reasonable range due to the control energy and hardware implementation limits. Thus the constraint can be written as

where lb,k ub,k lb,k ub,k *GG TT* , , , are the lower and upper bounds of the gains and time

Then the GA-SA is adopted to solve the problem. The following algorithm parameters have been chosen after running a number of trials: The weights in the fitness function: 123 4 *www w* 0.2, 0.4 ; population size = 20; crossover rate = 0.9; mutation rate = 0.1; number of generations = 30; cooling rate=0.95; and initial temperature = <sup>10</sup> 1\* / log (0.1) *f* ,

where *f* is the maximum margin of the fitness value of the initial population.

 41 41 61 C, , *d q* **<sup>X</sup> δ ω i u**

is replaced with a rational approximation

is

. For the target system, the time delay

lb,k ub, lb,k ub,k , *G GG T TT ki k* (6)

*3.4.1. The linearized open-loop system model* 

**3.4. System modeling** 

T T , , *<sup>T</sup>*

*3.4.2. The combined system model* 

further analysis and synthesis, the time delay *<sup>s</sup> e*

**3.5. Optimal design of SEDC based on GA-SA** 

in the frequency domain, i.e. <sup>2</sup> (1 / 2) *<sup>s</sup> e s*

system model and the SEDC controller.

constants.

**Figure 5.** Convergence of the GA-SA algorithm

### **3.6. Performance verification of the designed SEDC**

#### *3.6.1. Model study*

To verify the tuned SEDC, both eigenvalue analysis and EMT simulation were fulfilled on the close-loop system.

Eigenvalue analysis was conducted under the evaluation conditions, first in the absence and then in the presence of SEDC. The result is presented in Table 3. It shows that, without SEDC, modes 2 and 3 will incur negative damping, resulting in an unstable system. When SEDC is included into the control loop, the damping of torsional modes under all the specified conditions has been increased considerably and all unstable cases have been well stabilized.

To investigate how well the proposed controller can withstand large disturbances such as short-circuit faults, the nonlinear power system was simulated using EMT software "PSCAD/EMTDC". Step-by-step time-domain electromagnetic simulations were conducted to check the response of the SEDC controlled system during large disturbances.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 257

Mode 1 ( <sup>1</sup> *s*

With SEDC

SEDC

1 0 0 0 0 online offline 0.0312 0.0515 **-0.5603** 0.0873 0.0211 0.1078 2 0 0 0 0 online online 0.0367 0.0571 **-0.0429** 0.1074 **-0.0053** 0.0830 3 0 0 0 offline online offline 0.0338 0.0783 **-0.1992** 0.1381 **-0.0764** 0.1286 4 0 0 0 offline online online 0.0380 0.0632 0.0449 0.1148 0.0236 0.0864 5 0 0 offline offline online offline 0.0402 0.0827 **-0.0235** 0.1426 **-0.0921** 0.2030 6 0 0 offline offline online online 0.0442 0.0670 0.0628 0.1395 0.0290 0.0690 7 0 offline offline offline online offline 0.0506 0.0833 0.0594 0.1145 0.0267 0.0782 8 0 offline offline offline online online 0.0533 0.0694 0.0796 0.1051 0.0325 0.0554 9 40 40 40 40 online offline 0.0598 0.0897 **-0.4887** 0.1926 0.0511 0.1312 10 40 40 40 40 online online 0.0654 0.0966 0.0383 0.3859 0.0242 0.1022 11 40 40 40 offline online offline 0.0590 0.0910 **-0.1882** 0.4687 0.0229 0.1761 12 40 40 40 offline online online 0.0641 0.0973 0.0569 0.3889 0.0460 0.0890 13 40 40 offline offline online offline 0.0599 0.0945 **-0.0123** 0.4788 **-0.0864** 0.2073 14 40 40 offline offline online online 0.0646 0.1002 0.0748 0.3812 0.0567 0.0874 15 40 offline offline offline online offline 0.0664 0.1039 0.0684 0.4095 0.0529 0.0774 16 40 offline offline offline online online 0.0717 0.0889 0.0916 0.3418 0.0620 0.0877 17 100 100 100 100 online offline 0.0674 0.1008 **-0.5455** 0.0958 0.1344 0.2122 18 100 100 100 100 online online 0.0716 0.1076 0.0787 0.5216 0.1026 0.2274 19 100 100 100 offline online offline 0.0615 0.0985 **-0.1835** 0.5564 0.1035 0.2470 20 100 100 100 offline online online 0.0648 0.1044 0.1015 0.5413 0.1284 0.2119 21 100 100 offline offline online offline 0.0555 0.0974 0.0248 0.6364 **-0.0192** 0.3600 22 100 100 offline offline online online 0.0580 0.1022 0.1226 0.5553 0.1409 0.2083 23 100 offline offline offline online offline 0.0511 0.0997 0.1163 0.6081 0.1370 0.2097 24 100 offline offline offline online online 0.0540 0.1038 0.1414 0.5453 0.1467 0.2076

) Mode 2 ( <sup>1</sup> *s*

No SEDC With SEDC

) Mode 3 ( <sup>1</sup> *s*

No SEDC )

With SEDC

Oper. Cond. #

units online.

Generator output (%) Status of SPP-

Gen 1 Gen 2 Gen 3 Gen 4 Line 1 Line 2 No

Chengde lines

**Table 3.** Torsional damping with/without SEDC under the evaluation conditions

This test was also performed with and without SEDC, respectively. At the initial state, both Shangdu-Chengde lines were in service but only one FSCs was switched on (i.e., k1, k3, k4 were closed and k2 was open, see Fig.2). Breaker k3 was first opened to trip one Shangdu-Chengde line and then reclosed after five seconds. This test was meant to check the system response under the 3-machine and one-line condition. Fig.8 illustrates the dynamics of mode 2 during the operation. It can be seen that during the short period of the 3-machine one-line condition, without SEDC, unit 1 and 2 suffered diverging SSR; while for unit 4 the torsional oscillation converges, but with a very weak damping ratio. The reason is that units 1/2 have a higher mode-2 frequency than units 3/4 (see Table 2) and thus are exposed to a greater SSR risk in this operating condition. If SEDCs were applied, however, the torsional oscillation can be damped out soon and the SSR risk is avoided effectively for all

b. Tripping and reclosing of one Shangdu-Chengde line

In both of the two simulation experiments conducted, one Shangdu-Chengde line suffered a serious three-phase short-circuit fault. The faulting line was tripped 70 milliseconds later. In Experiment 1 all generators were 40% loaded while in Experiment 2 all generators were fully loaded. The result is summarized in Fig.6. As is shown in (a) and (c) of Fig. 6, in the absence of SEDC, the diverging delta mechanical speed of the high-pressure turbine indicates that the generators have a growing torsional vibration, which would probably lead to great damage on the shafts. When SEDC is applied in Fig.11 (b) and (d), subsynchronous oscillations in both experiments are successfully damped out.

Besides these two experiments, sufficient simulations have been carried out under other operating conditions and with large but different disturbances. Generally, SEDC is effective in improving the damping of all torsional modes and can satisfactorily mitigate the vibrations caused by SSR in large disturbances.

### *3.6.2. Practical applications and field tests*

After extensive laboratory tests, the proposed SEDC was then applied to the practical SSR problem of the target system. By the end of October 2008, all SEDCs were put into their places. The two FSCs of the Shangdu-Chengde lines were also ready for operation. Then, to further validate the effectiveness of the proposed countermeasure, the project team conducted a series of joint tests on the SEDCs and the series-compensated system. By planned operations of SEDCs, generators, series capacitors and transmission lines, the dynamic characteristics of SSR and the response of the SEDCs were fully investigated. To save space, only some typical test results, to illustrate the function of SEDC in damping TI-SSR, are discussed hereafter.

The initial operating condition of the tested system was as follows: Units 1, 2 and 4 were online and half-loaded; Unit 3 was out of service; both Shangdu-Chengde lines were in operation; FSCs were out of service but ready for operation.

a. Switching on/off FSCs with/without SEDC

First, with all SEDCs in service, the two FSCs were switched on and then they were switched off one by one. During these operations, the torsional oscillation was observed in real time and each operation was initiated only after the dynamics that was triggered by the previous operation completely died away. The next stage involved quitting all SEDCs and repeating the above switch-on/off operations of FSCs. The mode-2 dynamics of this test is depicted in Fig.7. It can be observed that the torsional oscillation of the online machines are convergent under both the SEDC-off and the SEDC-on conditions; while the SEDC can considerably improve the modal damping and make the oscillation converge much more quickly.


**Table 3.** Torsional damping with/without SEDC under the evaluation conditions

### b. Tripping and reclosing of one Shangdu-Chengde line

256 Simulated Annealing – Advances, Applications and Hybridizations

oscillations in both experiments are successfully damped out.

operation; FSCs were out of service but ready for operation.

a. Switching on/off FSCs with/without SEDC

vibrations caused by SSR in large disturbances.

*3.6.2. Practical applications and field tests* 

SSR, are discussed hereafter.

quickly.

To investigate how well the proposed controller can withstand large disturbances such as short-circuit faults, the nonlinear power system was simulated using EMT software "PSCAD/EMTDC". Step-by-step time-domain electromagnetic simulations were conducted

In both of the two simulation experiments conducted, one Shangdu-Chengde line suffered a serious three-phase short-circuit fault. The faulting line was tripped 70 milliseconds later. In Experiment 1 all generators were 40% loaded while in Experiment 2 all generators were fully loaded. The result is summarized in Fig.6. As is shown in (a) and (c) of Fig. 6, in the absence of SEDC, the diverging delta mechanical speed of the high-pressure turbine indicates that the generators have a growing torsional vibration, which would probably lead to great damage on the shafts. When SEDC is applied in Fig.11 (b) and (d), subsynchronous

Besides these two experiments, sufficient simulations have been carried out under other operating conditions and with large but different disturbances. Generally, SEDC is effective in improving the damping of all torsional modes and can satisfactorily mitigate the

After extensive laboratory tests, the proposed SEDC was then applied to the practical SSR problem of the target system. By the end of October 2008, all SEDCs were put into their places. The two FSCs of the Shangdu-Chengde lines were also ready for operation. Then, to further validate the effectiveness of the proposed countermeasure, the project team conducted a series of joint tests on the SEDCs and the series-compensated system. By planned operations of SEDCs, generators, series capacitors and transmission lines, the dynamic characteristics of SSR and the response of the SEDCs were fully investigated. To save space, only some typical test results, to illustrate the function of SEDC in damping TI-

The initial operating condition of the tested system was as follows: Units 1, 2 and 4 were online and half-loaded; Unit 3 was out of service; both Shangdu-Chengde lines were in

First, with all SEDCs in service, the two FSCs were switched on and then they were switched off one by one. During these operations, the torsional oscillation was observed in real time and each operation was initiated only after the dynamics that was triggered by the previous operation completely died away. The next stage involved quitting all SEDCs and repeating the above switch-on/off operations of FSCs. The mode-2 dynamics of this test is depicted in Fig.7. It can be observed that the torsional oscillation of the online machines are convergent under both the SEDC-off and the SEDC-on conditions; while the SEDC can considerably improve the modal damping and make the oscillation converge much more

to check the response of the SEDC controlled system during large disturbances.

This test was also performed with and without SEDC, respectively. At the initial state, both Shangdu-Chengde lines were in service but only one FSCs was switched on (i.e., k1, k3, k4 were closed and k2 was open, see Fig.2). Breaker k3 was first opened to trip one Shangdu-Chengde line and then reclosed after five seconds. This test was meant to check the system response under the 3-machine and one-line condition. Fig.8 illustrates the dynamics of mode 2 during the operation. It can be seen that during the short period of the 3-machine one-line condition, without SEDC, unit 1 and 2 suffered diverging SSR; while for unit 4 the torsional oscillation converges, but with a very weak damping ratio. The reason is that units 1/2 have a higher mode-2 frequency than units 3/4 (see Table 2) and thus are exposed to a greater SSR risk in this operating condition. If SEDCs were applied, however, the torsional oscillation can be damped out soon and the SSR risk is avoided effectively for all units online.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 259

ii. With SEDC applied, the damping of all torsional modes is significantly improved so that the unstable modes become stabilized, which guarantees the safety of the generator

iii. The field test includes various operations and covers different working conditions. Especially, the most dangerous conditions previously recognized (i.e., the condition of three/four machine and one Shangdu-Chengdu line) were tested sufficiently. Therefore

the effectiveness as well as the robustness of SEDC is very well validated.

**Figure 8.** Dynamics of mode 2 during switching operations of Shang-Cheng line #1 under the 3-

**4.1. A description of the target system and the multimodal SSR problem** 

The Jinjie Power Plant is located near Yulin city, Shanxi Province, about 500 kilometers west of Beijing city. As a mine-mouth power plant, it has four 600 MW turbine-generators connected to the North China Power Grid through 500 kV transmissions. Fig. 9 illustrates the one-line diagram of the equivalent system. To improve the transferring capability, 35% SC is applied to the parallel lines between Jinjie Power Plant, Xinzhou substation and Shibei substation. The neighboring Fugu Power Plant, with two 600 MW turbine-generators, is

Each of Jinjie and Fugu turbine-generators consists of four rotors, i.e., a high-andintermediate-pressure (HIP) turbine rotor, two low-pressure (LPA/LPB) turbine rotors and the generator rotor, thus resulting in three subsynchronous torsional modes. The modal frequencies were measured via field test and they are about 13.11 Hz (mode 1), 22.77 Hz

Similarly, a thorough evaluation of the SSR problem was conducted under all possible system conditions for the Jinjie system. As part of the results of eigenvalue analyses, Table 4

**4. Application of GA-SA for SVC-SSDC optimization** 

connected to the Xinzhou substation through two uncompensated lines.

(mode 2) and 28.22 Hz (mode 3) respectively.

shaft and the stability of the system.

machine condition (unit 1)

**Figure 6.** Delta mechanical speed of the high-pressure turbine

**Figure 7.** Dynamics of mode 2 during FSCs' switching-on/off operations under the 3-machine condition (unit 4)

From the results of the field tests, it can be concluded that:

i. There is real danger of SSR for the Shangdu series-compensated system, especially for mode 2, which exhibits instability when three or four machines are online while only one Shangdu-Chengde line is in service.

ii. With SEDC applied, the damping of all torsional modes is significantly improved so that the unstable modes become stabilized, which guarantees the safety of the generator shaft and the stability of the system.

258 Simulated Annealing – Advances, Applications and Hybridizations

**Figure 6.** Delta mechanical speed of the high-pressure turbine

From the results of the field tests, it can be concluded that:

one Shangdu-Chengde line is in service.

(unit 4)

(a) Without SEDC in Experiment 1 (b) With SEDC in Experiment 1

(c) Without SEDC in Experiment 2 (d) With SEDC in Experiment 2

**Figure 7.** Dynamics of mode 2 during FSCs' switching-on/off operations under the 3-machine condition

i. There is real danger of SSR for the Shangdu series-compensated system, especially for mode 2, which exhibits instability when three or four machines are online while only iii. The field test includes various operations and covers different working conditions. Especially, the most dangerous conditions previously recognized (i.e., the condition of three/four machine and one Shangdu-Chengdu line) were tested sufficiently. Therefore the effectiveness as well as the robustness of SEDC is very well validated.

**Figure 8.** Dynamics of mode 2 during switching operations of Shang-Cheng line #1 under the 3 machine condition (unit 1)

## **4. Application of GA-SA for SVC-SSDC optimization**

## **4.1. A description of the target system and the multimodal SSR problem**

The Jinjie Power Plant is located near Yulin city, Shanxi Province, about 500 kilometers west of Beijing city. As a mine-mouth power plant, it has four 600 MW turbine-generators connected to the North China Power Grid through 500 kV transmissions. Fig. 9 illustrates the one-line diagram of the equivalent system. To improve the transferring capability, 35% SC is applied to the parallel lines between Jinjie Power Plant, Xinzhou substation and Shibei substation. The neighboring Fugu Power Plant, with two 600 MW turbine-generators, is connected to the Xinzhou substation through two uncompensated lines.

Each of Jinjie and Fugu turbine-generators consists of four rotors, i.e., a high-andintermediate-pressure (HIP) turbine rotor, two low-pressure (LPA/LPB) turbine rotors and the generator rotor, thus resulting in three subsynchronous torsional modes. The modal frequencies were measured via field test and they are about 13.11 Hz (mode 1), 22.77 Hz (mode 2) and 28.22 Hz (mode 3) respectively.

Similarly, a thorough evaluation of the SSR problem was conducted under all possible system conditions for the Jinjie system. As part of the results of eigenvalue analyses, Table 4 gives a list of the real parts, or modal damping, of the three SSR modes under the 4-machine operating conditions (with different load levels and line status, see the "No SVC" column). Thus, the severity of SSR is quantified and the most risky situations are identified.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 261

mode has a unique least damped condition, making it a challenge to design a controller

Various countermeasures, including the blocking filter, SEDC and SVC were considered to handle the SSR problem of the system. However, SVC was finally chosen as the SSRdepressing device through detailed study and technical versus economic comparison.

The adopted six-pulse SVC (see Fig. 10) comprises a thyristor controlled reactor (TCR) and a passive LC filter tuned for the 5th harmonics. It is connected to the high-voltage (500kV)

Since its MW loss (generally less than 0.6%) is negligible, SVC can be modeled as a controllable inductive admittance [21]. Fig. 11 illustrates the control scheme and mathematical model that describes the dynamics of the fundamental admittance, wherein SVC circuit is represented by combination of a first-order plus dead-time model and a

0 0.5

; *X*T is the

(7)

TCR <sup>L</sup> *B X* 2 sin(2 ) /( ),

is the firing angle as determined by the reference 1ref *<sup>B</sup>* and <sup>1</sup> *<sup>f</sup>* ( )

transformer reactance, TCR FLT *B B*, are admittances of the TCR and the filter; *X*<sup>L</sup> is the full

With some simplification, the dynamics of fundamental admittance can be expressed

adaptable to all operating conditions.

**4.2. The SVC-based subsynchronous damping control** 

*4.2.1. The power-electronic circuit and mathematical model of SVC* 

side of the power plant through a step-up transformer.

**Figure 10.** The power-electronic circuit of the six-pulse SVC

, in series, i.e.

TCR FLT T *f BB X* ( ) 1 1( ) ,

nonlinear function *f*( )

reactance of the TCR.

concisely by:

where

**Figure 9.** The one-line diagram of the equivalent transmission system


**Table 4.** Real Parts of SSR modes without and with SVC-SSDC under the 4-machine operating conditions (Note: The "m+n" of the "lines" column means that there are "m" Jinjie-Xinzhou lines and "n" Jinzhou-Shibei lines in service.)

Through the evaluation study, characteristics of the SSR problem are summarized as follows:


mode has a unique least damped condition, making it a challenge to design a controller adaptable to all operating conditions.

#### **4.2. The SVC-based subsynchronous damping control**

260 Simulated Annealing – Advances, Applications and Hybridizations

**Figure 9.** The one-line diagram of the equivalent transmission system

"n" Jinzhou-Shibei lines in service.)

SSR problem is a multimodal one.

follows:

gives a list of the real parts, or modal damping, of the three SSR modes under the 4-machine operating conditions (with different load levels and line status, see the "No SVC" column).

Oper. Con. Mode 1 (s-1) Mode 2 (s-1) Mode 3 (s-1) Gen. output lines

4×0% 2+3 -0.0370 -0.0060 **0.3202**  4×40% 2+3 -0.0378 -0.0262 **0.3099**  4×100% 2+3 -0.0367 -0.0468 **0.3062**  4×0% 2+2 -0.0323 **0.0001 0.5708**  4×40% 2+2 -0.0333 -0.0200 **0.5620**  4×100% 2+2 -0.0326 -0.0404 **0.5649**  4×0% 1+3 -0.0156 **0.0223 0.3795**  4×40% 1+3 -0.0175 -0.0023 **0.3620**  4×100% 1+3 -0.0182 -0.01760 **0.3480**  4×0% 1+2 -0.0132 **0.0305 0.2301**  4×40% 1+2 -0.0152 -0.0105 **0.2108**  4×100% 1+2 -0.0161 -0.0090 **0.1902** 

**Table 4.** Real Parts of SSR modes without and with SVC-SSDC under the 4-machine operating conditions (Note: The "m+n" of the "lines" column means that there are "m" Jinjie-Xinzhou lines and

Through the evaluation study, characteristics of the SSR problem are summarized as

i. Mode 1 is well-damped in all operating conditions. Mode 2 is stable in most common operating conditions. However, it becomes weakly-damped or even unstable in some conditions when the generator output is relatively low (corresponding to a lower mechanical damping) or a part of the transmissions are out of service. Mode 3 is the worst damped and tends to be unstable for a number of operating conditions. Thus the

ii. Modal damping is affected by several factors, including the number of online generators and lines as well as the generators' output. What's more, each torsional

Thus, the severity of SSR is quantified and the most risky situations are identified.

Various countermeasures, including the blocking filter, SEDC and SVC were considered to handle the SSR problem of the system. However, SVC was finally chosen as the SSRdepressing device through detailed study and technical versus economic comparison.

#### *4.2.1. The power-electronic circuit and mathematical model of SVC*

The adopted six-pulse SVC (see Fig. 10) comprises a thyristor controlled reactor (TCR) and a passive LC filter tuned for the 5th harmonics. It is connected to the high-voltage (500kV) side of the power plant through a step-up transformer.

**Figure 10.** The power-electronic circuit of the six-pulse SVC

Since its MW loss (generally less than 0.6%) is negligible, SVC can be modeled as a controllable inductive admittance [21]. Fig. 11 illustrates the control scheme and mathematical model that describes the dynamics of the fundamental admittance, wherein SVC circuit is represented by combination of a first-order plus dead-time model and a nonlinear function *f*( ) , in series, i.e.

$$f(\alpha) = \begin{bmatrix} \int 1/(B\_{\text{TCR}} + B\_{\text{FLT}}) - X\_{\text{T}} \end{bmatrix}, \ B\_{\text{TCR}} = \begin{bmatrix} 2a + \sin(2a) - \pi \end{bmatrix} | (\pi X\_{\text{L}}), \ 0 \le \alpha \le 0.5\pi \tag{7}$$

where is the firing angle as determined by the reference 1ref *<sup>B</sup>* and <sup>1</sup> *<sup>f</sup>* ( ) ; *X*T is the transformer reactance, TCR FLT *B B*, are admittances of the TCR and the filter; *X*<sup>L</sup> is the full reactance of the TCR.

With some simplification, the dynamics of fundamental admittance can be expressed concisely by:

$$B\_1 = \frac{e^{-sT\_d}}{1 + sT\_s} B\_{1\text{ref}'} \quad B\_1 \in \left[ \frac{-(1 - B\_{\text{FL}}X\_{\text{L}})}{X\_{\text{T}}(1 - B\_{\text{FL}}X\_{\text{L}}) + X\_{\text{L}}}, 0 \right] \tag{8}$$

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 263

> HIP after

The proposed SSDC is a real-time control system that works through the SVC powerelectronic circuit to inject sub- and super- synchronous currents into the generator-stators and produces damping torque to ease SSR. Fig. 12 shows the block diagram of the 3-path SVC-SSDC designed for the Jinjie system. The average mechanical speed of HIP turbines of

proper conditioning. It is then passed to the low- & high-pass filter. Speed signal filtered next proceeds to three separate control paths. Each, tuned to a specific mode, comprises a modal filter, an amplifier *ki* and a unity-gain phase-shifter 2 2 (1 ) /(1 ) *k k sT sT* to generate the control signal for the corresponding mode. The control signals of all torsional modes are summarized, clipped and finally added to form the control signal, or SSR *B* , which drive the

The purpose of the low- & high-pass filter, with the transfer functions shown in (11), is to depress the unrelated signals, including the DC, low-frequency (<10Hz) and high-frequency

The modal filter is a series combination of a second-order band-pass filter and two second-

m 22 2 2 2 2

SSR HIP *B Hs*( )

/ 1 (/ ) 1 (/ ) ( ) 1 8 / (/ )1 6 / (/ )1 6 / (/ )

*ss s s s s* 

2 H

> 

P B 1 B 2

*ii i*

P P B1 B1 B2 B2

*ii i i i i*

 

L 

 2 40, <sup>H</sup> 

2 2

 

P2 B11 B32 2 

 

> 

=143.07 rad/s,

(13)

2 10 (11)

(12)

(>40Hz) components, to enable only related subsynchronous signals to pass through.

 

LL HH

*sss f s*

=82.38 rad/s,

 

L&H 2 2

 

=177.33 rad/s.

where *H s*( ) is the integrated transfer function of SVC-SSDC.

Then the SVC-SSDC can be reformulated into (13)

*<sup>s</sup> f s*

order band-stop filters, as described in (12):

P1 B21 B31 1 

*i*

P3 B12 B22 3 

where

<sup>1</sup> (/ ) ( ) , 1 / (/ )1 / (/ )

*ss s s*

the online machines provides the input, which becomes the standard signal

*4.2.2. Configuration of the proposed SVC-SSDC* 

SVC circuit as shown in Fig. 12.

**Figure 12.** Configuration of SVC-SSDC

where *<sup>d</sup> T* and *<sup>s</sup> T* represent the dead and delay times of the thyristor circuits. For the six-pulse SVC, generally d 0 *T T* / 24 , 0 / 6 *<sup>s</sup> T T* and 0 *T* =0.02 second.

**Figure 11.** The dynamic model of the SVC

To depress SSR, it is necessary for SVC to generate currents at the complementary subsynchronous frequencies. For this purpose, we propose the idea of modulating the fundamental admittance with subsynchronous frequencies, i.e., to make the reference value of the fundamental admittance ( 1ref *B* ) vary according to the subsynchronous frequencies, or

$$B\_{1\text{ref}} = B\_{10,\text{ref}} + B\_{\text{SSR}}, \quad B\_{\text{SSR}} = \sum\_{m=1}^{3} B\_{1\text{m}}(t) \cos(\alpha\_m t + \varphi\_{m,\text{ref}}) \tag{9}$$

where 10,ref *B* , SSR *B* denote the DC and subsynchronously modulated components of 1ref *B* ; *<sup>m</sup>* is the torsional frequency; *<sup>m</sup>*,ref is the initial phase.

Thus, by some derivation, the current of SVC can be derived as

$$\dot{i}\_S = \dot{i}\_1 + \sum\_{m=1}^3 \dot{i}\_{a\_0 - a\_m} + \sum\_{m=1}^3 \dot{i}\_{a\_0 + a\_m} + \sum\_{k=1}^\alpha \left(\dot{i}\_{pk \pm 1} + \dot{i}\_{(pk \pm 1)a\_b \mp a\_m} \right) + \dot{i}\_{\text{else}} \tag{10}$$

where *iB t* 1 1 10 0 U cos( ) is the fundamental current; <sup>0</sup> 1 1m <sup>0</sup> 0.5U ( )cos ( ) *<sup>m</sup> m m i Bt <sup>t</sup>* are the sub- and super- synchronous currents; *pk* <sup>1</sup> *<sup>i</sup>* denotes the characteristic harmonics; 0 ( 1) *<sup>m</sup> pk <sup>i</sup>* are fractional harmonics caused by the subsynchronous modulating control; else *i* represents the rest of the negligible components; and U is fundamental voltage. 1

It is observed that by modulating the fundamental admittance with subsynchronous frequencies, SVC produces sub- and super- synchronous currents with controllable amplitude and phase, which subsequently generate subsynchronous damping torque on generator shaft. This is the basic principle of SSR mitigation using SVC. As shown in (10), this method simultaneously causes fractional currents.

#### *4.2.2. Configuration of the proposed SVC-SSDC*

262 Simulated Annealing – Advances, Applications and Hybridizations

**Figure 11.** The dynamic model of the SVC

*<sup>m</sup>* is the torsional frequency; *<sup>m</sup>*,ref

where *iB t*

<sup>0</sup> 1 1m <sup>0</sup> 0.5U ( )cos ( ) *<sup>m</sup> m m i Bt <sup>t</sup>*

1 1 10 0 U cos( )

the subsynchronous modulating control; else

components; and U is fundamental voltage. 1

*pk* <sup>1</sup> *<sup>i</sup>* denotes the characteristic harmonics; 0 ( 1) *<sup>m</sup> pk <sup>i</sup>*

this method simultaneously causes fractional currents.

frequencies, or

  d

SVC, generally d 0 *T T* / 24 , 0 / 6 *<sup>s</sup> T T* and 0 *T* =0.02 second.

1 1ref 1

*B BB*

FLT L

(8)

s T FLT L L (1 ) , [ ,0] 1 (1 ) *sT e B X*

*sT X BX X*

where *<sup>d</sup> T* and *<sup>s</sup> T* represent the dead and delay times of the thyristor circuits. For the six-pulse

To depress SSR, it is necessary for SVC to generate currents at the complementary subsynchronous frequencies. For this purpose, we propose the idea of modulating the fundamental admittance with subsynchronous frequencies, i.e., to make the reference value of the fundamental admittance ( 1ref *B* ) vary according to the subsynchronous

> 3 1ref 10,ref SSR SSR 1m ,ref 1

*m*

where 10,ref *B* , SSR *B* denote the DC and subsynchronously modulated components of 1ref *B* ;

is the initial phase.

S 1 1 ( 1) else

*ii i i i i i*

0 0 <sup>0</sup>

 

are the sub- and super- synchronous currents;

It is observed that by modulating the fundamental admittance with subsynchronous frequencies, SVC produces sub- and super- synchronous currents with controllable amplitude and phase, which subsequently generate subsynchronous damping torque on generator shaft. This is the basic principle of SSR mitigation using SVC. As shown in (10),

<sup>+</sup> *m m <sup>m</sup> pk pk*

 

( )cos( ) *m m*

(10)

 

is the fundamental current;

are fractional harmonics caused by

*i* represents the rest of the negligible

(9)

 

,

1 11

 

*m mk*

 

Thus, by some derivation, the current of SVC can be derived as

3 3

*B B B B Bt t*

The proposed SSDC is a real-time control system that works through the SVC powerelectronic circuit to inject sub- and super- synchronous currents into the generator-stators and produces damping torque to ease SSR. Fig. 12 shows the block diagram of the 3-path SVC-SSDC designed for the Jinjie system. The average mechanical speed of HIP turbines of the online machines provides the input, which becomes the standard signal HIP after proper conditioning. It is then passed to the low- & high-pass filter. Speed signal filtered next proceeds to three separate control paths. Each, tuned to a specific mode, comprises a modal filter, an amplifier *ki* and a unity-gain phase-shifter 2 2 (1 ) /(1 ) *k k sT sT* to generate the control signal for the corresponding mode. The control signals of all torsional modes are summarized, clipped and finally added to form the control signal, or SSR *B* , which drive the SVC circuit as shown in Fig. 12.

**Figure 12.** Configuration of SVC-SSDC

The purpose of the low- & high-pass filter, with the transfer functions shown in (11), is to depress the unrelated signals, including the DC, low-frequency (<10Hz) and high-frequency (>40Hz) components, to enable only related subsynchronous signals to pass through.

$$f\_{\rm L\&H}(\mathbf{s}) = \frac{1}{1 + s/\,\alpha\_{\rm L} + \left(\mathbf{s} \,/\,\alpha\_{\rm L}\right)^{2}} \frac{\left\{\mathbf{s} \,/\,\alpha\_{\rm H}\right\}^{2}}{1 + \mathbf{s} \,/\,\alpha\_{\rm H} + \left(\mathbf{s} \,/\,\alpha\_{\rm H}\right)^{2}}, \ \alpha\_{\rm L} = 2\pi \cdot 40, \ \alpha\_{\rm H} = 2\pi \cdot 10 \tag{11}$$

The modal filter is a series combination of a second-order band-pass filter and two secondorder band-stop filters, as described in (12):

$$f\_{\rm m'i}(\mathbf{s}) = \frac{\mathbf{s} \;/\; o\_{\rm Pl}}{1 + 8\pi\mathbf{s} \;/\; o\_{\rm Pl}^2 + \left(\mathbf{s} \;/\; o\_{\rm Pl}\right)^2} \frac{1 + \left(\mathbf{s} \;/\; o\_{\rm Bi1}\right)^2}{1 + 6\pi\mathbf{s} \;/\; o\_{\rm Bi1}^2 + \left(\mathbf{s} \;/\; o\_{\rm Bi1}\right)^2} \frac{1 + \left(\mathbf{s} \;/\; o\_{\rm Bi2}\right)^2}{1 + 6\pi\mathbf{s} \;/\; o\_{\rm Bi1}^2 + \left(\mathbf{s} \;/\; o\_{\rm Bi2}\right)^2} \tag{12}$$

where P1 B21 B31 1 =82.38 rad/s, P2 B11 B32 2 =143.07 rad/s, P3 B12 B22 3 =177.33 rad/s.

Then the SVC-SSDC can be reformulated into (13)

$$
\Delta B\_{\rm SSR} = H(\text{s}) \Delta o\_{\rm HIP} \tag{13}
$$

where *H s*( ) is the integrated transfer function of SVC-SSDC.

#### *4.2.3. Linearized model of the controlled system*

To formulate the control-design problem, the whole system is linearized around a certain operating point to obtain an open-loop linear model [22-24], as described in (14).

$$
\Delta \rho\_{\rm HIP} = G(s) \Delta B\_{\rm SSR} \tag{14}
$$

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 265

min{ , , }, 1, 0

subject to: ( ) , , 0 *i i ii*

*i i i i*

 

min , Re( ) / Im( ) ,

*Sk kT T i j*

calculation of the SSR mode of the included transfer function; the positive weight coefficients 123 4 *www w* 0.2, 0.4 ; the subscripts *i*, *j* denote the number of SSR mode

In (17), *R* is the H∞-norm of *R* , which represents the peak gain of *R* across all frequencies and can be calculated with the formula (19); *R*<sup>m</sup> is the desired maximum value

> *R Rj* max ( )

Practically, the output of SSDC is restricted by the SVC capacity. Therefore, to limit the control output and to improve robustness to additive model uncertainty, the transfer function from disturbance to control output, i.e. *R s*( ) in (13), should not be too large. In the other side, SVC rating has great effect on the tuning of control gains and the effectiveness of the closed-loop system. Generally, larger size of SVC leads to higher gains and better torsional damping. However, equipment cost and operation loss increase accordingly. Thus, SVC size was determined for achieved the balance between cost and performance, 240MVA,

The control-design problem (17)-(18) can be solved again by the proposed GA-SA method. In this case, GA-SA converges to satisfactory results after 40 - 60 iterations, consuming about 10-15 minutes on a modern computer. The optimized SVC-SSDC parameters are listed as follows: 1*k* 25.46 , 2 *k* 26.03 , 3 *k* 25.02 , 1 *T* 0.0070s , 2 *T* 0.0053s , 3 *T* 0.0025s . The

A thorough eigenvalue analysis has been conducted, first in the absence and then in the presence of SVC-SSDC, to check the damping performance of the closed-loop system. Results of the specific operating conditions are presented in Table 5. Comparisons between damping values with and without SVC-SSDC suggest that all torsional modes see

*f ww w w*

 

3 4 4 123 1 1

*i i*

1 31 3

*i ij ij ij ij <sup>j</sup>*

where ,*i i k T* are control parameters; *<sup>j</sup> S* is the sensitivity function;

of *R* ; and ub, ub, , *i i k T* are the upper bounds of the gains and time constants.

 

60 1

*j*

 

*ij i j*

*Rj R k k T T*

( ~ , ~ ) ; 1,2,3; 1,...,60

max

 

about 10% of the rated power of all Jinjie machines.

maximum fitness value generated is 0.00185.

*4.4.1. Small-signal eigen-analysis* 

**4.4. Verification of the optimized SVC-SSDC** 

and the operating condition respectively.

*f*

m ub, ub,

 

(17)

(19)

(18)

*<sup>i</sup>* means the

where *G s*( ) represents the open-loop transfer function from SSR *B* to HIP .

The closed-loop system model is obtained by combining (13) with (14), as shown in Fig.13, where the control reference *r* is set to 0; *d* denotes noise and/or model uncertainties. With some manipulation, the transfer functions between the input *d* and the outputs *y* , SSR *B* can be derived as:

$$y(s) = S(s)d(s) \tag{15}$$

$$
\Delta B\_{\rm SSR}(\mathbf{s}) = R(\mathbf{s})d(\mathbf{s})\tag{16}
$$

where <sup>1</sup> *Ss GsHs* () 1 () () is the sensitivity function, and *Rs HsSs* () ()() .

**Figure 13.** The close-loop system

For the target system, the open-loop and the closed-loop system function, i.e., *Gs Ss* ( ), ( ) have a maximum order of 93 and 258. Obviously, their poles that correspond to torsional modes represent the stability or damping of these modes for the open-loop and the closedloop systems, respectively. Hence, by eigen-analysis, the relationship between torsional damping with system variables and SVC-SSDC parameters can be traced out. With predetermined filtering functions of SVC-SSDC, the selection of gains and phase-shifts or , ( 1,2,3) *i i kTi* in Fig.12 is the key to solve the multimodal SSR problem.

#### **4.3. Optimal design of the SVC-SSDC based on GA-SA**

For the target system, sixty representative conditions are selected as the "evaluation conditions" (partly listed in Table 4), which cover the full range of generator output levels as well as the status of transmission lines. While these evaluation conditions by no means limit the situations under which the power system operates, together they form the edge of practical operating conditions and the most unfavorable situations relevant to SVC-SSDC. Therefore, the derived SVC-SSDC is robust enough to stabilize the system under all possible conditions.

The control-design task of SVC-SSDC can also be formulated into a nonlinear constrained optimization problem similar to that of (4)-(5). Specifically, it is expressed by (17)-(18).

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 265

$$\begin{aligned} \max f\\ \text{subject to: } \left| R(jo) \right|\_{o} \le R\_{\text{m}}, \left| k\_{i} \right| \le k\_{\text{ub},i'} \quad 0 \le T\_{i} \le T\_{\text{ub},i} \end{aligned} \tag{17}$$

$$\begin{aligned} f &= \sum\_{i=1}^{3} w\_i \eta\_i + w\_4 \min\{\eta\_1, \eta\_2, \eta\_3\} \sum\_{i=1}^{4} w\_i = 1, w\_i > 0\\ \eta\_i &= \min\_{j=1}^{j=60} \{\sigma\_{ij}\} \text{ , } \sigma\_{ij} = -\text{Re}(\mathcal{A}\_{ij}) / \left| \text{Im}(\mathcal{A}\_{ij}) \right| \,, \\ \mathcal{A}\_{ij} &= \lambda\_i \{S\_j (k\_1 \sim k\_3, T\_1 \sim T\_3) \}; i = 1, 2, 3; j = 1, \dots, 60 \end{aligned} \tag{18}$$

where ,*i i k T* are control parameters; *<sup>j</sup> S* is the sensitivity function; *<sup>i</sup>* means the calculation of the SSR mode of the included transfer function; the positive weight coefficients 123 4 *www w* 0.2, 0.4 ; the subscripts *i*, *j* denote the number of SSR mode and the operating condition respectively.

In (17), *R* is the H∞-norm of *R* , which represents the peak gain of *R* across all frequencies and can be calculated with the formula (19); *R*<sup>m</sup> is the desired maximum value of *R* ; and ub, ub, , *i i k T* are the upper bounds of the gains and time constants.

$$\left| R \right|\_{\Rightarrow} = \max\_{o} \left| R(j\nu) \right| \tag{19}$$

Practically, the output of SSDC is restricted by the SVC capacity. Therefore, to limit the control output and to improve robustness to additive model uncertainty, the transfer function from disturbance to control output, i.e. *R s*( ) in (13), should not be too large. In the other side, SVC rating has great effect on the tuning of control gains and the effectiveness of the closed-loop system. Generally, larger size of SVC leads to higher gains and better torsional damping. However, equipment cost and operation loss increase accordingly. Thus, SVC size was determined for achieved the balance between cost and performance, 240MVA, about 10% of the rated power of all Jinjie machines.

The control-design problem (17)-(18) can be solved again by the proposed GA-SA method. In this case, GA-SA converges to satisfactory results after 40 - 60 iterations, consuming about 10-15 minutes on a modern computer. The optimized SVC-SSDC parameters are listed as follows: 1*k* 25.46 , 2 *k* 26.03 , 3 *k* 25.02 , 1 *T* 0.0070s , 2 *T* 0.0053s , 3 *T* 0.0025s . The maximum fitness value generated is 0.00185.

#### **4.4. Verification of the optimized SVC-SSDC**

#### *4.4.1. Small-signal eigen-analysis*

264 Simulated Annealing – Advances, Applications and Hybridizations

*4.2.3. Linearized model of the controlled system* 

can be derived as:

where <sup>1</sup> *Ss GsHs* () 1 () ()

**Figure 13.** The close-loop system

conditions.

To formulate the control-design problem, the whole system is linearized around a certain

HIP SSR

The closed-loop system model is obtained by combining (13) with (14), as shown in Fig.13, where the control reference *r* is set to 0; *d* denotes noise and/or model uncertainties. With some manipulation, the transfer functions between the input *d* and the outputs *y* , SSR *B*

is the sensitivity function, and *Rs HsSs* () ()() .

For the target system, the open-loop and the closed-loop system function, i.e., *Gs Ss* ( ), ( ) have a maximum order of 93 and 258. Obviously, their poles that correspond to torsional modes represent the stability or damping of these modes for the open-loop and the closedloop systems, respectively. Hence, by eigen-analysis, the relationship between torsional damping with system variables and SVC-SSDC parameters can be traced out. With predetermined filtering functions of SVC-SSDC, the selection of gains and phase-shifts or

For the target system, sixty representative conditions are selected as the "evaluation conditions" (partly listed in Table 4), which cover the full range of generator output levels as well as the status of transmission lines. While these evaluation conditions by no means limit the situations under which the power system operates, together they form the edge of practical operating conditions and the most unfavorable situations relevant to SVC-SSDC. Therefore, the derived SVC-SSDC is robust enough to stabilize the system under all possible

The control-design task of SVC-SSDC can also be formulated into a nonlinear constrained optimization problem similar to that of (4)-(5). Specifically, it is expressed by (17)-(18).

, ( 1,2,3) *i i kTi* in Fig.12 is the key to solve the multimodal SSR problem.

**4.3. Optimal design of the SVC-SSDC based on GA-SA** 

*Gs B* ( ) (14)

*y*() ()() *s Ssds* (15)

SSR *B s Rsds* () ()() (16)

HIP .

operating point to obtain an open-loop linear model [22-24], as described in (14).

where *G s*( ) represents the open-loop transfer function from SSR *B* to

A thorough eigenvalue analysis has been conducted, first in the absence and then in the presence of SVC-SSDC, to check the damping performance of the closed-loop system. Results of the specific operating conditions are presented in Table 5. Comparisons between damping values with and without SVC-SSDC suggest that all torsional modes see


considerable improvement in modal damping by using the optimized SSDC and all unstable modes are well stabilized.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 267

> (b) Delta mechanical speed of the HIP turbine (with SVC-SSDC)

To test the SVC-SSDC in the presence of large disturbances, a three-phase short-circuit is triggered first on the SSDC-free and then, on the SSDC-controlled system. The results are

(a) Delta mechanical speed of the HIP turbine (b) The mode- 3component of TCR current

(c) TCR current (with SVC-SSDC) (d) Spectrum of TCR current (with SVC-SSDC)

As shown in Fig.15 (a), in the absence of SSDC, the faulted system is unstable and soon collapses. However, when SVC-SSDC is put into service, the subsynchronous oscillation is successfully damped out (see Fig.15(b)). Fig.15(c) displays SVC's current output during the experiment. For a short period of time after the fault (about 5 seconds), the magnitude of the feedback is so large that the SSDC works nearly in a "bang-bang" mode due to the intense impact of the fault. As the vibration is weakened, the SSDC output becomes attenuated accordingly. From the spectrum of the TCR current shown in Fig.15(d), it can be seen that

**Figure 14.** System response during the switching of the Jin-Xin SCs and SVC-SSDC

(a) Delta mechanical speed of the HIP turbine (without SVC-SSDC)

**Figure 15.** System response following a three-phase fault

illustrated in Fig. 15.

**Table 5.** Real Parts of SSR modes without and with SVC-SSDC under the 4-machine operating conditions (Note: The "m+n" of the "lines" column means that there are "m" Jinjie-Xinzhou lines and "n" Jinzhou-Shibei lines in service.)

## *4.4.2. EMT study with real-time digital simulator*

To further investigate the performance of the proposed control scheme, real SSDC controllers were developed using digital signal processors. The controllers were then connected to real-time digital simulator (RTDS), which thoroughly incorporated all system nonlinearities, including the power electronic SVC. Real-time electromagnetic simulations were conducted to check the response of the controlled system in the presence of various disturbances to validate the effectiveness of the developed controllers. Some typical results are presented here.

Preliminary study shows that, when only three Xin-Shi SCs are in service, the system is SSRstable. However, once Jin-Xin SCs are switched on, the system becomes unstable. Fig. 14 depicts the responses of the system during consecutive switch-ons of two Jin-Xin SCs and SVC-SSDC. At the initial stage, Xin-Shi SCs are in service while Jin-Xin SCs and SVC-SSDC are out of service. When #1 Jin-Xin SC is switched on, the system is still stable, as indicated by the converging speed deviation in Fig.14(a). However, once #2 Jin-Xin SC is added in, SSR diverges rapidly and the generators experience violent torsional vibration, as indicated by the exponentially growing delta mechanical speed. This would probably lead to great damage to generator shafts. When SVC-SSDC is applied, SSR attenuates immediately. Fig.14(b) shows the current component of TCR corresponding to torsional mode 3, which is the most intractable among the three SSR modes.

To test the SVC-SSDC in the presence of large disturbances, a three-phase short-circuit is triggered first on the SSDC-free and then, on the SSDC-controlled system. The results are illustrated in Fig. 15.

**Figure 14.** System response during the switching of the Jin-Xin SCs and SVC-SSDC

266 Simulated Annealing – Advances, Applications and Hybridizations

modes are well stabilized.

"n" Jinzhou-Shibei lines in service.)

are presented here.

*4.4.2. EMT study with real-time digital simulator* 

the most intractable among the three SSR modes.

Gen. output

considerable improvement in modal damping by using the optimized SSDC and all unstable

lines No SVC With SVC No SVC With SVC No SVC With SVC

Oper. Con. Mode 1 (s-1) Mode 2 (s-1) Mode 3 (s-1)

4×0% 2+3 -0.0370 -0.1524 -0.0060 -0.3903 **0.3202** -0.7235 4×40% 2+3 -0.0378 -0.2174 -0.0262 -0.4796 **0.3099** -0.6081 4×100% 2+3 -0.0367 -0.3256 -0.0468 -0.5863 **0.3062** -0.4118 4×0% 2+2 -0.0323 -0.1566 **0.0001** -0.4559 **0.5708** -2.0244 4×40% 2+2 -0.0333 -0.2249 -0.0200 -0.5479 **0.5620** -1.6510 4×100% 2+2 -0.0326 -0.3376 -0.0404 -0.6581 **0.5649** -1.1766 4×0% 1+3 -0.0156 -0.1596 **0.0223** -0.6877 **0.3795** -0.5840 4×40% 1+3 -0.0175 -0.2437 -0.0023 -0.7942 **0.3620** -0.8436 4×100% 1+3 -0.0182 -0.3792 -0.01760 -0.9205 **0.3480** -1.2701 4×0% 1+2 -0.0132 -0.1645 **0.0305** -0.7615 **0.2301** -0.4166 4×40% 1+2 -0.0152 -0.2495 -0.0105 -0.8712 **0.2108** -0.6349 4×100% 1+2 -0.0161 -0.3864 -0.0090 -1.0017 **0.1902** -0.9860

**Table 5.** Real Parts of SSR modes without and with SVC-SSDC under the 4-machine operating conditions (Note: The "m+n" of the "lines" column means that there are "m" Jinjie-Xinzhou lines and

To further investigate the performance of the proposed control scheme, real SSDC controllers were developed using digital signal processors. The controllers were then connected to real-time digital simulator (RTDS), which thoroughly incorporated all system nonlinearities, including the power electronic SVC. Real-time electromagnetic simulations were conducted to check the response of the controlled system in the presence of various disturbances to validate the effectiveness of the developed controllers. Some typical results

Preliminary study shows that, when only three Xin-Shi SCs are in service, the system is SSRstable. However, once Jin-Xin SCs are switched on, the system becomes unstable. Fig. 14 depicts the responses of the system during consecutive switch-ons of two Jin-Xin SCs and SVC-SSDC. At the initial stage, Xin-Shi SCs are in service while Jin-Xin SCs and SVC-SSDC are out of service. When #1 Jin-Xin SC is switched on, the system is still stable, as indicated by the converging speed deviation in Fig.14(a). However, once #2 Jin-Xin SC is added in, SSR diverges rapidly and the generators experience violent torsional vibration, as indicated by the exponentially growing delta mechanical speed. This would probably lead to great damage to generator shafts. When SVC-SSDC is applied, SSR attenuates immediately. Fig.14(b) shows the current component of TCR corresponding to torsional mode 3, which is

**Figure 15.** System response following a three-phase fault

As shown in Fig.15 (a), in the absence of SSDC, the faulted system is unstable and soon collapses. However, when SVC-SSDC is put into service, the subsynchronous oscillation is successfully damped out (see Fig.15(b)). Fig.15(c) displays SVC's current output during the experiment. For a short period of time after the fault (about 5 seconds), the magnitude of the feedback is so large that the SSDC works nearly in a "bang-bang" mode due to the intense impact of the fault. As the vibration is weakened, the SSDC output becomes attenuated accordingly. From the spectrum of the TCR current shown in Fig.15(d), it can be seen that during the controlling process, the TCR current contains harmonics at the frequencies of <sup>0</sup> (2 1) *<sup>m</sup> n* . This is in line with the previous theoretic analysis.

Genetic Algorithm and Simulated Annealing: A Combined Intelligent Optimization Method and Its Application to Subsynchronous Damping Control in Electrical Power Transmission Systems 269

[4] IEEE Committee Report. First supplement to a bibliography for study of subsynchronous resonance between rotating machines and power systems. *IEEE Trans.* 

[5] IEEE Committee Report. Second supplement to a bibliography for study of subsynchronous resonance between rotating machines and power systems. *IEEE Trans.* 

[6] IEEE Committee Report. Third supplement to a bibliography for study of subsynchronous resonance between rotating machines and power systems. paper no. 90 SM 328-5-PWRS presented at the PES Summer Meeting, Minneapolis, Minnesota, July

[7] IEEE Subsynchronous Resonance Working Group of the System Dynamic Performance Subcommittee Power System Engineering Committee. Countermeasures to subsynchronous resonance problems. *IEEE Trans. Power Apparatus and Systems*. 1980;

[8] IEEE Committee Report, Terms, definitions and symbols for subsynchronous oscillations. IEEE Trans. Power Apparatus and Systems. 1985, 104(3): 1326-1334. [9] V. Pottakulath, E. P. Cheriyan, R. S. Kumar. Synthesis of power system model for SSR

[10] P. M. Anderson, B. L. Agrawal, J. E. Van Ness, *Subsynchronous Resonance in Power* 

[11] Liwei Wang, J. Jatskevich, H. W. Dommel, Re-examination of synchronous machine modeling techniques for electromagnetic transient simulations. *IEEE Trans. Power* 

[12] U. Karaagac, J. Mahseredjian, O. Saad, S. Dennetiere. Synchronous machine modeling precision and efficiency in electromagnetic transients. *IEEE Trans. Power Delivery*, 2011,

[13] Donghui Zhang, Xiaorong Xie, Shiyu Liu, Shuqing Zhang. An intelligently optimized SEDC for multimodal SSR mitigation. *Electric Power Systems Research 2009*, 79(7): 1018-

[14] Kit Po Wong, Yin Wa Wong. Combined genetic algorithm/simulated annealing/fuzzy set approach to short-term generation scheduling with take-or-pay fuel contract. *IEEE* 

[15] Mantawy AH, Abdel-Magid YL, Selim SZ. Integrating genetic algorithms, tabu search, and simulated annealing for the unit commitment problem. *IEEE Transactions on Power* 

[16] Thompson M, Fidler JK. Application of the genetic algorithm and simulated annealing to LC filter tuning. *IEE Proceedings-Circuits Devices and System* 2001; 148(4): 177-182. [17] Tang Renyuan, Yang Shiyou, Li Yan; Wen Geng, Mei Tiemin. Combined strategy of improved simulated annealing and genetic algorithm for inverse problem. IEEE

[18] Lee YK, Mohamed PS. A real-coded genetic algorithm involving a hybrid crossover method for power plant control system design. in *Proceedings of 2002 Congress on* 

analysis. in *Proc. TENCON 2010 IEEE Region 10 Conf.*, 2010, pp. 545–550.

*Systems*. New York, NY: Wiley-IEEE Press, Feb., 1999.

*Transactions on Power System* 1996; 11(1): 128-136.

Transactions on Magnetics 1996; 32(3): 1326-1329.

*Evolutionary Computation*, pp.1069– 1074, Honolulu, Hawaii 2002.

*Power Apparatus and Systems*. 1979, 98(6): 1872-1875.

*Power Apparatus and Systems*. 1985, 104(2): 321-327.

1990.

99(5): 1810-1818.

26(2): 1072-1082.

1024.

*Systems*, 2007, 22(3): 1221–1230.

*Systems* 1999; 14(3): 829-836.

Besides the exemplary experiment, simulations under other operating conditions have been conducted in the presence of different large disturbances. From Figs.14-15 and other results not presented here, it is concluded that SVC-SSDC is effective in improving torsional damping and can satisfactorily mitigate subsynchronous vibrations caused by any type of disturbance.

## **5. Conclusions**

In this chapter, a novel intelligent tuning technique based on the combined genetic algorithm and simulated annealing, or GA-SA, was proposed and explicated for the design of optimized controllers to depress SSR in practical multi-machine transmission systems compensated with fixed series capacitors. The advantage of the proposed method lies in that: by solving the constrained optimization problem, the multiple control parameters are simultaneously optimized and the obtained controller is robustified to damp multimodal SSR under a wide range of operating conditions. The proposed method is applied to the control-design problems of two practical power systems, i.e., SEDC of the Shangdu system and SVC-SSDC of the Jinjie system. The dynamics of the controlled system are investigated via eigenvalue analyses, electromagnetic simulation and/or field tests, the results of which fully demonstrate the effectiveness of the GA-SA tuned controllers.

## **Author details**

Xiaorong Xie *State Key Lab. of Power System, Department of Electrical Engineering, Tsinghua University, Beijing, China* 

## **Acknowledgement**

This work was supported in part by National Natural Science Foundation of China (Grant No. 51077080 and 51037002) and State Key Lab. of Power System (Grant No.SKLD11M02).

## **6. References**


[4] IEEE Committee Report. First supplement to a bibliography for study of subsynchronous resonance between rotating machines and power systems. *IEEE Trans. Power Apparatus and Systems*. 1979, 98(6): 1872-1875.

268 Simulated Annealing – Advances, Applications and Hybridizations

<sup>0</sup> (2 1) *<sup>m</sup> n* 

disturbance.

**5. Conclusions** 

**Author details** 

**Acknowledgement** 

*Trans.Power System*. 1992; 7(1):150-157.

*Systems*. 1976, 95(1): 216-218.

Xiaorong Xie

*Beijing, China* 

**6. References** 

during the controlling process, the TCR current contains harmonics at the frequencies of

Besides the exemplary experiment, simulations under other operating conditions have been conducted in the presence of different large disturbances. From Figs.14-15 and other results not presented here, it is concluded that SVC-SSDC is effective in improving torsional damping and can satisfactorily mitigate subsynchronous vibrations caused by any type of

In this chapter, a novel intelligent tuning technique based on the combined genetic algorithm and simulated annealing, or GA-SA, was proposed and explicated for the design of optimized controllers to depress SSR in practical multi-machine transmission systems compensated with fixed series capacitors. The advantage of the proposed method lies in that: by solving the constrained optimization problem, the multiple control parameters are simultaneously optimized and the obtained controller is robustified to damp multimodal SSR under a wide range of operating conditions. The proposed method is applied to the control-design problems of two practical power systems, i.e., SEDC of the Shangdu system and SVC-SSDC of the Jinjie system. The dynamics of the controlled system are investigated via eigenvalue analyses, electromagnetic simulation and/or field tests, the results of which

. This is in line with the previous theoretic analysis.

fully demonstrate the effectiveness of the GA-SA tuned controllers.

*State Key Lab. of Power System, Department of Electrical Engineering, Tsinghua University,* 

This work was supported in part by National Natural Science Foundation of China (Grant No. 51077080 and 51037002) and State Key Lab. of Power System (Grant No.SKLD11M02).

[1] Subsynchronous Resonance Working Group of the System Dynamic Performance Subcommittee. Reader's guide to subsynchronous resonance Power Systems. *IEEE* 

[2] J. W. Balance, S. Goldberg. Subsynchronous resonance in series compensated transmission lines. *IEEE Trans. Power Apparatus and Systems*. 1973, 92(5): 1649-1658. [3] IEEE Committee Report. A bibliography for study of subsynchronous resonance between rotating machines and power systems. *IEEE Trans. Power Apparatus and* 


	- [19] X. Xie, X. Guo, Y. Han. Mitigation of multimodal SSR using SEDC in the Shangdu series-compensated power system. *IEEE Trans. Power Systems*, 2011, 26(1): 384-391.

**Chapter 0**

**Chapter 13**

**Fuzzy c-Means Clustering, Entropy Maximization,**

Many engineering problems can be formulated as optimization problems, and the deterministic annealing (DA) method [20] is known as an effective optimization method for such problems. DA is a deterministic variant of simulated annealing (SA) [1, 10]. The DA characterizes the minimization problem of cost functions as the minimization of Helmholtz free energy which depends on a (pseudo) temperature, and tracks the minimum of free energy while decreasing temperature and thus it can deterministically optimize the function at a given temperature [20]. Hence, the DA is more efficient than the SA, but does not guarantee a global optimal solution. The study on the DA in [20] addressed avoidance of the poor local minima of cost function of data clustering. Then it was extensively applied to various subjects such as combinational optimization problems [21], vector quantization [4], classifier design

On the other hand, clustering is a method which partitions a given set of data points into subgroups, and is one of major tools for data analysis. It is supposed that, in the real world, cluster boundaries are not so clear that fuzzy clustering is more suitable than crisp clustering. Bezdek[2] proposed the fuzzy c-means (FCM) which is now well known as the standard

Then, after the work of Li et al.[11] which formulated the regularization of the FCM with Shannon entropy, Miyamoto et al.[14] discussed the FCM within the framework of the Shannon entropy based clustering. From the historical point of view, however, it should be noted that Rose et al.[20] first studied the statistical mechanical analogy of the FCM with the

To measure the "indefiniteness" of fuzzy set, DeLuca and Termini [6] defined fuzzy entropy after Shannon. Afterwards, some similar measures from the wider viewpoints of the indefiniteness were proposed [15, 16]. Fuzzy entropy has been used for knowledge retrieval

Tsallis [24] achieved nonextensive extension of the Boltzmann-Gibbs statistics. Tsallis postulated a generalization form of entropy with a generalization parameter *q*, which, in a

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

©2012 Yasuda, 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

© 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,

maximum entropy method, which was basically probabilistic clustering.

from fuzzy database [3] and image processing [31], and proved to be useful.

**and Deterministic and Simulated Annealing**

Additional information is available at the end of the chapter

Makoto Yasuda

**1. Introduction**

http://dx.doi.org/10.5772/48659

[13], pairwise data clustering [9] and so on.

cited.

technique for fuzzy clustering.

