Interaction between phase shifting transformers installed in the tie ...

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Electrical Power and Energy Systems 33 (2011) 1351–1360

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Interaction between phase shifting transformers installed in the tie-lines of interconnected power systems and automatic frequency controllers Desire Rasolomampionona a, Sohail Anwar b,⇑ a b

Warsaw University of Technology, Warsaw, Poland Pennsylvania State University, Altoona College, 3000 Ivyside Park, Altoona, PA 16601, USA

a r t i c l e

i n f o

Article history: Received 23 November 2009 Received in revised form 11 March 2011 Accepted 3 June 2011 Available online 22 July 2011 Keywords: Flexible AC transmission systems Power system stability Automatic generation control

a b s t r a c t FACTS devices like TCPAR can be used to regulate the power flow in tie-lines of interconnected power system. The transient state power flow occurring after power disturbances can be influenced by using TCPAR equipped with power regulator and frequency-based stabilizer. The analysis of a simple interconnected power system consisting of two power systems has shown that the control of TCPAR can force a good damping of both power swings and oscillations of local frequency. In the case of a larger interconnected power system consisting of more than two power systems, the influence of the control of TCPAR on damping can be more complicated. Strong damping of local frequency oscillations and power swings in one tie-line may cause larger oscillations in remote tie-lines and other systems. Hence the use of devices like TCPAR as tools for damping power swings and frequency oscillations in a large interconnected power system must be justified by detailed analysis of power system dynamics. In this paper, some results of time-domain simulations of a three-system area are presented. These results have proven that it is possible to obtain a good damping of tie-line power and frequency swings by optimizing the main parameters of TCPAR installed in tie-lines. The results have been confirmed by an eigenvalue analysis of linearized model of interconnected system consisting of three subsystems. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction Worldwide transmission systems are undergoing continuous changes and restructuring. They are becoming more heavily loaded and are being operated in ways not originally envisioned. Transmission systems must be flexible to react to more diverse generation and load patterns. In addition, the economical utilization of transmission system assets is of vital importance to enable utilities in industrialized countries to remain competitive and to survive. In developing countries, the optimized use of transmission systems investments is also important to support industry, create employment, and efficiently utilize scarce economic resources. Classical automatic generation and power control should be modified as a result of deregulation of electric power energy sector and introduction of market mechanism in power trade. The restructuring of management and propriety in the power system sector should be accompanied at least by a partly decentralized power system control. The concept of utilizing power electronic devices for power system control is widely accepted presently. Flexible AC Transmission Systems (FACTS) permit the independent adjustment of certain system variables like power flows. Flexible AC Transmission Sys⇑ Corresponding author. Tel.: +1 8149495181; fax: +1 8149495190. E-mail address: [email protected] (S. Anwar). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.06.001

tems (FACTS) is a technology that responds to these needs. It significantly alters the way transmission systems are developed and controlled along with the improvements in asset utilization, system flexibility, and system performance. FACTS devices are used for the dynamic control of voltage, impedance, and phase angle of high voltage AC transmission lines. According to several experts, FACTS devices can be used as tools in control decentralization. These devices are used for voltage and power regulation. Their role in the regulation processes of interconnected power systems is not fully analyzed yet. The introduction of far-reaching changes in power system control is a problem requiring a very careful analysis. There are some circumstances which lead to an apprehension that a full control decentralization can endanger the power system security. Any change in power system control, among others, the use of FACTS devices in transmission network should be preceded by detailed studies. It has been shown FACTS devices like TCPAR or UPFC can participate in the power system control in a perfect manner, both in steady-state and in transient states, for power post-disturbance swing damping. There is little information available regarding the influence of the use of these devices on slow dynamic system response accompanying the power and frequency regulation after a long-term imbalance of tie-line interchange power. New fast FACTS-based devices may be used for the compensation of sudden load change devices. Some authors [1–7], use SMES


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D. Rasolomampionona, S. Anwar / Electrical Power and Energy Systems 33 (2011) 1351–1360

devices in order to suppress frequency deviations through tie-line power regulation. Also, a case of using SSSC for frequency regulation is noted [8]. Thyristor Controlled Phase Shifter (TCPS) is expected to be an effective apparatus for the tie-line power flow control of an interconnected power system [9–11]. So far, only four papers [12–15] related to the TCPS application in the load frequency control have been published. In this paper, some results of a research financed by the Polish SCSR (State Committee for Scientific Research) are presented. The problem concerning the coordination of the FACTS devices like TCPAR, used for tie-line power control in interconnected power systems equipped with AGC is presented in this paper.

2. An interconnected power system composed of two subsystems Load frequency control (LFC) is one of the major requirements in providing reliable and quality operation in multi-area power systems. In the interconnected large power systems, variations in frequency can lead to serious large scale stability problems. Frequency is one of the stability criteria for large-scale stability of power networks. For stable operation, constant frequency and active power balance must be provided. Frequency is dependent on active power. Any change in active power demand/generation at power systems is reflected throughout the system by a change in the frequency. The problem of the interconnection exchange was analyzed first by Elgerd [16,17] and a critical review of the application of modern control theory to the AGC was presented by Carpentier [18]. Various components of power systems are in general, non-linear. Thus the operating point of a power system may change very much during a daily cycle. However, if the time horizon to be analyzed does not exceed a few minutes, linearized models are used [16,17,19]. Many authors have investigated the AGC problems using variable structure controllers. Chan and Hsu [20], Kumar et al. [21] analyzed a two-area power system with non-reheat and reheat thermal systems using variable structure controllers. Some authors [22,23] suggested the use of adaptive techniques in AGC design. However, a report [24] prepared under the auspices of IEEE Power System Engineering Committee, pointed out that these methods are unrealistic for the multiarea power systems. Most of the work related to the LFC of interconnected power systems is based on the tie-line bias control strategy [16,17,22,23]. Investigations conducted since the eighties are geared towards the use of new technologies in controlling the ACE. Supplementary controllers are used to regulate the ACE to zero in a more effective way. In the interconnected power systems, automatic generation control is implemented in such a way that each area, or subsystem, has its own central regulator. As shown in Fig. 1, the power system is in equilibrium if, for each area, the total power generation PT, the

total power demand PL, and the net tie-line interchange power Ptie satisfy the condition:

PT ðPL þ Ptie Þ ¼ 0

The objectives of each area regulator are to maintain frequency at the scheduled level (frequency control) and to maintain net tieline interchanges from the given area at the scheduled values (tieline control). The regulation is executed by changing the power output of the turbines in the area through varying Pref in their governing systems. Fig. 2 shows a functional diagram of the central regulator. Frequency is measured in the local low voltage network and compared with the reference frequency to produce a signal that is proportional to the frequency deviation Df. The information regarding the power flow in the tie-lines is sent via telecommunication lines to the central controller which compares it with the reference value in order to produce a signal proportional to the tie-line interchange error DPtie. In the past, the central control of frequency and power interchanges was implemented using large analog controllers. Later on, this has been replaced by a central computer, which also performed security assessment analysis and economic dispatch in normal operating conditions, i.e. allocation of the generated power among the power plants in a way that minimizes the production cost. Should an emergency occur, the computer may change the allocation of power in order to prevent system blackout. A detailed description of these facilities is beyond the scope of this paper. The DPref signals, resulting from the frequency and tie-line control, and the Pref signals, resulting from the economic dispatch or emergency control, are transmitted to the power plants using the same telecommunication links. In a deregulated electricity market, part of the system reliability is dependent on having adequate generation reserve in the system. The reserves are classified into regulation, 10-min spinning reserves, and 10- and 30-min non-synchronous reserves, depending on the ramping capability of the available generators. Many generators are operated quite conservatively because they have to observe temperature and pressure constraints. However, with the availability of advanced sensors, it is possible to use additional feedback control to reduce transients internal to turbines such that they can be more responsive and at the same time, operated more reliably. For example, if the pressure variable in the penstock of a hydro turbine is measured, then a pressure feedback loop can be implemented on the turbine valve to reduce the water column oscillations in the penstock, thus allowing the hydro turbine to have a faster ramp rate. Having faster generator response rates will generally result in lower energy and reserve prices, allowing a power system to operate more efficiently. The frequency regulation process can be approximately analyzed using a simplified mathematical model, which considers the dynamics of rotors, turbines and their respective prime movers [25–27]. With regard to this fact and the derived equations, after

f ref

PT

control area

P tie

remainder control areas

ð1Þ

P tie

f + – Σ

ref

– Σ + P tie

Δf

λR

ΔP tie

ΔP f – ΔP ref ACE PI Σ –

α2 αi αn

PL Fig. 1. Power balance of a control area.

α1

Fig. 2. Functional diagram of a central regulator.

ΔP ref1 ΔP ref2 ΔP refi ΔP refN



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introducing the following symbols: GT(s), GH(s), 1k which are, respectively: the equivalent transfer functions of turbines, prime movers, secondary controllers (frequency controllers), the derivation of the appropriate relations and performance of the adequate transformations leads to the block diagram shown in Fig. 3. Maintaining a primary regulation margin at the set level is a centrally co-ordinated service provided by the transmission system operator (TSO). The primary control goal is to automatically increase/decrease the power of primary control ranked sources within a few seconds (in the scope of set control margin) in order to balance frequency deviations. The primary control has a proportional character and contributes to maintaining a balance between generation and consumption using a turbine power or speed regulator. The right part of Fig. 4 represents the power system with its turbines and prime mover control (primary regulation). The amplification of the feed-back loop is indicated by the symbol R. The left part is the frequency control (secondary regulation). The secondary control goal is to maintain frequency at the nominal (required) value and balance with the interconnected systems at the required value. Secondary control must be harmonized with the primary control. The primary control possibilities are preferred to frequency deviations, and secondary control is applied when a frequency deviation persists or in the case of a deviation from the agreed balance. The relation between the tie-line power DPtie and frequency deviation Df in the interconnected subsystems can be determined using the model described in [25–27]. Tie-line power flow deviations depend on the terminal line voltage angle and are strictly linked to frequency deviations. Interconnection may necessitate reinforcement of existing networks for stability or other reliability considerations. The reinforcement may be assisted by the deployment of Flexible AC Transmission System (FACTS) devices and computer-controlled energy management system (EMS). Flexible AC Transmission System technologies are aimed to install power electronic devices at the proper locations of the existing AC systems to improve their steady-state and dynamic behavior and to keep the preset power transfer. Generally, the objectives of FACTS technology are to enhance system controllability and to increase power transfer limit. Detailed simulation of possible steady state, dynamic state and transient state operations must be studied based on the proposed interconnection schemes, taking into account, the applications of various FACTS devices.

where c is the TCPAR quadrature component, EA, EB, XR are the equivalent emfs of both of the subsystems and the equivalent reactance of the whole transmission system (the TCPAR device reactance included), respectively. The emfs’ arguments are dA, dB respectively. In the expression (2), the angle d = (dA dB). The prefault values of the variables at the steady-state are marked by a ‘‘hat’’ (^) inserted on the top of the appropriate symbol. They are, b tie ; ^ ^. Differentiating the expression (2) in the respectively P d; and c neighbourhood of the given values yields:

2.1. The mathematical model

DPtie ¼

In order to set the symbols used in this paper, the corresponding block diagram for an interconnected power system composed of

After some simple mathematical transformations, the following expression is obtained

two subsystems A and B, including AGC has been used. GT(s), GRT(s) are the equivalent transfer functions of turbines and governors of the same system. GSEE(s) is the equivalent transfer functions of the power system and is associated with the rotor dynamics. R is the droop of the speed-drop characteristic resulting from the feed-back loop of the primary control. GRf(s) is the central regulator transfer function. kR is the setting value of frequency bias factor, representing the frequency deviation in the frequency regulation loop of the central regulator. More detailed information about numerical values, forms and derivations of different transfer functions is given in [25–27]. The right-middle part of the model presented in Fig. 4 shows the interconnected system tie-line. If a TCPAR device is installed on this line, the tie-line model should take into account the TCPAR model and its control. The interchange power increment DPtie is referred to the power reference setting value. The power system model including the TCPAR model can be obtained from the incremental matrix equation of active power including the TCPAR equation. The equivalent system obtained above is presented in Fig. 5. The TCPAR device can be replaced by an equivalent reactance [28] and a transformation ratio g = b + jc. The direct component b of this ratio influences the voltage level control and the reactive power flow. Similarly, the quadrature component c influences the voltage angle control and the active power flow. The TCPAR device is much simpler than UPFC device and controls the quadrature component c only. Using the nodal method in the power system in Fig. 2, it can be demonstrated as in [28] that the interchange tie-line active power is given by the following expression:

Ptie ¼ bsind cb cos d;

b¼

jEA jjEB j XR

@Ptie @Ptie D d þ Dc @d d¼^d @ c c¼c^

ð3Þ

secondary control

primary control

Δf (s )

1

λ

Δf (s )

−

KI s

ΔPref (s ) +

Σ

ΔPL (s )

ΔPT (s )

-

GH

GT

+

G p (s ) -

Σ

Kp 1 + sT p

Fig. 3. Power system frequency control block diagram (Kirchmayer (1959)).

ð2Þ

Δf (s )


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f 0 -f

A

-1 Δf A 1 RA

System A

λ RA

+ Σ

GRfA (s)

+

+ Σ

GRTA (s) GTA (s)

-

+

Σ

+ Σ

GRfB (s)

+

+

Σ

GRTB (s) GTB (s) +

1 RB

λ RB

Δf B

f A -f 0 A GSEE (s)

Δf

A

-

Δf A + ΔPtie Tie-line equipped with TCPAR device – Δf B B + ΔPtie Δf B GSEE (s) Σ + f B -f 0 ΔP0 B

ΔPtie ΔPtie

System B

ΔP0A -

-1 f 0 -f

B

Fig. 4. Block diagram of frequency regulation in a two-area power system composed of subsystems A and B.

{L A }

{L B } B P tie

A Ptie

“a”

GA

“b”

GB

TCPAR N Fig. 5. Illustration of the interconnection of two power system areas, with a TCPAR device installed on the tie-line.

h i b tie þ c b tie Dc b tie Dd H ^P DPtie ¼ ð1 þ c^2 Þ H

ð4Þ

TCPAR control model

where

b tie H

@Ptie ¼ ¼ b cos ^d @d d¼^d;c^¼0

ð5Þ

Δγ

The model corresponding to the expression (4) is shown in Fig. 6.

ΔPtie 3. Simulation results for an interconnected power system composed of three subsystems Basic data of the three systems are presented in Fig. 8. The block diagram of this system is built in the same way as for the two subsystem model presented in Fig. 4, but it is based on three subsystems with three different AGC systems and three tie-lines. A full presentation of the block diagram can be found in [25]. Simulation results have also shown that the best influence on transient time responses has been obtained when the TCPAR was equipped with a power control system having a frequency-based PSS (Fig. 8). At the beginning of the simulation analysis, it has been assumed that only one TCPAR device is installed in the tie-line between the subsystems A and B. For powers of different subsystems equal to power reference values, this TCPAR can have an influence upon the loads of the tie-lines A–B and A–C to balance the total power flowing in those tie-lines. In other words, the TCPAR can unburden the line A–B at the cost of line A–C and vice-versa. In this case, and for different configuration of TCPAR parameters and localization, it

Δf

H tie _

A

2π s

(1 + γ )H 2

+

Σ

Δδ

Σ +

Δδ A

tie

γ Ptie

Σ Δδ B 2π s Δf

B

Fig. 6. Model of a tie-line with TCPAR device.

was not possible to find out a satisfactory solution to the problem of damping and elimination of oscillations occurring in the frequency and tie-line power output signals [14,15]. After having changed the localization of the TCPAR controllers in the tie-lines and manipulating their parameters, no special improvement in stabilization enhancement has been observed. It has been decided to use special algorithms of optimization to find the optimal parameters for getting better solutions.


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The performance of optimization based on minimization of a multiobjective function has been of help in finding an adequate solution in [29]. The method is based on classical optimization algorithm. Optimization techniques are used to find a set of design parameters, x = {x1, x2, . . ., xn}, that can in some way be defined as optimal. In a simple case, this might be the minimization or maximization of a system characteristic that is dependent on x. In a more advanced formulation, the objective function, f(x), to be minimized or maximized, might be subject to constraints in the form of equality constraints. A General Problem (GP) description is stated as

min x 2 fRg

f ðxÞ

ð6Þ

An efficient and accurate solution to this problem depends not only on the size of the problem in terms of the number of constraints and design variables but also on characteristics of the objective function and constraints. When both the objective function and the constraints are linear functions of the design variable, the problem is known as a Linear Programming (LP) problem. Quadratic Programming (QP) deals with the minimization or maximization of a quadratic objective function that is linearly constrained. For both the LP and QP problems, reliable solution procedures are readily available. More difficult to solve is the Nonlinear Programming (NP) problem in which the objective function and constraints can be nonlinear functions of the design variables. A solution of the NP problem generally requires an iterative procedure to establish a direction of search at each major iteration. This is usually achieved by the solution of an LP, a QP, or an unconstrained subproblem [30]. The rigidity of the mathematical problem posed by the general optimization formulation given in (6) is often different from that of a practical design problem. Rarely does a single objective with several hard constraints adequately represents the problem being considered. More often, there is a vector of objectives

FðxÞ ¼ F 1 ðxÞ; F 2 ðxÞ; :::; F n ðxÞ

ð7Þ

that must be traded off in some way. The relative importance of these objectives is not generally known until the system’s best capabilities are determined and tradeoffs between the objectives fully understood. As the number of objectives increases, tradeoffs are likely to become complex and less easily quantified. There is much reliance on the intuition of the designer and his or her ability to express preferences throughout the optimization cycle. Thus, requirements for a multiobjective design strategy are to enable a natural problem formulation to be expressed, yet to be able to solve the problem and enter preferences into a numerically tractable and realistic design problem. In the case of the system shown in Fig. 8, it can be stated that the optimization is multiobjective. Let DfA(t), DfB(t), DfC(t) be the frequency deviations of system A, B and C, respectively, and DPWAB(t), DPWBC(t), DPWCA(t) be the respective tie-line power exchanged between the three systems. The main objective of the optimization is to minimize the frequency DfA(t), DfB(t), DfC(t) and tie-line powers DPWAB(t), DPWBC(t), DPWCA(t) signals. If the objective of the system is a one-time computation of the objective function (i.e. computation of all the optimized values of DfA(t), DfB(t), DfC(t) and DPWAB(t), DPWBC(t), DPWCA(t) at an instant t), the optimization would be a single objective. In this case, a vector containing all the optimized data for the instant t is built after the adequate optimization algorithm is run. However, for this case, each optimized value should be computed for each quantity (DfA, DfB, DfC and tie-line power DPWAB, DPWBC, DPWCA), and again the time parameter should be taken into account in the computation

process. For this case, the optimization becomes multiobjective. One of the Matlab functions, which can be used for solving this problem, is lsqnonlin, for which the description can be found in The Matlab Documentation. This function implements Gauss– Newton and Levenberg–Marquardt methods for nonlinear leastsquares (LS) optimization. Another function, which can be used, is Minimax. This method makes it possible to closely mimic Newton’s method for constrained optimization just as is done for unconstrained optimization. At each major iteration, an approximation is made of the Hessian of the Lagrangian function using a quasi-Newton updating method. One of the methods, that has been efficiently used in the analysis is the minimization of output signals, composed of DfA(t), DfB(t), DfC(t) and DPWAB(t), DPWBC(t), DPWCA(t). Details of this optimization process are presented in [29]. This method has been applied to the Matlab model of the system presented in Fig. 8 and the results shown in Figs. 9–11 were obtained. During steady-state, the influence of TCPAR device depends on the system at which the power imbalance has occurred. If the disturbance in power imbalance occurs in the subsystem A, instantaneous local frequency deviations of system A result in a very efficient response of the TCPAR regulator and a very strong damping of power and frequency swings. This phenomenon is illustrated in Fig. 9, in which it can be observed that frequency oscillations of all three subsystems are lower after the TCPAR device has been installed. Also inter-area power swings have been damped in such a way that power time responses do not contain typical inter-area oscillations but low aperiodic power deviations forced by AGC of different subsystems. If the disturbance in power imbalance occurs in the subsystem B, power control system with a frequency-based PSS makes the subsystem A local frequency oscillations strongly damped. However, this situation has no significant influence on subsystem B and C local frequency oscillation. Additionally, the subsystem A local frequency oscillation damping has an unwanted back effect consisting of stronger power deviation in other tie-lines, in particular, lines A–C and B–C. The same situation occurs when the disturbance in power imbalance takes place in subsystem C. Power control system with a frequency-based PSS dampens only subsystem A local frequency oscillations, but has no significant influence on the subsystems B and A local frequency oscillation. Quite strong power swings, comparable to those of system without TCPAR occur in all tie-lines. This problem has been solved after applying an optimization method based on (6), for which the multiobjective function was based on a few parameters of TCPAR devices installed in tie-lines of the systems. The results for disturbance occurrence in system A are shown in Figs. 9–11. Thin curves indicate the obtained responses in case of lack of TCPAR stabilizing devices in the system. Bold curves

Δf

Ptie Pref

_ Σ

1 fn

-

Δf fn

Kf 1+Tf s γMAX

+ ΔP tie

KP

+ _

Σ

1 TP s γ min

γMAX +

+

γ

Σ γ min

KF Fig. 7. Control system composed of power controller and frequency-based PSS.


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99 500 MW

19 700 MW

100 km 40 Ω

100 200 MW

200 MW

300 MW

S zw = 7 000 MVA X C = 39,6 Ω E C = 430 kV Tm = 8 s

,,B”

150 km 60 Ω

200 150

PTA [MW]

500 MW

S zw = 10 000 MVA X B = 27,7 Ω E B = 420 kV Tm = 7 s

100 50

200 km 80 Ω

0

15 000 MW

Fig. 8. Basic data of the interconnected power system composed of three subsystems.

180 160 140 120 100 80 60 40 20 0

5

-5

PTC [MW]

fA [mHz]

40

60

80

100

0

20

40

60

80

100

80

100

time [s]

0

-10 -15 0

20

40

60

80

100

time [s] 0 -1 -2 -3 -4 -5 -6 -7 -8 -9

40 35 30 25 20 15 10 5 0

0

20

40

60

time [s] Fig. 10. Time responses of generating power signals after a power imbalance occurs in the subsystem A.

0

20

40

60

80

100

time [s] 5

PTB [MW]

fB [mHz]

20

,,C”

15 100 MW

fC [mHz]

0

time [s]

PTB [MW]

S zw = 7 000 MVA 20 500 MW X A = 39,6 Ω TCPAR E A = 440 kV ,,A” Tm = 8 s

0 -5

180 160 140 120 100 80 60 40 20 0

0

20

40

60

80

100

time [s]

-15 0

20

40

60

80

100

time [s] Fig. 9. Time responses of frequency signals after a power imbalance occurs in the subsystem A.

PTC [MW]

-10

indicate the responses obtained after having performed an optimization of selected TCPAR parameters.

P tieAB[MW]

The aim of the analysis is to optimize the influence of TCPAR controller parameters on the eigenvalue loci of the linearized model of the system shown in Fig. 7. The objective of the optimization is to move the considered eigenvalues thus to left, and thus to maximize the damping ratio as much as possible. General information concerning oscillation trajectories and all power system quantities connected to them can be obtained by solving the system characteristic equation, and by solving the

0

20

40

60

80

100

time [s]

4. Eigenvalue analysis 4.1. General information

40 35 30 25 20 15 10 5 0

0 -20 -40 -60 -80 -100 -120 -140 -160 -180 0

20

40

60

80

100

time [s] Fig. 11. Time response of tie-line powers after a power imbalance occurs in the subsystem A Control parameters KP = 0.002, TP = 5 s, Kf = 2000, Tf = 5, KF = 1, cMAX = +0.2, cMIN = 0.2 (see Fig. 7)


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eigenproblem. The shape of state variable trajectories depends [25–27] on each eigenvalue kj of the system matrix, that is, the output signal time signals. In the general case, transient trajectories are associated with the conjugated complex eigenvalues of the form kj ¼ rj jtj , where the imaginary part tj part determines the angular frequency whereas the real part rj is associated with the damping ratio. Therefore, it is important to make all the eigenvalues lie as far as possible from the imaginary axis in the left half plane. ðRefkj g < 0Þ. In such a situation, the system damping is increased and the return to equilibrium is faster. One of the convenient ways to determine the influence of system parameters (among others, the TCPAR parameters) on eigenvalue loci located on the complex plane is the sensitivity analysis of the eigenvalues. This analysis allows the determination of participation factors cij of the ith state variable Dxi in the ith eigenvalue kj . Participation factors express the sensitivity of eigenvalue kj to the variation of the element aii of the state matrix A. The sum of all the participation factors associated with one eigenvalue is equal to 1. In the general case, participation factors can have complex values. Therefore, the module jcji j of the complex participation factor jcji j is considered as the amount of participation of the ith state variable in the jth eigenvalue. 4.2. Eigenvalue computation The optimization presented in [29] has led to the conclusion that there is a possibility of verification of the results by performing an eigenvalue analysis of the interconnected system shown in Fig. 7, during which, the sensitivity analysis will be carried out for the interval of values taken into account during the performance of time simulation for the optimization process. A few assumptions should be taken into account to this end. In order to perform eigenvalue computation, there are a few built-in Matlab tools which can be used. Among others, Matlab has a graphic tool, which allows to display the obtained eigenvalues in the complex plane. Without preparing a special code, this tool cannot be used for root locus representation for MIMO systems, which is the case for this example. For this reason, a few cases have

been computed using the Matlab software, and thereafter, the results have been output to external files. Finally, the external programs (MathCad, Excel) can be run to display and verify the obtained results. The main algorithm is as follows: 1. Eigenvalues are computed and written to an external file. 2. Some parameters of the control devices are modified and then the previous step is repeated. 3. After having reached the optimal criterion defined in [28], the computation is stopped. The obtained results are read as input data to MathCad. This program was also used for the verification of the accuracy of the obtained eigenvalues. In calculating the eigenvalues of the system, the linearized DAE system equations can be used instead of the reduced system state matrix. This is commonly known as the generalized eigenvalue problem. Its major advantage is that sparse matrix techniques can be used to speed up the computation. Furthermore, the extended eigenvector can be used to identify the dominant algebraic variable associated with the critical mode. In this model, all system tie-lines can be equipped with TCPAR devices. The FACTS device model is located in tie-line mathematical model. The test procedure in [28] was classified as follows: First, the test system is evaluated without TCPAR devices. As shown in Fig. 12, the examined system is stable, but a few sensitive eigenvalues, corresponding to the modes are located near the imaginary axis Im. After optimization of TCPAR parameters presented in [28], some parameters of TCPAR can be increased step by step, using an algorithm defined in [28]. The observation of the obtained results leads to the conclusion that even without coordinated tuning, the power system becomes more stable, that is, the less sensitive eigenvalues shift towards the LHP. The time-domain results presented in the previous section have been confirmed by an eigenvalue analysis performed on a linearized model of the system depicted in Fig. 7. In [31], several tests of sensitivity analysis of the system are presented. Only a few significant

Im Before optimisation

TP = 3

TP = 3..1.25 4

ΔδC ( C ) , Δf C

3 Tp = 2

2

Tp = 1.5

TP = 3..1.25

TCPAR, TGA

1

ΔδA ( A ) , Δf A

HPA , TGA T = 3..1.25 P

TGC

LPC 0 -3.8 -3.6 -3.4 -3.2

-3

IPB, Δf B

TCPAR _ 2_ mod_2

-2.8 -2.6 -2.4 -2.2

-2

-1.8 -1.6 -1.4 -1.2

-1

LPC - low-pressure part of turbine C

HPA - high-pressure part of turbine A

TGC - turbine C governor

TGA - turbine A governor

IPB - intermediate-pressure part of turbine B

TCPAR_2_mod_2 - mod nr 2 of TCPAR nr 2

-0.8 -0.6 -0.4 -0.2 -0

Re

Fig. 12. Influence of time constant TP on the root locus of interconnected system model (TCPAR devices are installed in AB and CA tie-lines).


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cases are described in this paper. Analyzing the influence of the most characteristic parameters of TCPAR devices, can perform a reliable qualitative and quantitative evaluation of the influence of TCPAR devices on the eigenvalue loci. Eigenvalue analysis conducted in this section consists of tracking their migration to the intervals of parameters to be optimized defined for the time-domain optimization process detailed in [29], which led to the optimized curves presented in Figs. 9–11. The procedure is as follows: that is, when the parameter TP (Fig. 7) was optimized, it has been stated that its value is in the interval [15]. When the time-domain optimization process was run, the initial value of TP was 5. Its final value was equal to 1. Having regard to this fact, also this initial value was set for the eigenvalue tracking process. Eigenvalues were computed, a new smaller value of TP was set, then eigenvalues were computed again, until the moment when TP was equal to the value for which the optimization criterion was fulfilled. An example of this eigenvalue analysis is shown in Fig. 12. The following intervals are also taken into consideration, according to the above presented assumptions: 1. For 0.05 6 Tf 6 0.15. 2. For 1 6 TP 6 5. 3. For 600 6 KF 6 1200.

3. Eigenvalues corresponding to power angle and frequency modes of the system C (DdC(C), DfC) are the most sensitive to (TP). A change of TP values in the interval [3..1.25], for which all other TCPAR parameters remain constant, results in shifting the corresponding eigenvalues about 300% to the left in the LHP. 4. Also, eigenvalues corresponding to power angle and frequency modes of the system A (DdA(A) DfA), moved slightly in the LHP, as TP values have changed in the interval [3..1.25]. 5. All other eigenvalues have changed their locations in such a way that the overall stability has been improved. For example, the conjugated eigenvalues corresponding to turbine governor (TGA) and the first TCPAR device modes also move left in the LHP, as TP values decreases in the interval [3..1.25]. The same situation is observed for the eigenvalue corresponding to the second TCPAR device mode. 6. Eigenvalues corresponding to turbine B intermediate-pressure part (IPB) and frequency B modes (DfB) remain motionless. Considering TCPAR device settings resulting from the previously performed optimization [6], the influence of these devices on stability enhancement is positive. A progressive decreasing of power-dependent element feed-back loop element (TP) results in moving the most sensitive roots towards 1.

4.3. Influence of Tp on eigenvalue loci 4.4. Influence of TP and Kf on eigenvalue loci Only the time constant of TCPAR power-dependent element feed-back loop (TP) has been changed during the performance of this test. The TCPAR devices are installed in AB and CA tie-lines. The following conclusions can be drawn from the root loci analysis of Fig. 12: 1. The obtained linearized system is stable because all eigenvalues are situated in the LHP. 2. The time constant of the power-dependent element (TP) has a significant influence on the root loci of the three-system model.

Both the time constants of TCPAR power-dependent element feed-back loop (TP, interval of change: 3..0.7) and the gain coefficients of TCPAR power-dependent element feed-back loop lower element (KF, interval of change: 950..600) have been changed during the performance of this test. The TCPAR devices are installed in AB and CA tie-lines. It is obvious that a drop in the gain increases the damping of the (DdC(C), DfC) modes. The following conclusions can be drawn from the root locus analysis of Fig. 13:

Fig. 13. Influence of Kf and TP on the root locus of interconnected system model (TCPAR devices are installed in AB and CA tie-lines).



1. Eigenvalues corresponding to power angle and frequency modes of the system C (DdC(C), DfC) are the most sensitive to (TP). A change of TP values in the interval [3..1.25], for which all other TCPAR parameters remain constant results in shifting the corresponding eigenvalues about 300% to the left in the LHP. It is also observed that a progressive decreasing of TP accompanied with a progressive decreasing of KF results in a significant increasing of the damping generated by this mode. Damping is increased (eigenvalue moves toward 1) and oscillations get lower (their imaginary part is getting much lower, so the oscillations are smaller). 2. Also eigenvalues corresponding to power angle and frequency modes of the system A (DdA(A) DfA) slightly move to the left in the LHP, toward 1 for a while, then they move back towards the imaginary axis. Eigenvalues corresponding to the turbine C low-pressure part (LPC) and its governor (TGC), which are real at the beginning of the test, get closer to each other, as TCPAR feed-back loop time constant (TP, interval of change: 3..0.7) and gain coefficient (KF), interval of change: 950..600) drop then the two real roots coalesce to form a complex conjugate pair, and then move towards 1 in the LHP. 3. Complex eigenvalues corresponding to the turbine A governor (TGA) and TCPAR modes move first towards 1 in the LHP, then they become real. The TGA eigenvalue couples with the high-pressure part mode of turbine A (HPA) and then the two real roots coalesce to form a complex conjugate pair, which moves towards 1 in the LHP. 4. Eigenvalues corresponding to the turbine B intermediatepressure part (IPB) and frequency B modes (DfB) remain motionless. 5. Conclusions The strongest effect of TCPAR action on frequency and power swing damping is obtained when using power control system with a frequency-based PSS presented in Fig. 7. The analysis of results of the simulation of an interconnected power system consisting of the three subsystems has brought less optimistic observations. However, the TCPAR action can enforce the tie-line interchange power, in particular, in the tie-lines located relatively far away from the TCPAR device. For these reasons, the use of TCPAR based on power control system with a frequency-based PSS in very large interconnected power systems, requires more care and performance of more detailed analysis of the power system. A satisfactory level of control has been obtained using a simple optimization procedure. This problem has been solved after having applied a multiobjective optimization method, for which the multiobjective function was based on a few parameters of TCPAR devices installed in tie-lines of the systems. The obtained results have proven that it is possible to obtain a good damping of tie-line power and frequency swings by optimizing the main parameters of TCPAR installed in tie-lines. Also, they have been confirmed by an eigenvalue analysis of linearized model of interconnected system consisting of three subsystems. The eigenvalue analysis of the system has shown that: 1. The sensitivity analysis of the multi-area model could be conducted after performing a preliminary identification of different modes. In this paper, the method of participation factors has been used to this end. 2. The results lead to the conclusion that a more detailed qualitative and quantitative analysis of the slow phenomena occurring in the multi-area system can be conducted using more precise tools like sensitivity analysis of eigenvalues of the system. 3. Frequency and power angle modes are the most sensitive to TCPAR parameters. These are modes, which are connected directly to frequency and tie-line powers.

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4. Theoretical analysis and performance of frequency and timedomain analysis are very helpful in the determination of localization and settings of TCPAR devices used as stabilization devices in the multi-area power systems. The greater the number of stabilizing devices, the more difficult is their coordination and settings. Further work in this domain will be directed towards non-linear nature of the power system. A particular range of system operating conditions will also be studied to verify the capability of the controller settings. References [1] Banerjee S, Chatterjee JK, Tripathy SC. Application of magnetic energy storage unit as load frequency stabilizer. IEEE Trans Energy Convers 1990;5(1):46–51. [2] Demiroren A. Automatic generation control using ANN technique for multiarea power system with SMES units. Electr Power Compon Syst 2004;32(2):193–213. [3] Ngamroo I. An optimization technique of robust load frequency stabilizer for superconducting magnetic energy storage. Energy Convers Manage 2005;46(18–19):3060–90. [4] Ngamroo I, Mitani Y, Tsuji K. Application of SMES coordinated with solidstate shifter to load frequency control. IEEE Trans Appl Supercond 1999;9(2):322–5. [5] Tripathy SC, Bak-Jensen B. Automatic generation control of multi-area power system with superconducting magnetic storage unit. IEEE Power Tech Proc 2001;3(Porto, 10–13):6. [6] Tripathy SC, Balasubramanian R, Chandramohanan Nair PS. Adaptive automatic generation control with superconducting magnetic energy storage in power system. IEEE Trans Energy Convers 1992;EC 7(3):434–41. [7] Tripathy SC, Balasubramanian R, Chandramohanan Nair PS. Effect of superconducting magnetic energy storage on automatic generation control considering governor deadband and boiler dynamics. IEEE Trans Power Syst 1992;3(7):1266–73. [8] Ngamroo I, Kongprawechnon W. A robust controller design of SSSC for stabilization of frequency oscillations in interconnected power systems. Electr Power Syst Res 2003;67(3):161–76. [9] Robak S, Rasolomampionona DD. Block diagram transfer function model of generator – infinite busbar system including TCPAR. In: Proc international symposium modern electric power systems MEPS’02 Wroclaw; September 11– 13, 2002. [10] Robak S, Rasolomampionona DD. Advanced modelling of generator-infinitebusbar system including thyristor controlled phase shift transformer. Arch Electr Eng 2003;LII(2):201–19. ISSN 0004 – 0746. [11] Machowski et al. – Współdziałanie ARCM oraz urza˛dzen´ FACTS w regulacji systemu elektroenergetycznego w warunkach rynkowych. (AGC and FACTS Stabilization Device Coordination for Interconnected Power System Control in Market Conditions) Grant of Polish SCSR Nr8T10 B05818, Warszawa; 2002. [12] Abraham RJ, Das D, Patra A. AGC of a hydrothermal system with thyristor controlled phase shifter in the tie-line. In: Proc IEEE power India conference; April 2006. p. 7–14. [13] Abraham RJ, Das D, Patra A. Effect of TCPS on oscillations in tie-power and area frequencies in an interconnected hydrothermal power system IET Generation. Transm Distrib 2007;1(4):632–9. [14] Rasolomampionona DD. AGC and FACTS stabilization device coordination in interconnected power system control. IEEE Bologna PowerTech; June 23–26, 2003. 0-7803-7967-5/03/$17.00. [15] Rasolomampionona D. A modified power system model for AGC analysis. 2009 IEEE Bucharest PowerTech conference; 28th June to 2nd July 2009. [16] Elgerd I, Fosha CE. Optimum megawatt-frequency control of multiarea electric energy systems. IEEE Trans Power Appar Syst 1970;PAS-89(4):556–63. [17] Fosha CE, Elgerd I. The megawatt-frequency control problem: a new approach via optimal control theory. IEEE Trans Power Appar Syst 1970;PAS89(4):556–77. [18] Carpentier J. State of the art review: ‘To be or not to be modern’ that is the question for automatic generation control. Electr Power Energy Syst 1985;7(2):81–91. [19] Cavin RK, Budge MC, Rasmussen P. An optimal linear system approach to loadfrequency control. IEEE Trans Power Appar Syst 1971;PAS-90(Nov./ Dec.):2472–82. [20] Chan WC, Hsu YY. Automatic generation control of interconnected power systems using variable-structure controllers. IEEE Proc Gener Transm Distrib 1981;128(5):269–79. [21] Kumar A, Malik OP, Hope GS. Variable-structure-system control applied to AGC of an interconnected power system. IEE Proc Gener, Transm Distrib 1985;132(1):23–9. [22] Pan CT, Liaw CM. An adaptive controller for power system load-frequency control. IEEE Trans Power Syst 1989;4(1):122–8. [23] Shoults RR, Jativa Ibarra JA. Multi-area adaptive LFC developed for a comprehensive AGC simulator. IEEE Trans Power Syst 1993;8(2):541–7.


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[24] Jaleeli N et al. Understanding automatic generation control. IEEE Trans Power Syst 1992;7(3):1106–22. [25] Machowski J, Bialek J, Bumby J. Power system dynamics and stability. Chichester, New York: John Wiley & Sons; 1997. ISBN 0471 97174 X. [26] Kirchmayer LK. Economic control of interconnected systems. John Wiley & Sons; 1959. [27] Kundur P. Power system stability and control. New York: MacGraw – Hill, Inc.; 1993. [28] Januszewski M. Urza˛dzenia FACTS jako s´rodek poprawy stabilnosći systemu elektroenergetycznego. (FACTS devices as tools for power system stability enhancement). PhD Dissertation, Warsaw University of Technology; 2001.

[29] Machowski et al. Interakcje automatyk regulacyjnych w systemie elektroenergetycznym z uwzgle˛dnieniem urza˛dzen´ FACTS (power system control devices interactions). Part I: Time-domain optimisation analysis. Grant of Polish SCSR nr 4 T10B 014 23. Annual report; 2003. [30] The Matlab documentation. . [31] Machowski et al. Interakcje automatyk regulacyjnych w systemie elektroenergetycznym z uwzgle˛dnieniem urza˛dzen´ FACTS FACTS (power system control devices interactions). Part II: Steady-state stability analysis. Grant of Polish SCSR 4 T10B 014 23. Annual report; 2004.