coherent generator based transient stability analysis of...

Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (3): 456-461 (ISSN: 2141-7016)

456

Coherent generator based Transient Stability Analysis of the 16 machines,

330KV �igeria Power System

Izuegbunam F. I., Okafor E. �. C., Ogbogu S. O. E.

Department of Electrical and Electronic Engineering Federal University of Technology, P.M.B. 1526, Owerri, Nigeria

Corresponding Author: Izuegbunam F. I

___________________________________________________________________________ Abstract

This paper assesses the transient stability analysis of the expanded 16 machines, 330KV �igeria Grid System

through coherent generators aggregation based on equal acceleration and velocity concept. The technique

involves performing the electrical proximity test on the system generators under fault condition to select the

study area generators, subjecting these generators to coherency check through inertia and damping indices

conditions; and construction of dynamic equivalents representing each pair of coherent generators. The result

led to reduction of 16 machines system to 13, thereby simplifying the network complexity and requires minimal

computer time as well as memory space.

__________________________________________________________________________________________ Keywords: transient stability, coherent generators, dynamic equivalents, expanded grid, swing curve.

__________________________________________________________________________________________ I�TRODUCTIO�

The large interconnections that constitute the present day power networks has further added to its complexity hence requiring a simplified but accurate method of transient stability analysis with minimal computing time and memory space. The method of transient stability analysis of a large power system via simplified dynamic equivalents is attracting increased attention by researchers across the globe (Rau and Hussian, 1998). The coherent generator technique is an elegant and powerful tool for system simplification through construction of simplified reduced order dynamic models which can represent the entire system without loss of system significant characteristic. The reduction is based on the impact of large disturbances in a particular area known as the study system, while system external to study area are not of direct interest in stability analysis; and are thus represented by dynamic equivalents in recognition of its influence on the response of the study area to disturbances. Available literature shows that different authors applied variant principles of determining coherent generators. Lee and Schweppe, (Lee and Schweppe, 1993), used the concepts of distance measure such as admittance distance, reflection distance and acceleration distance to identify coherent generators for construction of dynamic equivalents. Also the concept of power variation of the generators at the instant of initiated fault is applied to classify the machines into inner circle (study area) machines and outer circle (external area) machines. Spalding, Yee, and Goudie, (Spalding et al., 1977), applied the principle of singular points in coherency identification. The singular points include the post-fault Stable Equilibrium Point (SEP) and Unstable Equilibrium Point (UEP). This approach

compares system’s differential equations describing change in relative machine angles from the stable equilibrium point to that of unstable equilibrium point corresponding to the system instability anticipated from the fault condition under investigation. Podmore, (1978), applied a simplified linear model of power system known as clustering Algorithm using linearized network and dynamic equations deemed clustered to determine the angular deviation of the machines. Al-Fuhaid, (1987), deployed the properties of exponential matrix and Cayley-Hamilton theorem using the state transition matrix to determine the linearized system dynamic equations. Rudmick, Patino, and Brameller, (1981), applied the principle based on rate of change of kinetic energy, relating the power system behaviour to its potential energy and kinetic energy transition. Krishnaparandhama et al., (1981), and Sankaranarayanan et al., (1983), applied coherent generators identification based on equal acceleration principle using a linearized acceleration equation with the damping coefficient neglected. In this paper, the coherent generator based transient stability analysis of the 16 machines expanded Nigeria power system was conducted through the following steps: electrical proximity test, coherent generator identification, construction of dynamic equivalents of coherent generator pairs, and re-integration of the aggregated generators with the rest of system machines. The purpose is to assess the transient stability condition of the expanded Nigerian 330KV Grid system using a technique that reduces the model order of the Network without altering the characteristic structure and it is justified because

Journal of Emerging Trends in Engineering and Applied Sciences (JETEAS) 2 (3): 456-461 © Scholarlink Research Institute Journals, 2011 (ISSN: 2141-7016) jeteas.scholarlinkresearch.org


457

most of the generating plants and transmission lines in this grid are still under construction, but its response to large disturbance has been simulated. A practical power network has been used to test the effectiveness of the coherent generator technique.

Multimachine Power System Modeling

Figure 2.1 represents a large power network Figure 2.1 Large power network The rotor dynamic equation for ith machine of an n-generator system is given by,

(2.1) For ith

and jth generators to be coherent,

(2.2) where,

Pre-fault value of the difference between the ith and jth rotor angles. Linearizing equation (2.1) on the assumption that the power variation is small and that the generator coherency is independent of the magnitude of the disturbance (Podmore, 1978), gives,

(2.3) Where,

= the change in acceleration of the ith machine

= change in the velocity of the ith machine The mechanical power input to the machine is assumed constant for a short duration of the fault, hence the term in equation (2.3) becomes zero. The change in electrical power of the ith

generator due to a small change in the angle of the jth generator evaluated at the time of initiating the fault (Lee and Schweppe, 1993), gives,

(2.4)

(2.5)

(2.6)

The following assumptions are made: 1. Internal voltages of the machines are equal

to 1.0 pu 2. The angular difference is within

300.

3. The network is highly reactive, while the conductances are not neglected. The above assumption simplifies equation (2.6) to,

(2.7)

Substituting equation (2.7) into (2.3) gives, (2.8)

Let , and (2.9)

Putting (2.9) into (2.8) gives,

(2.10)

Electrical proximity for any two machines depends upon their mutual admittance (Sankaranarayanan et al., 1983). As the mutual admittance between two machines becomes larger, that pair of machines is considered more close to each other (Rau and Hussian, 1998). Therefore, the mutual admittance between a pair of machines is a measure of coupling between them (Rau and Hussian, 1998), (Monticelli, 1999).

Electrical Proximity index

Electrical proximity index αįj is defined by the expression,

(2.11)

The generators that satisfy equation (2.11) qualifies for coherency pair check with the ith machine. The modified linearized acceleration equation for ith and jth machines can be written as,

(2.12)

(2.13)

For equal acceleration of the ith and jth machines, equations (2.14) and (2.15) must be satisfied.

(2.14) Where,

(2.15)

From the equations (2.14) and (2.15) the two vital indices β and γ for coherency check are obtained as :

(2.16)

(2.17) where, = Inertia index for ith and jth machines

= Damping index for ith and jth machines

X΄d1 E

΄1

X΄dn E

΄n

1

2

3

X΄

d2 E΄2

X΄

d3 E΄3

X΄

d4 E14

Fictitious Point


458

In a situation where machines i and j are absolutely coherent, both β and γ becomes equal to zero. But when one of the inertias tends to infinity, then both β and γ will tend to unity (1.0), which indicates the machines deviation from each other. The tolerance values of 0.2 for β and 0.5 for γ are found to give acceptable results (Rau and Hussian, 1998). In a situation whereby the damping coefficient is zero or the ratio D/M becomes uniform, the coherency grouping is conducted based on β criterion alone.

Construction of Dynamic Equivalents

The concept of dynamic equivalent construction technique deployed in this text is based on the power invariance at the terminal buses and at the internal buses of the coherent machines (Kimbark, 1948), (Sankaranarayanan et al., 1983), (Hussian and Rau, 1993). When the coherent generators are not on the same bus-bar (not parallel), fictitious points are inserted between the reactance and the source of internal voltage of each generator, and the machines are tied together at that point. The parallel voltage sources are then replaced by a single source (dynamic equivalent) which is adjusted to deliver the same active and reactive power to the network as the sources it replaced.

Mathematical Formulation

The apparent power received by the ith terminal bus from the corresponding coherent machine and the internal voltages of ith machine are given as:

(2.18)

(2.19) Where, = terminal voltage of ith machine = current flowing to the terminal bus of ith machine. The sum of the powers received by p buses of the coherent group is given by,

(2.20) The equivalent terminal bus of the coherent group must be such that it can receive the total power ST and current IT from the equivalent machine replacing the coherent group of machines, therefore;

(2.21)

Where, = the vector sum of the currents (2.22)

= the number of the machine in the coherent group. The equivalent machine internal voltage, Ee is given by,

(2.23)

The equivalent machine transient reactance is given by,

(2.24)

The equivalent inertia constant, of the coherent group is the sum of the inertia constants of the individual machines, and is given by,

(2.25) The equivalent damping coefficient, of the coherent group is the sum of the damping coefficients of the individual machines, and is given by,

(2.26) The equivalent mechanical power, is the sum of mechanical powers required to accelerate the individual machines, and is given by,

(2.27) Figure 2.2 shows the algorithm for coherency identification and construction of dynamic equivalents.

Figure 2.2 Algorithm coherent generators identification and construction of dynamic equivalent (Izuegbunam, 2010).

Start

Perform electrical proximity test to

choose coherent machines

candidates: αįj≥ 1.0

Perform the β - index check for the

coherent group candidates. βįj < 0.2

Evaluate the fault on - index

check for coherent groups.

Choose another pair of

machines for coherency

check

Re-evaluate

Post fault γ-index check

for coherent groups.

Is γįj < 0.5 ?

Construct dynamic equivalents

of the coherent machine groups.

Yes

Stop

No

Input data for machines with more

than 30% Power variation.


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RESULTS A�D DISCUSSIO�

Figure 3.1 shows a 16 machines Nigeria 330KV Power Network. Figure 3.1: Expanded Nigeria 330KV Power Network (16 machines – 49 bus System) (Izuegbunam, 2010). The system’s response for a three - phase fault impressed on bus 3, line 3-44 evaluated through direct method as shown in figure 3.2 indicated loss of Shiroro, Geregu and Kainji generators even at clearing time of 0.16s. Beyond the critical clearing time of 0.21s and stability margin of 0.238, Omoku generator becomes unstable. Results of Generators-Coherency Check And

Aggregation For A Three-Phase Fault At Bus 3

Line 3-44.

The fault on power variations was carried out using the reduced bus admittance matrix, Ŷ, and the generators with real power variations of less than 30% were classified as external area generators, while those with higher power variations became study area generators. Electrical proximity index check was thereafter conducted on external area machines to determine the degree of coupling among the generators which is a function of mutual admittances. The machines that satisfied the index condition qualify for coherent groups check. The β- and γ- indices checks were subsequently performed on the machines with the following generators found to reasonably form coherent groups: Group I: Delta (2) and Eyaen (16) Group II: Afam (7) and Okpai (8) Group III: Papalanto (10) and Omotosho (11)

Figure 3.3 shows the swing curves for group I coherent generators, while figure 3.4 shows their equivalent swing curve after aggregation. Figure 3.5 on the other hand shows the swing curves for group II coherent generators, while figure 3.6 shows their equivalent swing curve after aggregation. Also figure 3.7 shows the swing curves for group III coherent generators, while figure 3.8 shows their equivalent swing curve after aggregation. Figure 3.9 shows the swing curves of the new network after integrating the aggregated generators to the rest of the power system, resulting to network reduction from 16 machines system to 13 machines. Hence, through coherent generators aggregation, the complexity of the original network is simplified, and the computational time for the system simulation reduced.

Figure 3.2 Swing curves for fault at bus 3, line 3 – 44 (clearing time: 0.16s)

Figure 3.3 Swing curves for machines 2 and 6 for fault at bus 3, line 3 – 44

1

1

2 2

1

2

2

4 2

4

4

4

4

3

6 4

4

3

44

4

1

4

3

2

5

2

6

8

2

9

3

1

1

1

13

3

3

3

1

3

3

3

5

3

0

9

2

1

3

2

7

1

5

2

1

∼

∼


460

Figure 3.4 Equivalent swing curve for machines 2 and 6





Figure 3.9 Swing curves of New Network after aggregation for fault at bus 3, line 3 – 44/41 CO�CLUSIO� This paper deployed the dynamic generators aggregation technique to perform transient stability analysis of 16 machines Nigeria power system. The principle is based on rotor angle swing coherency of the candidate generators, dependent on electrical proximity test which defines the degree of coupling between two or more generators. The inertia (β) and damping (γ) indices were in addition used to identify coherent generators for possible aggregation through construction of dynamic equivalents. The technique proved useful since it helped in reducing the 16 machines network to 13 machines system for a three-phase fault at bus 3 (line 3-44), resulting in the reduction of network complexity, ensuring power invariance at the aggregated generators terminals before and after aggregation. REFERE�CES

Al-Fuhaid A. S. (1987), ‘Coherency Identification for Power System’, Inter. Jour. of Elect. Powerand Energy Sys. Vol. 9, No. 3, July, pp. 149-56. Hussian M. Y. and Rau V. G. (1993), ‘Coherency Identification and Construction of Dynamic Equivalent for large Power System’, Proceedings of 2nd Inter. Confr. on Advances in Power Sys. Control, Operation and Management, Hongkong, Dec.


461

Izuegbunam F. I. (2010), ‘Transient Stability Analysis of Nigeria Power System: Multimachine Approach’, Ph.D Thesis, Federal University of Technology, Owerri, Nigeria. Kimbark E. W. (1948), ‘Power System Stability, vol. 1: Synchronous Machine’, John Wiley & Sons. Krishnaparandhama T., Elvangovan S., and Kuppurajulu A. (1981), ‘Method for Identifying Coherent Generators’, vol. 3, No. 2, pp. 85-90, April. Lee S. T. Y. and Schweppe F.C. (1993), ‘Distance measures and Coherency Recognition for Transient Stability Equivalents’, IEEE Trans. Vol. PAS-82, Sept./Oct., pp. 1550-1557. Monticelli A. (1999), ‘State Estimation in Electric Systems, A Generalized Approach’, Kluwer Academic Publishers, USA. Podmore R. (1978), ‘Identification of Coherent Generators for Dynamic Equivalents’, IEEE Trans., vol. PAS-97, No. 4, pp. 1060-1069. Rau V. G. and Hussian M. Y. (1998), ‘Coherent Generators’, Allied Publishers Limited, New Delhi. Rudnick H., Patino R. I., and Brameller A. (1981), ‘Power System Dynamic Equivalents: Coherency Recognition via the rate of change of Kinetic Energy’, Proceedings of the IEE, vol. 128, pt. C, No. 6, pp. 325-33, Nov. Sankaranarayanan V., Venugopal M., Elangovan S., and Dharma R. N. (1983), ‘Coherency Identification and Equivalents for Transient Stability Studies’, Elect. Power Sys. Res. Vol. 6, pp. 51-60. Spalding B. D., Yee H. and Goudie D. B. (1977), ‘Coherency Recognition for Transient Stability Studies using singular points’, IEEE Trans., vol. PAS-96, pp. 1368-75.

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