Time domain dynamic simulation and Eigen value

G Naveen Kumar et al. / International Journal of Engineering Science and Technology (IJEST)

Time domain dynamic simulation and Eigen value sensitivity analysis of an interconnected power system in MATLAB G NAVEEN KUMAR Assistant Professor, Electrical and Electronics Engineering Department, VNRVJIET, Hyderabad, India

Dr. M. SURYA KALAVATHI Professor, Electrical and Electronics Engineering Department, Jawaharlal Nehru Technological University, Hyderabad, India

Abstract: Due to the continuous expansion of power systems and its inter-connection the stability of the power systems has become a major problem for its successful operation. Such a system suffers with instability problems like voltage instability, inter-area oscillations, etc. The timely monitoring of the power system is very important in identifying and rectifying the instability problems which can prevent many serious accidents and to maintain stability. We perform a Timed Domain Analysis along with Eigen value analysis in PSAT using MATLAB. Newton Rapson Method is used for analysis in the power flow computation. Keywords: Eigen values, Time domain analysis, PSAT, stability, instability. 1. INTRODUCTION Many scientists and engineers have defined the term power system stability [7] as the ability of an interconnected electric power system, for a given initial operating condition, to regain a state of operating equilibrium after being subjected to a physical disturbance, the most system variables bounded so that system integrity is preserved. This physical disturbance can in the form of a fault, generators perturbation, increasing angular swings of generators leading to loss of synchronism with other generators, load variations etc. In this paper, it is the stability and control of voltage [1-3] that is the issue, rather than the maintenance of synchronism. This type of instability can also occur in the case of loads covering an extensive area in a large system. This is analyzed using a IEEE 14-bus system in MATLAB7.1 using PSAT toolbox [6]. The simulation is performed on Intel core 2 duo processor with 2.13GHz clock speed. 2. IMPLEMENTATION The analysis done on the IEEE 14 bus system is carried out in two steps the first being Eigen value analysis and the second being Time domain analysis based on Newton Rapson iterative Method [1]. The Eigen value analysis is considered as a proximity indicator for analysis of a power system online or offline. 2.1 Eigen value calculation: The tool box [6] used here uses the basic Eigen matrix technique to solve for the Eigen value dynamic order. Total buses taken for this analysis are 14. Generating stations considered are 5. An appropriate power system model including machine and load dynamics is required for small signal stability analysis. x’ = f(x, u) y = g(x, u)

ISSN : 0975-5462

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G Naveen Kumar et al. / International Journal of Engineering Science and Technology (IJEST) Where x’ the vector of system state variables is, refers to system input variables, refers to system output variables, describes the dynamics of the system, and includes equality conditions such as power flow equations of the system. By combining all linear state equations of dynamic devices, we can obtain the augmented state equation = Aa

(2)

Where Aa is the augmented state matrix, which is of high sparsity. The system state matrix can be obtained as As = JA-JBJD-1JC

(3)

Because of the reduction process in (3), the state matrix As is a dense matrix. The objectives of small signal stability analysis are to calculate the Eigen values of augmented state matrix Aa and state matrix As and to analyze the nature of the modes by analyzing Eigen values and eigenvectors [5] of Aa and As. i) Finding Eigen Values and Vectors Given a matrix M implementing a linear transformation, what are its Eigen vectors and values? Let the vector x represent an Eigen vector and let l be the Eigen value. We must solve x*M = lx. Rewrite lx as x times l times the identity matrix and subtract it from both sides. The right side drops to 0, and the left side is x*M-x*l*identity. Pull x out of both factors and write x*Q = 0, where Q is M with l subtracted from the main diagonal. The Eigen vector x lies in the kernel of the map implemented by Q. The entire kernel is known as the Eigen space, and of course it depends on the value of l. If the Eigen space is nontrivial then the determinant of Q must be 0. Expand the determinant, giving an n degree polynomial in l. (This is where we need a field, to pull all the entries to the left of l, and build a traditional polynomial.) This is called the characteristic polynomial of the matrix. The roots of this polynomial are the Eigen values. There are at most n Eigen values. Substitute each root in turn and find the kernel of Q. We are looking for the set of vectors x such that x*Q = 0. Let R be the transpose of Q and solve R*x = 0, where x has become a column vector. This is a set of simultaneous equations that can be solved using Gaussian elimination. In summary, a somewhat straightforward algorithm extracts the Eigen values, by solving an n degree polynomial, and then derives the Eigen space for each Eigen value. Some Eigen values will produce multiple Eigen vectors, i.e. an Eigen space with more than one dimension. The identity matrix, for instance, has an Eigen value of 1, and an n-dimensional Eigen space to go with it. In contrast, an Eigen value may have multiplicity > 1, yet there is only one Eigen vector. This is illustrated by [1,1|0,1], a function that tilts the x axis counterclockwise and leaves the y axis alone. The Eigen values are 1 and 1, and the Eigen vector is 0,1, namely the y axis. ii) The Same Eigen Value: Let two Eigen vectors [5] have the same Eigen value. Specifically, let a linear map multiply the vectors v and w by the scaling factor l. By linearity, 3v+4w is also scaled by l. In fact every linear combination of v and w is scaled by l. When a set of vectors has a common Eigen value, the entire space spanned by those vectors is an Eigen space, with the same Eigen value. This is not surprising, since the Eigen vectors associated with l are precisely the kernel of the transformation defined by the matrix M with l subtracted from the main diagonal. This kernel is a vector space, and so is the Eigen space of l. Select a basis b for the Eigen space of l. The vectors in b are Eigen vectors, with Eigen value l, and every Eigen vector with Eigen value l is spanned by b. Conversely, an Eigen vector with some other Eigen value lies outside of b. iii) Different Eigen Values: Different Eigen values always lead to independent Eigen spaces.

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Suppose we have the shortest counterexample. Thus c1x1 + c2x2 + … + ckxk = 0. Here x1 through xk are the Eigen vectors, and c1 through ck are the coefficients that prove the vectors form a dependent set. Furthermore, the vectors represent at least two different Eigen values. Let the first 7 vectors share a common Eigen value l. If these vectors are dependent then one of them can be expressed as a linear combination of the other 6. Make this substitution and find a shorter list of dependent Eigen vectors that do not all share the same Eigen value. The first 6 have Eigen value l, and the rest have some other Eigen value. Remember, we selected the shortest list, so this is a contradiction. Therefore the Eigen vectors associated with any given Eigen value are independent. Scale all the coefficients c1 through ck by a common factor s. This does not change the fact that the sum of cixi is still zero. However, other than this scaling factor, we will prove there are no other coefficients that carry the Eigen vectors to 0. If there are two independent sets of coefficients that lead to 0, scale them so the first coefficients in each set are equal, then subtract. This gives a shorter linear combination of dependent Eigen vectors that yields 0. More than one vector remains, else cjxj = 0, and xj is the 0 vector. We already showed these dependent Eigen vectors cannot share a common Eigen value, else they would be linearly independent; thus multiple Eigen values are represented. This is a shorter list of dependent Eigen vectors with multiple Eigen values, which is a contradiction. If a set of coefficients carries our Eigen vectors to 0, it must be a scale multiple of c1 c2 c3 … ck. Now take the sum of cixi and multiply by M on the right. In other words, apply the linear transformation. The image of 0 ought to be 0. Yet each coefficient is effectively multiplied by the Eigen value for its Eigen vector, and not all Eigen values are equal. In particular, not all Eigen values are 0. The coefficients are not scaled equally. The new linear combination of Eigen vectors is not a scale multiple of the original, and is not zero across the board. It represents a new way to combine Eigen vectors to get 0. If there were two Eigen values before, and one of them was zero, there is but one Eigen value now. However, this means the vectors associated with that one Eigen value are dependent, and we already ruled that out. Therefore we still have two or more Eigen values represented. This cannot be a shorter list, so all Eigen vectors are still present. In other words, all our original Eigen values were nonzero. Hence a different linear combination of our Eigen vectors yields 0, and that is impossible. Therefore the Eigen spaces produced by different Eigen values are linearly independent. These results, for Eigen values and Eigen vectors, are valid over a division ring. The previous Eigen value analysis and the preceding time domain analysis is performed on the IEEE 14 bus system shown in figure 2.2 Time Domain Analysis: In order to study the behaviour of the system under small perturbations, a time domain simulation was performed. A disturbance was created in the total bus system. The voltages at different buses were observed and the simulation was perturbed after creating a disturbance in the ratings of a few generators. The line diagram of the IEEE 14-bus system on which the experimentation was performed is shown in figure 1. This has 5 generating stations. Power System Analysis Toolbox [6] was taken in MATLAB to perform the above two tasks. Momentary disturbances were created at various buses and the response of the entire system was noted down at various points. Newton Rapson Method [1] is used for analysis in the power flow computation. Trapezoidal integration is used in time domain simulation. Power flow dynamic simulation is completed in 0.297seconds. The results enclosed with system stable, the time domain simulation was done till the complete 20 seconds. When these results are compared to the unstable system, where the same bus results are compared, we can observe that the voltage becomes unstable after 1.5 seconds.

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Figure 1: Line Diagram of IEEE 5-machine 14- bus system on which the testing is done

Figure 2: Eigen value analysis for the 14-bus system without disturbance

3. RESULTS AND ANALYSIS In the following figures, the Eigen value analysis can be observed. The Eigen values are plotted in the PSAT toolbox [6] for the 14- bus system. The dynamic order is observed to be 60 for the system without disturbance. The positive and negative Eigen’s are in equal ratio. When the disturbance is created, the dynamic order is observed to be reduced to be halved to 30. The positive and negative Eigen’s are not in equal ratio in this case. There are more of negative Eigen’s which shows instability in the system..

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Figure 3: Eigen value analysis for the 14-bus system with disturbance

The voltages in per unit values are observed at different buses with and without stability. The following are the results obtained during the simulation. It can be seen from the results enclosed in section 6.2.1 that the time domain simulation is done till the complete 20 seconds. When these results are compared to the ones in section 6.2.2., where the same bus results are shown, we can observe that the voltage becomes unstable after 1.5 seconds. i) Time Domain simulation with system stable: Following results show the system operating at equilibrium with any sort of disturbance. Each result has been recorded in per unit values of voltage at respective bus versus time in seconds.

Figure 4: Time domain analysis for bus2 with stability

Table: 1

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Time(s)

Voltage(v)

0

1

1.8

0.7

3.8

1.2

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Figure 5: Time domain analysis for bus 4 with stability

Table: 2

Time(s)

Voltage(v)

0

0.86

2

0.36

7

0.6


Table: 3

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Time(s)

Voltage(v)

0

1

3.5

1.25

5.8

0.7

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Table: 4 Time(s)

Voltage(v)

0

0.73

1.9

0


Table: 5

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Time(s)

Voltage(v)

0

0.81

1.8

0.41

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Table: 6 Time(s)

Voltage(v)

0

0.74

3.8

0.4

ii) Time Domain Simulation with system going Unstable: Following results show the reaction of the system when being subjected to voltage changes. Each result has been recorded in per unit values of voltage at respective bus versus time in seconds.

Figure 10: Time domain analysis for bus2 without stability Table: 7

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Time(s)

Voltage(v)

0

1

1.2

0.45

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Time(s)

Voltage(v)

0

0.86

1.5

0.25

Figure 12: Time domain analysis for bus 6 without stability Table: 9

Time(s)

Voltage(v)

0

1

1.2

0.75

Figure 13: Time domain analysis for bus9 without stability

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Table 10

Time(s)

Voltage(v)

0

0.75

1.2

0.5


Time(s)

Voltage(v)

0

0.82

1.2

0.58


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Time(s)

Voltage(v)

0

0.72

0.93

0.25

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Below is a consolidated data of voltage variations at all the fourteen buses before and after the application of disturbance. The voltage data about stability and instability conditions given is measured in perunit values at the respective buses.

Table: 13 Variations at all 14 buses

Bus number

Voltage in p.u at 1.2s (with stability)

1 2 3 4 5 6 7 8 9 10 11 12 13 14

0.78 0.75 0.75 0.44 0.45 0.75 0.55 0.76 0.45 0.55 0.6 0.65 0.65 0.55

Voltage in p.u at 1.2s (without stability) 0.1 0.45 0.6 0.35 0.02 0.7 0.5 0.85 0.84 0.4.5. .0.5 0.65 0.6 0.45

4. CONCLUSION The above results show that the dynamic order that was 60 before a disturbance occurred went on to become 30 post disturbance. This clearly gives us an idea how a power system reacts to a disturbance indicating the need to design the power system in such a way as to resist the faults or disturbances that are short term. The disturbances induced to test the model using the Power System Analysis toolbox were of small duration. The same is the case with Time Domain simulation. The above analysis gives us a very clear picture as to how we protect the power system using these results in observation of static and dynamic behaviour of a power system. REFERENCES [1] Power System voltage Stability by Carson. W. Taylor. [2] “Performance Operation and control of EHV transmission system” by A. Chakrabarthy, D.P. Kotari, A.K. Mukopadyay [3] Power System Dynamics: Stability and Control by K. R. Padiyar., BS Publications. [4] Short course on “Power System Voltage Stability” by Dr. Prabha Kundur, Cuernavaca, Morelos. [5] “Electrical power systems” by C.L. Wadhwa. [6] Small Signal Stability Assessment with Online Eigen value Identification Based on Wide Area Measurement System. Xiaorong Xie, Member, IEEE, Shuqing Zhang, Jinyu Xiao, Jingtao Wu, Ying Pu. [7] An open source power system analysis toolbox, Prof. Dr. Federico Milano. [8] “Power System Stability” by Edward Wilson Kimbark. [9] Power Generation, Operation and Control by Allen. J. Wood, Bruce F. Wollen berg.

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