Electrical Power and Energy Systems 53 (2013) 810–817

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Electrical Power and Energy Systems

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Time domain transient state estimation using singular valuedecomposition Poincare map and extrapolation to the limit cycle

0142-0615/$ - see front matter � 2013 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.ijepes.2013.06.003

⇑ Corresponding author.E-mail addresses: rcisneros@faraday.fie.umich.mx (R. Cisneros-Magaña),

amedinr@gmail.com (A. Medina).

Rafael Cisneros-Magaña, Aurelio Medina ⇑Facultad de Ingeniería Eléctrica, División de Estudios de Posgrado, U.M.S.N.H., Ciudad Universitaria, C.P. 58030 Morelia, Michoacán, Mexico

a r t i c l e i n f o

Article history:Received 8 May 2012Received in revised form 29 April 2013Accepted 6 June 2013

Keywords:Limit cycleMeasurement equationObservabilityPower qualityState estimationTransients

a b s t r a c t

This paper proposes an alternative methodology based on the Singular Value Decomposition (SVD) toevaluate the Transient State Estimation (TSE) in the time domain of a power system. Transient phenom-ena such as faults, sags and load changes can be estimated with the TSE, covering over, normal and under-determined cases, according to the number of measurements and state variables related through themeasurement state estimation equation; proposed as being a function of measurements and their deriv-atives. The system observability can be determined by means of SVD. If the system is unobservable theSVD can determine which parts of the network are observable. The TSE takes partial measurements, cal-culates the best estimated state variables and completely determines the system state representing atransient condition. The periodic steady state following a transient condition is obtained through a New-ton method based on a Numerical Differentiation (ND) process, Poincaré map and extrapolation to thelimit cycle; this method is applied before and after the simulated transient. The TSE results are validatedthrough direct comparison against those obtained with the Power Systems Computer Aided Design/Elec-tromagnetic Transients Program including Direct Current (PSCAD/EMTDC) simulator.

� 2013 Elsevier Ltd. All rights reserved.

1. Introduction

The Power Quality State Estimation (PQSE) is an important topicfor the power system operation to deliver the electrical energywith good quality indexes, the TSE can be classified as part of thePQSE [1,2].

The methods to solve the time domain TSE are the recursivenormal and weighted least squares, least mean squares [3],SVD, Kalman filter [4], optimization methods [5] and neural net-works. The least square methods have disadvantage to solve thestate estimation mainly in under-determined cases or ill condi-tioned matrices; Kalman filter requires the initial previous stateand the covariance matrices; the optimization methods and neu-ral networks demand a high computational effort when are ap-plied in the time domain. A method based on the SVD isproposed to solve the TSE in association with the ND method toobtain the periodic steady state of the power system. The SVDadvantages are the solution of ill conditioned matrices andunder-determined cases also the intrinsic observability analysisduring the state estimation [6].

The TSE can be seen as a transient simulation reverse process.Fig. 1 shows this relationship, the main objective is to calculate

the best estimates of the state variables under a transient conditionwhere they are not monitored. An electrical system can be statespace represented by a first order ordinary differential equations(ODE) set and the output equation. The state estimation measure-ment equation (z = Hx) is formulated based on the state spaceequations and on the network topology relating the measurementsto the state variables; each measurement adds an equation to setup the measurement equation [7,8]. From the TSE results, it is pos-sible to identify the disturbance location by inspecting the currentand voltage mismatches within the network. TSE can be used toquickly locate the disturbances to take prompt corrective actions[9,10].

TSE uses a limited number of measurements; these data can becontaminated with noise and can possibly have gross measure-ment errors. The choice of measurement points and quantities tomeasure are important aspects to take into account as this willinfluence the system observability [12] and if the measurementequation is over, normal or under-determined [8,13].

TSE estimates fluctuations in the waveforms; one approach is touse the criterion of minimizing the squared error sum between themeasured and estimated values (least square estimation LSE). Thiserror indicates the state estimation accuracy [14,15]. The statespace formulation matrices are used to follow the system dynam-ics in the time domain and to evaluate the state and output vari-ables; with these values any variable in the system can becalculated [16].

Fig. 1. Reverse process relating the transient simulation and transient stateestimation [11].

Table 1Variable options as measurements.

Measurement System variable Definition

y Cx + Du Output variabledy/dt CAx + CBu + Du Output variable derivative

R. Cisneros-Magaña, A. Medina / Electrical Power and Energy Systems 53 (2013) 810–817 811

The conventional state estimation includes in its measurementvector, nodal voltages, power flows, power injections and currentmagnitudes. The measurement equation is nonlinear and demandsan iterative solution. If the measurement equation is linear, as inTSE, a direct solution is performed. In state estimation a useful cri-terion is the weighted least squares; this can be used with the nor-mal equation to solve the estimation, but this formulation failswhen the measurement matrix is ill conditioned; under thiscondition an alternative is to use the SVD [16].

A Newton method based on a ND process, Poincaré Map andextrapolation to the limit cycle is used to find the periodic steadystate before and after the transient state, the TSE is solved underthe periodic steady state to be sure that only one transient is pres-ent in the network; the ND method reduces the execution time andthe computational effort of the simulated transient mainly whenthe considered system has long time constants or is under-damped, this is possible due to the reduction of the cycles to beprocessed [17].

The rest of the paper is organized as follows: Section 2 detailsthe proposed methodology for the TSE; with three subsections toexplain the SVD decomposition, the ND method and the numericalderivative. Section 3 presents the case studies, for over, normal andunder-determined cases to verify the TSE. Section 4 presents themain conclusions drawn from this research work.

2. Methodology

The methodology includes the next four steps to solve the timedomain TSE using the SVD and the ND method,

(1) Definition of H measurement matrix selecting output vari-ables and their derivatives as measurements and SVDdecomposition of H to define the observability and the sys-tem condition.

(2) ND determines the periodic steady state for the network toremove the initial transient and obtains a more convenientinitial condition to evaluate the TSE avoiding divergenceproblems.

(3) SVD solves the TSE including over, normal and under-deter-mined cases. The estimated state variables, measurementsand the physical laws are used to evaluate the rest of vari-ables in the network.

(4) ND obtains the new periodic steady state of the power sys-tem after the estimated transient condition.

The state space system model is:

_x ¼ Axþ Bu ð1Þ

y ¼ Cxþ Du ð2Þ

The state estimation measurement equation can be structured withvarious options, Table 1 shows the variables that can be taken asmeasurements [10].

This work proposes to take as measurements a combination ofoutput variables and their derivatives; this ensures a linear mea-surement equation. If a state variable can be measured accordingto the proposed model, this variable can be used to set up themeasurement equation. The measurements set defines themeasurement equation, i.e.,z ¼ Hx ð3Þ

This equation is solved using the least squares criterion for the esti-mation error when the H matrix is non-singular, by means of thenormal equation formulation or using the SVD [7,9]:

x ¼ ðHT HÞ�1HT z ¼ VW�1UT z ð4Þ

When H is ill conditioned or under-determined, SVD gives a solu-tion using the pseudo-inverse,

xþ ¼ VWþUT z ð5Þ

Eq. (1) relates the state variables with the system dynamics; it canbe linear or nonlinear depending of the system and its components.In this work the linear time invariant case is analyzed. Eq. (3) can beadded with a vector v representing the measurements noise [7,9]and is related with the state space output Eq. (2) as,

z ¼ Hxþ m ð6Þ

The state estimation error is e = z � zestimated = z � Hxestimated

2.1. Singular value decomposition [18]

The SVD gives a unique solution when the system is over or nor-mal-determined (m > n or m = n, m-measurements, n-states), butwhen the system is under-determined (m < n), the SVD can givea solution with minimum norm and establish the system observ-ability. The observable states are estimated and the unobservableareas of a power system can be delimited [19]. SVD can be usedto verify the system observability before or during the state esti-mation [20], [21]. SVD factorizes the measurement matrix H, i.e.,

H ¼ UWVT ð7Þ

The condition number of a matrix is the division of the largest to thesmallest singular value. A matrix with a large condition number isill conditioned. If a singular value is zero or near zero, a zero isplaced in the corresponding diagonal element of W�1 instead of 1/w, then if,

W ¼S 00 0

� �ð8Þ

Its pseudo-inverse is defined by:

Wþ ¼ S�1 00 0

" #ð9Þ

The pseudo-inverse of H is,

Hþ ¼ VWþUT ð10Þ

This gives the pseudo-inverse solution,

xþ ¼ Hþz ¼ VWþUT z ð11Þ

Fig. 2. IEEE 14 bus test system.

Table 2Pre-fault conventional TR and ND methods convergence process.

NC TR ND

1 6.447332e�01 6.447332e�012 5.973785e�02 5.973785e�02

..

. ... ..

.

34 1.269982e�06 6.469466e�0335 1.259118e�06 3.652759e�11

..

. ...

1117 9.992150e�11

812 R. Cisneros-Magaña, A. Medina / Electrical Power and Energy Systems 53 (2013) 810–817

The pseudo-inverse yields the solution of the least-squared error; ifmultiple solutions exist, it provides the solution with minimumnorm [18,19].

2.2. Numerical differentiation method (ND) [22]

The ND method is used to efficiently calculate the periodicsteady state in the time domain. If a first order ODE set is the math-ematical model of a system and this has a periodic steady statesolution, this can be calculated using the Poincaré Map and anextrapolation process of the state variables to the limit cyclethrough Newton methods [23]. The ODE set is represented by:

_x ¼ f ðx; tÞ ð12Þ

The state variables at the limit cycle x1 can be evaluated as in [22],i.e.,

x1 ¼ xi þ Cðxiþ1 � xiÞ ð13Þ

C ¼ ðI �UÞ�1 ð14Þ

where xi is the state variables vector at the base cycle beginning, xi+1

is the state variables vector at the base cycle ending, U is the iden-tification (state transition) matrix, C is the iteration matrix,I unit matrix.

The ND is a Newton method used to identify the state transitionmatrix U [22]. It is applied to speed-up the state variables conver-gence to the limit cycle and thus to achieve a fast periodic statesolution of the power system. The method can be summarized asfollows: a base cycle x(t) is obtained, after the time domain statevariables solution over a number of periods of time (cycles). Witha perturbed value at the base cycle beginning, the differences be-tween the last two values at the end of the cycle are then calcu-lated to obtain Dxi+1, which allows to identify U by columns; C isevaluated, to calculate x1, which represents an estimate of thestate variables at the limit cycle. This process concludes once thedifference between two consecutive evaluations of x1meets a con-vergence criterion, e.g., 10�10. For a detailed explanation of thisnumerical method please refer to [22], [24].

For the case studies to follow, this technique is applied after twoinitial cycles, followed by a base cycle and one ND applications andagain is applied after the transient is removed to reach the finalperiodic steady state; the initial state variable values are set tozero. After the initial application of the ND method, a conventionalnumerical integration method based on the trapezoidal rule (TR) isapplied. The obtained results with the proposed TSE method arevalidated through direct comparison against the PSCAD/EMTDCsimulator response [25] and the actual system response, obtainedwith the TR conventional method. This limit cycle extrapolation al-lows a reduction of the processed cycles and the computation ef-fort in the TSE solution, consequently, the program executiontime is reduced for a possible real time state estimation, specifi-cally when the system has long time constants or is under-damped. The ND method is useful when is of interest to simulatethe transient condition under the periodic steady state and withoutthe influence of other system transients being the simulated tran-sient the only one in the system [24], [26].

2.3. Numerical derivative

The measurements numerical derivative is approximatelycalculated, as:

f 0ðxÞ � ðf ðxþ hÞ � f ðxÞÞ=h ð15Þ

The accuracy of this expression depends on the f function and thetime step h [18]. The measurements and their derivatives are used

to define the measurement equation, z = Hx, using the state spaceformulation, as it was illustrated in Table 1.

3. Case studies

Fig. 2 shows the IEEE 14 bus test system analyzed in the casestudies detailed next, the system data are taken from reference[27], considering a base power of 100 MVA and a base voltage of132 kV; real values are used to validate the results by directcomparison against the PSCAD/EMTDC simulation.

The lines 1–2, 1–5, 2–3, 2–4, 2–5, 3–4 and 4–5 are representedby a pi model and by a series impedance the rest of lines; thetransformers 4–9, 5–6, 4–8-9, are represented by an inductivereactance, according to the IEEE 14 bus test system.

A transient condition is simulated with a fault at node 5,reducing the rated resistance load from 2320.1 to 74.4 ohms, start-ing at the 10th cycle (0.066 s) and ending at the 12th cycle (0.1 s) ofsimulation. This change originates a transient condition that can beestimated from a partial set of measurements.

The integration TR is used to solve the 31 ODE set, taking linecurrents and nodal voltages as state variables, and with a step sizeof 512 points per period, i.e., 32.5 ls, for a frequency of 60 Hz. Thepre-fault plus fault time are 0.1 s. The ND method is applied oncetwo initial periods of time (cycles) and a base cycle are calculated;

Table 3Final conventional TR and ND methods convergence process.

NC TR ND

1 9.855697e�01 9.855697e�01

..

. ... ..

.

33 1.280940e�06 1.849277e�0134 1.269982e�06 2.580195e�1035 1.259118e�06 3.247402e�15

..

. ...

1134 9.993077e�11

Table 4Over-determined case.

Measurement Measured variables Estimated variables

R. Cisneros-Magaña, A. Medina / Electrical Power and Energy Systems 53 (2013) 810–817 813

then the state transition matrix U is kept constant during subse-quent applications of (13) to obtain x1, as shown in Table 2 usingpu values. This will give the pre-fault periodic steady state solutionof the power network. If the ND method is applied 35 cycles areprocessed, if TR method is used 1117 cycles must be processedfor the considered case to fulfill a 10�10 tolerance (Table 2). Afterthe fault is removed, an initial cycle and a base cycle are calculatedbefore the ND is applied. With U kept constant, (13) is appliedtwice to update x1 and to obtain the final periodic steady state,as shown in Table 3. For this case, 1134 full cycles (NC) were re-quired by the TR method, while 35 were needed by the ND method,or just 3.08% of NC calculated with the conventional TR method toobtain the periodic steady state. For this particular case and usingreal quantities during the iterative solution of the assumed testcase, the maximum error in the state variables using the conven-tional process based on the TR numerical integration method couldnot be reduced below 10�7. The number of cycles to process de-pends on each case study and mainly of the system characteristics,i.e. if it is under, normal or over-damped system and of its compo-nent time constant values [22]. The processed cycles vary for eachcase but if the system is under-damped, the ND extrapolation tothe limit cycle reduces the computational effort needed to obtainthe TSE solution, prior to a periodic steady state operationcondition.

The measurement equation z = Hx is defined by selecting theoutput variables and their derivatives as measurements and defin-ing elements of the z measurements vector, measurement matrix Hand state variables vector x; relating the measurements with thestate variables.

Fig. 2 shows the measurements set; chosen to have an observa-ble system condition, being enough to evaluate the TSE by meansof the SVD using the pseudo-inverse solution (11). They are currentmeasurements to estimate the set of state variables defined in themeasurement equation [28]. With the measurements set, esti-mated variables and their derivatives, and using the Kirchhoff’sand Ohm’s laws, it is possible to calculate the rest of the systemstate and output variables. The complete system state can be esti-mated from a partial set of measurements [29].

The measurements set is taken from a Matlab transient simula-tion using the ND method. The results obtained with the proposedTSE method are validated against the response given by the PSCAD/EMTDC simulator. A close agreement was obtained in all cases, asis illustrated in the following sections.

The measurement linear equation is solved using the SVD algo-rithm covering the over, normal and under-determined cases. Thesystem observability and the numerical condition of the measure-ment matrix H are analyzed.

z1 x2 Line current 2–3 x20 Node voltage 3z2 x3 Line current 2–4 x21 Node voltage 4z3 x1 Line current 1–5 x22 Node voltage 5z4 x4 Line current 2–5 x22 Node voltage 5z5 x9 Line current 5–6 x23 Node voltage 6z6 x7 Line current 4–7 x24 Node voltage 7z7 x10 Line current 6–11 x28 Node voltage 11z8 x11 Line current 6–12 x29 Node voltage 12z9 x12 Line current 6–13 x30 Node voltage 13

3.1. Over-determined case

This case presents the condition when there are more availablemeasurements from the power system than state variables to beestimated, normally the system is observable and all the state vari-ables can be estimated. The measurements are correct and free of

errors or excessive noise and the received values are adequate tosolve the state estimation.

The TSE applies the SVD taking nine measurements from thetime domain system solution and estimates eight state variables,with these measured and estimated state variables, the completesystem solution can be determined using the equilibrium systemequations, Kirchhoff’s and Ohm’s laws. The measurement pointsare indicated in Fig. 2; the z vector is formed using expressionscombining the measurements and their derivatives, as it wasestablished in the methodology section. The following expressionsdefine this measurement set:

z1 ¼ m2 � x2R23 � L23 _x2 ¼ x20 ð16Þ

z2 ¼ m2 � x3R24 � L24 _x3 ¼ x21 ð17Þ

z3 ¼ m1 � x1R15 � L15 _x1 ¼ x22 ð18Þ

z4 ¼ m2 � x4R25 � L25 _x4 ¼ x22 ð19Þ

z5 ¼ L56 _x9 ¼ x22 � x23 ð20Þ

z6 ¼ x7R47 þ L47 _x7 ¼ x21 � x24 ð21Þ

z7 ¼ x10R611 þ L611 _x10 ¼ x23 � x28 ð22Þ

z8 ¼ x11R612 þ L612 _x11 ¼ x23 � x29 ð23Þ

z9 ¼ x12R613 þ L613 _x12 ¼ x23 � x30 ð24Þ

Table 4 shows the measured and estimated variables.This definition gives the following linear measurement

equation,

z1

z2

z3

z4

z5

z6

z7

z8

z9

266666666666666664

377777777777777775

¼

1 0 0 0 0 0 0 00 1 0 0 0 0 0 00 0 1 0 0 0 0 00 0 1 0 0 0 0 00 0 1 �1 0 0 0 00 1 0 0 �1 0 0 00 0 0 1 0 �1 0 00 0 0 1 0 0 �1 00 0 0 1 0 0 0 �1

266666666666666664

377777777777777775

x20

x21

x22

x23

x24

x28

x29

x30

266666666666664

377777777777775

ð25Þ

This over-determined linear measurement equation is solved usingthe SVD inverse (4) evaluating the state variable vector x (nodalvoltages); the system is completely observable. Fig. 3 shows theactual and estimated solutions and their difference. The maximumdifference is negligible, only at the instant of the transient and inthe voltage node with fault (node 5), it is of the order of 4%; thisis due to the sudden change in the load condition of this node.

As selected variable, the estimated voltage at node 5 (x22) isshown in Fig. 4; this is the faulted node. The actual, estimated

Fig. 3. Actual, proposed TSE and difference for nodal voltages from 0.05 to 0.1 s ofsimulation, the transient starts at 0.066 s and ends at 0.1 s including the final cyclescorresponding to the final ND method application, 1 represents the limit cycle.

Fig. 4. Actual, proposed TSE and PSCAD/EMTDC responses, voltage node 5,transient: 0.066–0.1 s.

Table 5Applied methods in the numerical process.

Cycle Method Condition

1–2 TR Initial cycles3 ND Base cycle4 ND U constant5–10 TR Pre-fault11–12 TR Fault13 TR Initial cycle14 ND Base cycle15–16 ND U constant

Fig. 5. Calculated line current 3–4, state variable 5, over-determined systemcondition.

Table 6Normal-determined case.

Measurement Measured variables Estimated variables

z1 x2 Line current 2–3 x20 Node voltage 3z2 x3 Line current 2–4 x21 Node voltage 4z3 x1 Line current 1–5 x22 Node voltage 5z4 x9 Line current 5–6 x23 Node voltage 6z5 x7 Line current 4–7 x24 Node voltage 7z6 x10 Line current 6–11 x28 Node voltage 11z7 x11 Line current 6–12 x29 Node voltage 12z8 x12 Line current 6–13 x30 Node voltage 13

814 R. Cisneros-Magaña, A. Medina / Electrical Power and Energy Systems 53 (2013) 810–817

and PSCAD/EMTDC simulation results are shown, giving a veryclose agreement between the obtained waveforms.

Fig. 4 and Table 5 show two initial cycles, a base cycle and a NDapplication to obtain the pre-fault periodic steady state; then theconventional TR of integration is applied to simulate the transient,after this a base cycle and two ND applications are applied to ob-tain the final periodic steady state. Tables 2 and 3 show the NDconvergence process for the pre-fault and final periods. The NDmethod process 35 cycles and the TR method process 1134 cyclesto fulfill a 10�10 tolerance (Table 3). Fig. 5 shows the waveforms for

the line current 3–4 for the actual, estimated and PSCAD/EMTDCsimulation. A close agreement between the responses is obtained.The PSCAD/EMTDC response is compared from 0 to 0.116 s (6–13cycles), when the TR method is applied and the transient loadcondition is present.

The state variable 5, line current 3–4 is not a measured orestimated variable, it must be calculated taking into account themeasured, estimated state variables and the equilibrium laws inthe system.

3.2. Normal-determined case

This case is present when the available measurements are equalin number to the estimated state variables, regularly the system isobservable and all the state variables can be estimated. This condi-tion can be originated by measurement redundancy or lost mea-surements in reference to an over-determined case due tofailures of instruments or communication links.

The measurement Eqs. (16)–(24) set, has a redundancy in theover-determined case; the measurements z3 and z4 are the same(z3 = z4 = x22). If the measurement z4 is eliminated, the number ofmeasurements and the state variables is the same and equal toeight; this condition is the normal-determined case. The stateestimation using the SVD gives the same results of Figs. 3–5.The system has a complete observable condition with thisnormal-determined measurement set. Table 6 presents the mea-sured and estimated variables.

The TSE simulation can be used to determine the location orsource of transients in a power system. Fig. 6 shows the differencein nodal voltages as a function of time between the simulations of

Fig. 6. TSE nodal voltage difference, with disturbance minus without disturbance,indicating the node where the transient is originated, node 5 has the largest change.

Fig. 8. Voltage node 13, state variable 8, cannot be estimated with the under-determined condition. The actual value is in reference to the high voltage side.

Table 7Under-determined case.

R. Cisneros-Magaña, A. Medina / Electrical Power and Energy Systems 53 (2013) 810–817 815

TSE, with and without the disturbance. It can be observed fromFig. 6 that the node with the largest change in voltage is thenode 5, indicating that this node is the transient source. Bothsimulations, with and without transient, were simulated under anormal-determined condition.

Measurement Measured variables Estimated variables

z1 x2 Line current 2–3 x20 Node voltage 3z2 x3 Line current 2–4 x21 Node voltage 4z3 x1 Line current 1–5 x22 Node voltage 5z4 x9 Line current 5–6 x23 Node voltage 6z5 x7 Line current 4–7 x24 Node voltage 7z6 x10 Line current 6–11 x28 Node voltage 11z7 x12 Line current 6–13 x29 Node voltage 12

x30 Node voltage 13

3.3. Under-determined case

This condition is present when the available measurements arefewer than the state variables to be estimated, in most cases thesystem is unobservable and the unobservable areas have to bedelimited, only the observable state variables can be estimated,e.g., when the number of installed measurements is limited andless than the state variables or loss of measurements in referenceto a normal or over-determined case due to instrument errors,communication faults or excessive noise.

If the measurement z8 is eliminated, in reference to thenormal-determined case, there are only seven measurements toestimate eight state variables; this condition gives the under-determined case. The SVD is able to obtain a solution under thiscondition for the first seven state variables, but the state variable8, corresponding to the eliminated row in the measurement equa-tion cannot be estimated, and consequently, the associated statevariables cannot be calculated, creating an unobservable island in

Fig. 7. Actual, proposed TSE and difference values for the under-determined casefrom 0.05 to 0.116 s and the final period, 1 represents the limit cycle.

the system. The SVD estimation result is shown in Fig. 7. From thisfigure, it is observed that the state variable 8 is not estimated dueto the measurement equation condition. Fig. 8 shows the state var-

Fig. 9. Unobservable islands due to lost measurements.

Fig. 11. Proposed TSE measurements estimation from noisy measurements.

816 R. Cisneros-Magaña, A. Medina / Electrical Power and Energy Systems 53 (2013) 810–817

iable 8, actual and estimated values, the estimated value is equal tozero. All the other state variables are well estimated under thiscondition, despite of the under-determined measurement equationand the unobservable system, again, a close agreement is observed.This is a SVD advantage with respect of the normal equation for-mulation that gives no solution with this condition. Table 7 showsthe measured and estimated variables.

If the current measurement in line 6–12 (z8) is lost, an unob-servable island is originated where the current lines and nodalvoltages cannot be estimated, as shown in Fig. 9. It can also be ob-served that if the measurements line 6–13 (z9), line 6–11 (z7), andline 5–6 (z5) are sequentially lost, unobservable islands areoriginated.

This under-determined case illustrates the special situationwhen the measurement matrix H is ill conditioned. If the originalmeasurement in row 6 (z6) is repeated in row 7, as a redundantmeasurement and the SVD is applied; one of the singular valuesresults equal to zero, the corresponding rank of H matrix isrank = 6 and the state variables are n = 7; consequently, there isone singularity, the condition number is infinite. However, theSVD is able to estimate the 1–6 state variables, but the 7 and 8state variables cannot be estimated due to the singularity andthe under-determined system condition. Fig. 10 shows the actual,proposed TSE and their difference under this condition. Please ob-serve that the maximum difference is 8 kV or 4% in transient statefor the state variable 5.

Besides, the simulations were performed without measure-ment noise. However, during the measurements acquisitionand transmission from the system to the control center, theyare susceptible to be noise-contaminated, due to malfunctioningequipment or a noisy transmission environment. This noise ad-versely affects the state estimation accuracy, but it does not af-fect its solvability. Under this situation, the under-determinedcase for the TSE is again calculated. The results for the actualand estimated measurements are illustrated in Fig. 11; theseare calculated using (6) and e = z � zestimated = z � Hxestimated;the measurements are contaminated with 1% of noise, which isgenerated by means of a stochastic process. From the results ob-tained and especially from the drawn overall difference, it isclear that the noise affects in a direct proportion the TSE evalu-ation. For this case, the estimated maximum difference is of2.5 amps when the peak current is of 270 amps; this represents

Fig. 10. Proposed TSE with ill conditioned matrix H, the state variables 7 and 8cannot be estimated due to a singularity and the under-determined systemcondition, 1 represents the limit cycle.

an approximate average error of 0.93% according with the 1% ofmeasurements noise.

4. Conclusions

A time domain transient state estimator TSE has beendeveloped and implemented using SVD to estimate the systemtransient state and to analyze the system observability, reviewingthe singular values and the null space vectors to define the obser-vable and unobservable areas in the analyzed system. From the ob-tained results, it has been clearly evident the possibility of using acombination of an output variable and its derivative to form thestate estimation measurement equation. The ND Newton methodhas been applied to obtain the periodic steady state for the systemunder study when the initial and final transients are simulated, toprovide convenient state variable values to efficiently obtain thepre-fault and the final periodic steady state solution, once the faultis removed, reducing the cycles to be numerically processed whenis of interest to simulate the transient, prior to the periodic steadystate, without the influence of other system transients and for apossible real time implementation, especially when the system isunder-damped.

The over, normal and under-determined conditions of the sys-tem have been analyzed by means of the SVD given a solutionfor each condition. The condition number of the measurement ma-trix H must be analyzed to determine if this matrix is ill condi-tioned, and if this is the case, it is required to verify the systemobservability and to establish the observable and unobservablesystem islands.

The proposed TSE method has been compared against the actualsystem response obtained with the conventional numerical inte-gration TR method and that obtained with the widely acceptedPSCAD/EMTDC simulator, yielding a close agreement betweenthem, in all the analyzed cases, validating the proposed TSEmethod.

It has been shown that the TSE can be applied to locate thesources of transients in a power system, comparing the TSE solu-tions with and without the disturbance condition. From the differ-ence in the state variables, the location of the transient source canbe determined when the system is observable. When the system ispartially observable, the comparison can be made by only consid-ering the nodes that are in the observable part of the system; forthis case, it is not possible to analyze the elements and nodes thatare in the unobservable part of the system.

R. Cisneros-Magaña, A. Medina / Electrical Power and Energy Systems 53 (2013) 810–817 817

It has been illustrated that a measurement noisy condition ad-versely affects the TSE response, in direct proportion to the per-centage of noise. Any condition of this type must be corrected asmuch as possible to avoid an inaccurate state estimation.

5. Acknowledgments

The authors gratefully acknowledge the UniversidadMichoacana de San Nicolás de Hidalgo through the División deEstudios de Posgrado, Facultad de Ingeniería Eléctrica (DEP-FIE),Morelia, México, for the facilities granted to carry out this investi-gation. The first author wishes to thank the scholarship granted bythe Consejo Nacional de Ciencia y Tecnología of México (CONACYT)to finance his doctoral studies at the DEP-FIE.

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