05994153.pdf

QV and PV curves as a Planning Tool of Analysis Pablo Guimarães,

Ubaldo Fernandez Tito Ocariz Administración Nacional de Eletricidad (ANDE) Assunción, Paraguai [email protected] [email protected], [email protected] Fritz W. Mohn Câmara dos Deputados Brasilia – DF, Brazil [email protected]

A. C. Zambroni de Souza Federal University at Itajubá Itajubá, MG – Brazil [email protected]

Abstract- This paper deals with the problem of system voltage security. In this sense, load margin, QV curve and system loss reduction are focused. The idea is to use these tools in the planning scenario to determine the best locations for installation of distributed generation . For this purpose, from a base case, the system load margin and its losses are calculated. Besides that, the reactive power margin of each bus is obtained by using the QV curve, such that all the necessary information may be collected. In particular, the application of these techniques to the Paraguayan system is appealing, since this system contains a metropolitan system radially connected to a weak area, which raises some important issues associated with voltage stability. Keywords: voltage stability, QV curve, loss reduction I. INTRODUCTION Planning the expansion of a power system is a complex task that involves different issues like transient and voltage stability, reliability and protection schemes. The computational tools developed may include dynamic and static aspects of the system, covering a wide range of likely studies. As for voltage stability studies, in general a voltage collapse point is associated with a singular Jacobian. At this singularity point, known as saddle node bifurcation, the power flow Jacobian matrix has a zero eigenvalue. No solution may be obtained beyond this critical point, which is the system maximum loadability point. Induced bifurcation may also occur, due to sudden exhaustion of the reactive power sources. Voltage collapse is a local or, at most, a regional phenomenon . Hence, identifying the system critical bus (or buses), i.e., the

bus where the problem is originated, or mostly related to, is important. These critical buses constitute the set of candidate buses for reinforcement against voltage collapse. In addition to these buses another set of buses of interest are the ones most likely for loss reduction when reactive power compensation is considered. The effects of the reinforcement in the voltage stability may be assessed by the system load margin, obtained with the help of the continuation method. Such a method is based on a predictor and a corrector step and traces the bifurcation path from a base case. The load/generation increase direction may consider different scenarios, so it may emulate interesting operating situations. Another way of assessing the system voltage security is by the means of loss reduction. This idea comes from the fact that voltage collapse may be associated with high values of loss. In this sense, reducing the loss may keep a system away from voltage collapse. As discussed in the literature, this is not correct, since loss reduction and voltage collapse may be different problems. However, reducing the loss in the system critical area may produce good results, and loss reduction should be considered during the voltage security studies. Combining voltage stability and loss reduction studies provide interesting results that may be evaluated by load margin calculation, so the system robustness is focused. Another way of analyzing the system robustness is by using the QV curve [1]. This curve provides the reactive power margin of each bus and is obtained by transforming a PQ bus into a PV one and varying its voltage level. For each voltage level considered the reactive power generated is stored, generating a plot

A C. Zambroni de Souza would like to thank CNPq, CAPES and FAPEMIG for financial support.

978-1-4577-0365-2/11/$26.00 ©2011 IEEE

1601

representative of the bus robustness with respect to reactive power support. This paper proposes a strategy for planning scenario based on the theoretical tools described above. For this sake, the real Paraguayan system is employed, with all its limits taken into consideration. The results are obtained under real operating possibilities and the effectiveness of the proposed actions is discussed. II- TECHNIQUES TO CALCULATE LOAD MARGIN AND LOSS REDUCTION

This section describes how the continuation method works, and the importance of the tangent vector calculated during the process. After presenting the continuation method, a discussion about the system loss and the critical area takes place. The sensitivity technique used to identify the generators to act in the redispatch is also presented, as well as the idea of shunt compensation. 2.1. Continuation Method and Special Features Continuation methods may be used to trace the path trodden by a power system from a stable equilibrium point up to a bifurcation point [2], [3]. Such a methodology is based on the system model f(x, λ) = 0

where ||.|| stands for tangent vector norm. From this expression, the steeper the curve, the smaller the predictor step, and vice versa. The method takes bigger steps as the system is far from the bifurcation point, and smaller steps as the bifurcation is approached. The actual operating point is obtained with the help of the corrector stage. Corrector step Corrector step is obtained by the inclusion of an extra equation. Such equation arises from the fact that the predictor and corrector vectors are perpendicular to each other. An alternative to this step is obtained when the new operating point is obtained using the predictor step as the initial guess for a load flow program. In general, it converges in few iterations. III- LOSS SENSITIVITY BASED ON TANGENT VECTOR Tangent vector calculated in equation (2) is used here as a new tool for system losses sensitivity. This novel approach is based on the information provided by tangent vector, i.e., how state variables vary as a function of system parameter. The total system active power losses are given by: nl

Psystem =

(1)

∑V

Vjk ( Gk ( cos( δ (ij)k ) + cos( δ (ji)k )) - Gk (Vik

ik

k=1

where x represents the state variables and λ is a system parameter used to drive a system from one equilibrium point to another. This type of model has been employed on numerous voltage collapse studies, with λ being considered as the system load/generation increase factor or the power transfer level. Two steps move the system along the bifurcation path: predictor step and corrector step.

2

+Vjk )

(8)

where nl is the number of transmission lines, Vik and Vjk are the voltage level at ends (i) and (j) of transmission line k.

Predictor step

Gk is the transmission line k susceptance.

Predictor step is used to indicate a direction to move. Tangent vector may be used for this purpose, and is given by

δ (ij)k represents the phase angle between buses i and j .

⎡ Po ⎤ ⎡ Δθ ⎤ 1 -1 TV= ⎢ =J ⎢ ⎥ ⎥ ⎣Qo ⎦ ⎣ΔV ⎦ Δλ

If equation (8) is derived in relation to system parameter λ , it is obtained:

(2)

where J is the load flow Jacobian, θ and V the state variables (angle phase and voltage magnitude, respectively), and Po and Qo are the net active and reactive powers connected to each bus. TV is the shortage for tangent vector. The length of predictor step is then given by Δλ = 1/||TV||

2

d P system = dλ 2 Gk (Vik where

1602

nl

∑

d V ik

k =1

d V ik dλ

(

+

dλ

d V jk dλ

Vjk +

Vjk )

d V jk dλ

Vik ) A + Vik Vjk

(4)

dA dλ

A = Gk ( cos( δ (ij)k ) + cos( δ (ji)k )

where:

dA/dλ = Gk (2 sin ( δ (ji)k ) (d δ (i)k /dλ − d δ (j)k /dλ))

k is a scalar value used to speed up or slow down the computation.

Equation (4) shows how active power losses vary as a function of system parameter. Notice that all the partial derivatives of equation (4) consist on tangent vector components, known from equation (4). Therefore, computing equation (4) is not time consuming. Assume in equation (4) that the right-hand side is slightly perturbed through a capacitor installation at a generic load bus “l”. The new tangent vector may be obtained with no need of calculating the new operating point. If equation (4) is then calculated, the active power losses variation as a function of parameter λ (capacitor installation at bus “l” ) is known. Taking “l” as all system load buses, one by one, computation of equations (2) and 4) indicates the load bus, or buses, whose capacitor installation reduces at most the system active power loss. Similarly, the right-hand side may be modified by considering an active power injection at a bus “l”, simulating the input of a generator into the system. Note that this generator may also produce reactive power, enhancing the system voltage profile. Notice that in this process, a power flow program is executed only for the base case. IV. QV CURVE ANALYSIS The QV curve method [4],[5]has been used as a planning tool by many utilities, a practice that should often be complemented by dynamic studies [6],[7] The QV curve analysis should be performed in conjunction with PV curves. Using the QV curve may help engineers to identify critical buses in the system as well as the reactive power injections needed at those buses to ensure voltage security. The possibility of reducing the computational cost associated with the calculation of QV curves is analyzed here. As stressed for the PV curve, the whole curve is not the focus. Rather than that, the point of minimum is meant, yielding the reactive power margin for the bus analyzed. For this purpose, a novel approach, named QV continuation method is employed, as described next QV Continuation Method The idea is to trace the QV curve with the help of controlled steps until reaching the minimum. The step size is given by:

k Δλ = TVQ

||.|| denotes the tangent vector norm, and TVQ is calculated as:

TVQ = J −1Q1

(6)

In equation (6), J is the power flow Jacobian. Q1 is a vector with all zeros except for the reactive power associated with the bus under study. It is important to mention that the reactive power limits for all other PV buses are taken into account. Computing equation (5) provides the step length. The voltage level at the PQ buses is updated as:

ΔV = k *

TVQ TVQ

(7)

Calculating the step length in equation (5) and updating the state variables by equation (7) cause no problem. During the power flow convergence process, the bus under study is considered as a PV bus. However, the step length calculation and the voltage level correction are executed considering this bus as a PQ bus. This happens regardless the original type of the bus under study. The computational cost for obtaining the step length and the correction term is, however, very low since it is only executed after convergence of the Newton method. At this stage, the Jacobian matrix is known, and has already been factorized. As a consequence, the computational load is slightly increased. V. METHODOLOGY Because the Paraguayan system presents some weak areas with respect to voltage stability, the idea is to assess how the system responds when a small generation unit is installed. Distributed generation becomes a valid short-term solution of improving the system performance, while the transmission reinforcements, expected to secure the load serving capacity for such areas, overcome licensing and procurement problems. The effects in the system load margin (LM), system losses and reactive power margin (RPM) are evaluated. In this sense, the following options buses are considered for installation: Critical buses identified by tangent vector,

(5)

Buses with the least and larger reactive power margin, calculated by the QV curve,

1603

Buses candidate for active power injection aiming the system losses reduction. The results obtained are compared, so a good planning strategy is obtained. VI. TEST RESULTS IEEE Test Systems The proposed methodologies have been firstly tested by using the IEEE test systems. Because the qualitative results obtained are similar for all the test systems, only the 57 bus system is discussed. The idea is to consider small generating units in the buses indicated by the following indices: Tangent vector, base case (TVB): The bus with the largest entry in the tangent vector calculated at the base case. Reactive Power Margin (RPM): Bus with the least reactive power margin at the base case. Sensitivity to loss reduction (SLR): Bus most sensitive with respect to loss reduction when an active power injection is considered.

15

57

33

57

31

33

57

20

57

32

53

53

53

30

25

31

33

30

42

53

56

30

31

33

53

53

42

33

From Table I one can see that adopting the RPM as the criterion of compensation, injection above 30 MW does not produce good results. If the other indices are used to identify the best buses for active power injection, a better distribution of the injected MW is observed. This also happens for the IEEE 14 and 30 bus systems, as depicted in Fig. 1. Fig.1 depicts the results obtained when 30 MW are distributed according to the proposed methodologies. Note that the largest increase in the LM as well as the largest loss reduction is obtained by the MixPV Group. These results are similar to the ones obtained by the SLR and TVC groups. This flags an interesting signal for planners, since a reduced set of PV buses is considered for control actions. In order to assess the methodology proposed, the small units summing 30MW have been either randomly placed or located in some buses with low voltage levels. The results obtained

Tangent vector at the voltage collapse point (TVC): The most critical bus at the voltage collapse point indicated by the tangent vector. Mixing the techniques (Mix): A set of buses is obtained by mixing the above techniques. This yields to a reduction in the number of buses to be considered. The bus which produces the higher global system RPM is chosen., Mixing the techniques for PV buses (MixPV): Just like the previous index, but the gain in the RPM is focused only for the PV buses. Table I depicts the results obtained:

Fig. 1: Comparison of the results obtained (30 MW) for the IEEE 57 bus

Cumulative gen. capacityer

TABLE 1: Location of Generators by the proposed techniques Bus location for Additional generator Case 1: TVB

Case 2: RPM

Case 3: SLR

Case 4: TVC

Case 5: MIXPV

Case 6: MIX

5

33

33

31

33

31

33

10

33

32

33

57

57

31

and discussed above have a better performance, rendering the proposal as effective. In general, for the IEEE test system, the best results are obtained when the MixPV, TVC and SLR groups are considered to experiment the control actions. Results for the ANDE System

1604

This section presents the results obtained when the Paraguayan Subsystem 1 (SS1) is employed.

only described here for academic purposes, but has not been considered for planning scenarios.

Employing the continuation method from the base case yields a small load margin about 5%, as shown in Figs. 2 and 3, for some buses of 220 and 66 kV, respectively.

It is important to mention that the two only reactive power sources of the Metropolitan Region present a negative RPM since the base case. This may have some impact if contingency analysis is carried out [8]. The ANDE system is dependent on these reactive power reserves, so if they reach their limits, the system becomes unstable.

Convergence problems are observed when the load/generation increase produces an operating point close to the bifurcation. This point is reached because bus SVC installed at SLO

As a function of this “strong dependence” of these sources, an extra criterion has been proposed. It considers bus candidates based on the impact of generation injection to the system static var compensators (SVCs) performance. Figs 4 to 6 show that better results are obtained when the increase of the RPM of these PV buses is the focus, alleviating the system voltage control.

Fig. 2: PV Curves for buses of 220 kV Fig. 4: RPM analysis of the SVCs

Fig. 5: LM increase of the ANDE system

Fig. 3: PV Curves for buses of 66 kV

substation reaches its limit of 150 MVAr. From this point on, the only reactive power reserve for the Metropolitan Region is the SVC installed at LIM Substation (250 MVAr). When this source is depleted, the system voltage control is lost, reaching an induced bifurcation point [9]-[11]. A saddle-node point may be reached, for this system, if the reactive power limits are neglected and tap changers work in order to control the voltage level. This unrealistic situation is

1605

[11] R. A. Schlueter, “A Voltage Stability Security Assessment Method”, IEEE Transactions on Power Systems, Vol. 13, No. 4, pp. 1423-1438, November 1998.

Fig. 6: Loss reduction in the ANDE system

VII. CONCLUSION This paper proposed a planning tool based on some voltage stability criteria. For this purpose, a combination of loss reduction, reactive power margin and load margin increase are focused. Small generation units are considered as control actions aiming to analyze different operating scenarios and some groups of buses are considered. First, the IEEE academic systems have been tested and the encouraging results stimulated to employ the proposed technique in the Paraguayan system. This system has been chosen because of its voltage security fragility, so the results could be better assessed. Once again, good results have been obtained and some characteristics not observed in the academic systems could be explored in the Paraguayan system. The methodology proposed here may be used for other systems with no restriction, since different groups of buses are identified and tested as candidates to host the proposed control actions. Comparison with criteria currently used shows the methodology as effective. VIII. REFERENCES [1]

Fritz W. Mohn, A. C. Zambroni de Souza, Tracing PV and QV Curves with the Help of a CRIC Continuation Method. IEEE Transactions on Power Systems, v.21, p.1104 - 1114, 2006.

[2]

D. A. Alves, L. C. P. da Silva, C. A. Castro e V. F. da Costa., Continuation fast decoupled power flow with secant predictor, IEEE Transactions on Power Systems, Vol. 18, No. 3, Aug. 2003, pp. 1078 - 1085.

[3]

D. A. Alves, L. C. P. da Silva, C. A. Castro, and V. F. da Costa, Parameterized fast decoupled load flows for tracing the power systems bifurcation diagrams, 1999 Proc. IEEE PES Summer Meeting, pp. 708-713.

[4]

P. Kundur, Power System Stability and Control. Palo Alto: McGraw-Hill, 1994.

[5]

T. V. Cutsem, C. Vournas, Voltage Stability of Electric Power Systems, Kluwer Academic Publishers, 1998.

[6]

B. H. Chowdhury, C. W. Taylor, “Voltage stability analysis: V-Q power flow simulation versus dynamic simulation”, IEEE Transactions on Power Systems, vol. 15, No. 4, pp. 1354-1359, Nov. 2000.

[7]

Show-Kang Chang, Vladimir Brandwajn., “Adjusted solutions in fast decoupled load flow”, IEEE Transactions on Power Systems, Vol.3, N. 2, pp. 726-733, May 1988.

[8]

A . C. Zambroni de Souza, Fritz W. Mohn, Isabella F. Borges, Tito R. Ocariz, “Using PV and QV Curves with the Meaning of Static Contingency Screening and Planning”, accepted for publication by the Electric Power Systems Research.

[9]

V. Venkatasubramanian, “Singularity Induced Bifurcation and the van der Pol Oscillator”, IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, Vol. 41, No. 11, pp. 765-769, November 1994.

[10] M. K. Pal, “Voltage Instability Considering Load Characteristics”, IEEE Transactions on Power Systems, Vol. 7, No. 1, pp. 243-249, February 1992.

1606