A Simple Contingency Selection for Voltage Stability Analysis

The paper deals with a selection of potentially critical contingencies from the voltage stability view-point. Consideration of all contingencies is impractical. Before contingency analysis starts, one can state, that certain contingencies need not be taken into account. In the paper, after a short overview of the existing papers, in which the problem of the selection of contingencies is considered, the new method for selection of contingencies is presented. A selection of contingencies is based on results of testing of possible power transfers in a power network after the contingencies occur. During the tests, assessments of extreme values of active and reactive power flows, which change when power system state changes, are taken into consideration. In the paper, a computational example of utilization of the described method is given. At the end, the most important features of the method are pointed out. DOI: http://dx.doi.org/10.5755/j01.eee.19.10.5887


I. INTRODUCTION
One of the purposes of dispatcher is to ensure a secure operation of a power system.Among calculations made in a dispatcher centre to achieve this purpose there is also the contingency analysis, i.e. analysis of the loss/failure (L/F) of a small part of the power system (e.g. a transmission line), or L/F of individual equipment such as a generator or transformer [1].
The exhaustive investigations of contingencies (assuming consideration of all possible cases) are time-consuming.They comprise many cases of contingencies, which can be recognized as insignificant already at the beginning of investigation [1].There exist many papers, in which the methods for contingency selection are presented [2]- [10].Desired features of the contingency selection methods are high computation speed and effectiveness.The existing algorithmic methods are featured by short computation time but also by effectiveness, which is reduced by their drawbacks, e.g.dependence on masking effect of different contingencies.To overcome disadvantages of the algorithmic methods, different artificial intelligence techniques are used to develop more beneficial procedures for contingency selection, e.g.[7]- [9].Analyzing existing papers, it can be noted, that new methods, which would be Manuscript received March 12, 2013; accepted August 23, 2013.
better from the view point of the mentioned desired features, are still searched.
The aim of the paper is presentation of a new method for the contingency selection from the voltage-stability view point.In the paper, a new approach for the mentioned contingency selection is presented.It is an algorithmic one.It assumes examination of the necessary condition of fulfilment of power flow equations for a considered power system, what ensures possibility of detection of all critical contingencies.

II. GENERAL IDEA OF SEEKING CRITICAL CONTINGENCY IN A POWER NETWORK
For a stable power system, power flow equations are satisfied.In this paper, the necessary condition of that fact (further called Condition N) is formulated as: "The power delivered to a node is equal to the power received from it".When L/F of a branch in a power system occurs, the following cases can be distinguished: (i) there are changes of values of power flows in the power system but there are no changes of directions of these flows (Case 1), (ii) there are changes of values of power flows in the power system as well as at least some of directions of these flows (Case 2), (iii) there is no state of a power system for which power flow equations are satisfied (Case 3).In Case 3, there is no possibility to find solution of the power flow equations for the considered power system.
In the paper, it is assumed, that when L/F of any branch has place: (i) there are no changes of loads in the considered power system, (ii) Case 1 is taken into account, (iii) a possibility of delivery of power to a node is examined with use of knowledge about extreme values of power flows on branches of a power system.The way of calculation of the mentioned extreme values of power flows is described in Section V.
If under earlier-given assumptions, for L/F of a branch, it is not possible to satisfy Condition N for at least one of the nodes with which the mentioned branch is connected, then the considered contingency is qualified as a candidate for the contingency analysis.In the set of such contingencies, there are L/Fs of branches for which in a power system, Case 2 or Case 3 can have place.However, in the paper, we follow the principle (further called as Principle G): "It is better to consider more contingencies in the contingency analysis than to ignore a critical contingency in this analysis".

III. IDEA OF SIMPLE SEEKING CRITICAL CONTINGENCY IN A POWER NETWORK
In a radial power network, if we observe L/F of an individual branch and in consequence there is not possibility to meet Condition N, such power as before cannot be delivered to the appropriate load nodes.In this situation, there is not possibility to find a state of a power system in which such powers as initially could be received from all load nodes.The mentioned L/F of a branch is a contingency, which should be taken into account in contingency analysis.
Generally, in a looped power network, one can point out such load nodes which are powered by more than one branch.In not all cases, L/F of an individual branch leads to unfulfillment of Condition N for the considered node.If there are several nodes which are connected with the examined load node (let us assume that it is the node Nx), then the power delivered to all these nodes should cover the power required by the node Nx and power losses on branches connecting the node Nx with neighboring nodes.Further, the last sentence is called as Condition MN.If from any reasons Condition MN is not fulfilled, then possible to find a state of a power system in which such power, as before an event violating Condition MN occurs, can be received from the node Nx.If the earlier-mentioned event is L/F of the individual branch connected to any node (for example the node Ny) neighboring with the node Nx, then unfulfillment of Condition MN is a consequence of unfulfillment of Condition N for the node Ny.It should be underlined, that not always unfulfillment of Condition N for the node Ny leads to unfulfillment of Condition MN.
Searching for the set Ω, being a set of events which potentially cause unfulfillment of Condition MN, can be performed by examination of events causing unfulfillment of Condition N for all nodes which are neighbors to the node Nx,.In that situation, the set Ω can contain events, for which in fact Condition MN is fulfilled.According to Principle G, that idea is utilized in the paper.The set Ω is a set of contingencies which in this paper are proposed to consider in contingency analysis.The important feature of the mentioned idea of searching the set Ω is its simplicity.
The earlier-described idea of seeking events, which violate the condition of delivery of appropriate power to the node Nx, can be generalized for cases when the nodes, for which Condition N is examined, are not directly connected with the node Nx.

IV. INDICES USED FOR EXAMINATION OF THE CONDITION N
FOR A DISTINGUISHED NODE To find, that the required active power P or the required reactive power Q can occur on the branch between nodes i and j (i.e. the branch i-j) in a power system, we should know the extreme possible values of the active power or the reactive power for the branch i-j.These extreme values for particular branches are utilized during examination of Condition N for the nodes of the considered power system.
For the purposes of examination of Condition N for the node j, the following indices for particular branches connected with the mentioned node are tested: ,min \ , when , , when , where Jp+, Jp-are the sets of the numbers of nodes connected with the j-th node by the branches, at which ends at the j-th node the active power flows are positive and negative, respectively; Jq+, Jq-are the sets of the numbers of nodes connected with the j-th node by the branches, at which ends at the j-th node the reactive power flows are positive and negative, respectively; l is the number of node connected with the j-th node by the branch, of which L/F is considered; Pji, Qji are, respectively, active and reactive power flow on the branch i-j at the node j; If any of the earlier-defined indices is less than 0, then the branch l-j should be taken into account during the contingency analysis.L/F of that branch causes unfulfillment of Condition N for the node j.

V. EXTREME VALUES OF THE POWER FLOWS ON THE BRANCH
In a stable state of a power system, the power flow equations are satisfied.The mentioned statement means that for every branch, the active as well as reactive power flow does not exceed appropriate extreme value.The extreme value of active or reactive power flow on the considered branch can be found analysing equation of power flow for that branch.
In the paper, the pi-model of the branch is taken into consideration (Fig. 1).The complex power flow on the branch i-j (Fig. 1) at the j-th node is as follows: where Vi is a magnitude of Vi; Zj, φj are a magnitude and an argument of Zj, respectively; Zij, θij are a magnitude and an argument of Zij, respectively.The extreme values of Pji and Qji with respect to Zj are as follows: Pji,extr = Sji,extr cos φj, Qji,extr = Sji,extr sin φj, where where

VI. IDEA OF ADVANCED SEEKING CRITICAL CONTINGENCY
IN A POWER NETWORK If many branches deliver power to the considered node (e.g. the node j), Condition N can be fulfilled but in fact, L/F of certain branch (let us assume the branch l-j) from the mentioned ones is critical contingency.To reveal such contingency, testing branches on paths of delivery of required power to the node j, but not comprising the branch l-j, should be performed.During that test, possibility of transmission of the increased power by every considered branch is verified, using the following indices: where Pxy, Qxy are, respectively, active and reactive power flow on the branch x-y at the node x before L/F of the branch l-j occurs; Pxy, Qxy are, respectively, increase of the active and reactive power flow on the branch x-y at the node x as a result of L/F of the branch l-j.
It is assumed, that whole power delivered by the branch l-j to the node j is taken into account for calculation of Pxy and Qxy.Hereby, in the paper, the worst conditions of a transfer of the power required at the node j are considered.If there is a path comprising branches, for which the indices ηP1,xy, ηQ1,xy are positive and a generator feeding path can the required power then there is no basis to state that L/F of the branch l-j is a critical contingency.
If at least one of the indices ηP1,xy, ηQ1,xy for a branch x-y is negative, then the appropriate power cannot be delivered by the considered path to the node j, i.e. for this path the test of possibility of delivery of the required power to the node j is negative.If for the node j, only such power delivery paths can be found, then L/F of the branch l-j is a critical contingency.

VII. PRINCIPLE OF THE METHOD
The method comprises the following steps: 1. Calculation of extreme values of power flows at the ends of the branches of the considered power system.
2. Simple examination of fulfilment of Condition N for L/Fs of separate branches and creation of the set Scc.The set Scc is a set of contingencies proposed for taking into account in contingency analysis.If at least for one of terminal nodes of the branch i-j, its L/F entails unfulfillment of Condition N, then the mentioned contingency is qualified as a potentially critical one and it is inserted into the set Scc.
3. Advanced examination of fulfilment of Condition N for L/Fs of separate branches for which the examination in the Step 2 does not give the base to ascertain that the considered contingency is potentially critical one.If contingency is recognized as potentially critical one, it is inserted into the set Scc.

VIII. COMPUTATIONAL EXAMPLE
The example shows a part of calculations for the IEEE 118-bus test system.During the calculations, the modifications of loads in considered test system, which are shown in Table I, were assumed.In this example, the part of the test system is considered (Fig. 2).The power flows at the ends of the branches shown in Fig. 2 are given in Table II.Positive value of an active power flow or an inductive reactive power flow indicates that this flow enters the node, at which it is considered.In the example, L/Fs of the branches connected to the node 23 (Fig. 2) are taken into account.
At the beginning of the utilized procedure, the extreme values of power flows at the ends of branches of the test ystem (see Section V) are calculated.Results of these calculations are shown in the Table II.
The indices: ηP,ij, ηQ,ij, ηP,ji and ηQ,ji for branches connected to the node 23, which are shown in Table III, are calculated using the data from Table II   To verify effectiveness of the presented method, the n-1 contingency analysis for the test system has been realized.The results of that analysis for the considered part of the test system are in Table III, as well.The n-1 analysis was performed using the polar coordinate system (PCS) and the rectangular coordinate system (RCS) in power flow calculations.One can point out cases, when calculations in different coordinate systems give different results.We have the cases in which the results of power flow calculations: (i) are in a permissible area (PA), (ii) are close to the permissible area, (CP), (iii) are far from a permissible area (FP), (iv) can not be obtained, as the calculations are nonconvergent (NC).In the cases PA, CP, FP, magnitudes of the node voltages differ from the nominal value not more than 10 %, 10-20 % and more than 20 %, respectively.The n-1 analysis shows that only L/Fs of branches 22-23 and 23-24 can be recognized as critical contingencies.When L/F of the branch 23-32 is taken into account, the result of the n-1 analysis cannot be considered as pointing to this contingency as a potentially critical one.In this case, the nonconvergence of the power flow calculations in PCS should be treated as an effect of numerical problems.
Analysis of the whole test system allows noticing, that a number of branches, of which L/Fs are qualified by described method as potentially critical contingencies is essentially less than total number of branches in the system.The mentioned number of potentially critical contingencies is about 67 %.

IX. CONCLUSIONS
The presented method allows a selection of contingencies (a contingency screening) for the voltage stability analysis, i.e. solving one of the problems existing before a contingency analysis, which are the contingency screening and the contingency ranking.The method is based on the original approach assuming utilization of the necessary condition of fulfilment of power flow equations for a power system, which is as follows: "The power delivered to a node is equal to the power received from it".It is possible to make examination of particular L/F of branches from the view point of the mentioned necessary condition if extreme values of the power flows on the branches of a power system are known.In the paper, the formulas for these extreme power flows are derived assuming the pi-model of a branch.In other papers, a branch is characterized only by resistance and inductive reactance.The earlier-indicated formulas have been derived for a branch as the element of a looped power network and not as the element of a radial power network what is presented in the literature.
Utilization of the above-given necessary condition assures that a list of selected contingencies can be larger than the list of only critical contingencies, but the obtained list comprises all critical contingencies.That fact is an important feature of the presented method.Other feature of the method is that calculations are performed separately for particular branches using only relatively simple operations on results of load flow calculations.

Fig. 1 .Fig. 1 .
Fig. 1.The assumed pi-model of the branch.Zij = Rij + j Xij, Bij is a half of the capacitive susceptance of the branch.

Fig. 2 .
Fig. 2. The considered part of the test system.

TABLE I .
MODIFICATIONS OF LOADS IN THE CONSIDERED TEST SYSTEM.
. Analysing Table III, we can see that for each considered branch at least one of the mentioned indices is negative.In this situation, L/F of each branch given in Table III should be considered as potentially critical contingency.

TABLE II .
POWER FLOW DATA.

TABLE III .
RESULTS OF ANALYSES.