eeh - ETH Zürich · List of Acronyms OPF Optimal Power Flow SCOPF Security Constrained Optimal...

eeh power systemslaboratory

Bing Li

An Optimal Power Flow Formulation

Including Risk of Cascading Events

Master ThesisPSL1313

Department:EEH – Power Systems Laboratory, ETH Zurich

Examiner:Prof. Dr. Goran Andersson, ETH Zurich

Supervisor:Dipl.-Ing. Line Roald, ETH Zurich

Dr. Frauke Oldewurtel, ETH Zurich

Zurich, September 2013

Acknowledgement

I would like to gratefully acknowledge the enthusiastic supervision of LineRoald, who was abundantly helpful and offered invaluable assistance, sup-port and guidance during this work. Deepest gratitude are also due toDr. Frauke Oldewurtel, without whose knowledge and assistance this thesiswould not have been successful.

In addition, I would like to thank Prof. Goran Andersson for providing methe opportunity to work on my thesis in his research group.

Besides, I am grateful to all my friends from ETH Zurich and Politecnico diTorino, for being the surrogate family during the many years I stayed hereand for their continued support there after.

Finally, I wish to express my love and gratitude to my beloved families; fortheir understanding and endless love, through the duration of my studies.

Bing LI

Abstract

This thesis introduces a novel risk measurement tool which can facilitate thedaily decision making process of power system operators. Current optimiza-tion routines, using the ’N-1’ criterion, suffer from the drawback of beingunable to differentiate contingencies of differing probabilities and severity,hence seeking to avoid them all equally, and hence sub-optimally. A risk-based Optimal Power Flow (RBOPF) algorithm is proposed that constrainsthe probability of secondary line tripping in lieu of the traditional ’N-1’ se-curity constraints. A full cascade simulation is built and used to illustratethe performance advantages of RBOPF against existing security assessmenttools, such as Security Constrained Optimal Power Flow (SCOPF), in bothcost reduction and risk mitigation. A case study provides supporting evi-dence of risk reduction in terms of lower operating stress in the system andless exposure to occurrence of cascading outages.

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Contents

1 Introduction 11.1 Background of Cascading Outages . . . . . . . . . . . . . . . 11.2 Need for Risk Assessment of Cascading Outages . . . . . . . 2

2 Review of Existing Methods 42.1 Deterministic Approach . . . . . . . . . . . . . . . . . . . . . 4

2.1.1 Security Constrained Optimal Power Flow (SCOPF) . 42.2 Risk-based Approaches . . . . . . . . . . . . . . . . . . . . . . 6

2.2.1 Overview of Risk Indices . . . . . . . . . . . . . . . . . 62.2.2 Model of Contingency Probability . . . . . . . . . . . 72.2.3 Model of Severity . . . . . . . . . . . . . . . . . . . . . 8

3 Risk-based Optimal Power Flow 113.1 Risk Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . 113.2 Incorporate Risk Measure into an OPF . . . . . . . . . . . . . 153.3 Choice of Risk Limit . . . . . . . . . . . . . . . . . . . . . . . 183.4 Iterative Optimization . . . . . . . . . . . . . . . . . . . . . . 193.5 Alternate Form . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Probabilistic Cascade Simulation 244.1 Overview of the Method . . . . . . . . . . . . . . . . . . . . . 244.2 Initial Contingency Simulation . . . . . . . . . . . . . . . . . 27

4.2.1 Initial System State . . . . . . . . . . . . . . . . . . . 274.2.2 Simulate Single Outage . . . . . . . . . . . . . . . . . 274.2.3 Post-contingency Flow Calculation . . . . . . . . . . . 27

4.3 Calculate Prij . . . . . . . . . . . . . . . . . . . . . . . . . . 274.4 Monte Carlo Simulation . . . . . . . . . . . . . . . . . . . . . 27

4.4.1 Generation of Random Outages . . . . . . . . . . . . . 284.4.2 Number of Monte Carlo Runs . . . . . . . . . . . . . . 284.4.3 Island Detection . . . . . . . . . . . . . . . . . . . . . 294.4.4 Restore Load-Generation Balance . . . . . . . . . . . . 294.4.5 Overload Simulation . . . . . . . . . . . . . . . . . . . 30

4.5 Severity Evaluation . . . . . . . . . . . . . . . . . . . . . . . . 31

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CONTENTS iv

5 Case Studies 335.1 Description of Test System . . . . . . . . . . . . . . . . . . . 335.2 Risk Assessment with Fixed Power Load and Risk Limit in

RBOPF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2.1 Evaluate Risk of Cascading Events for SCOPF Solution 355.2.2 Risk Assessment up to Secondary Cascade Stage . . . 375.2.3 Risk Assessment of Full Cascade . . . . . . . . . . . . 40

5.3 Risk Assessment with Fixed Power Load and Varying RiskLimit in RBOPF . . . . . . . . . . . . . . . . . . . . . . . . . 455.3.1 Effect of Risk Limit on Generator cost . . . . . . . . . 455.3.2 Risk Assessment up to Secondary Cascade Stage . . . 475.3.3 Risk Assessment of Full Cascade . . . . . . . . . . . . 48

5.4 Risk Assessment for Varying Risk Limit and Power Load . . . 505.5 Probability of N-1 Constraint Violation . . . . . . . . . . . . 52

6 Conclusions and Future Work 53

Appendix 56

A Data of RTS 96 56

B DC Power Flow 60

List of Acronyms

OPF Optimal Power FlowSCOPF Security Constrained Optimal Power FlowRBOPF Risk-based Optimal Power FlowLODF Line Outage Distribution FactorGGDF Generalized Generation Shift Distribution FactorTSO Transmission System OperatorRTS Reliability Test SystemLOLP Loss Of Load ProbabilityRBSA Risk-based Security AssessmentEENS Expected Energy Not ServedVOLL Value Of Lost LoadMTTF Mean Time To FailMTTR Mean Time To RepairCLT Central Limited TheoryIEEE Institute of Electrical and Electronics EngineersRES Renewable Energy SourceCEI Catastrophic Expectation Index

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List of Symbols

CG Generation bidPG Power generated by each generatorPres Power generated by renewable energy sourcePL Load powerP0,j Pre-contingency flow on jth linePL,j Post-contingency flow on jth linePmax,j Maximum limit of jth linePtrip,j Trip limit of jth lineEi Post-contingency state of ith contingencyθ Voltage angleB Bus admittance matrixx Line impedancek Indicator of system statePri Probability of ith contingencyPrij Secondary tripping probabilitySev Severity of contingencyRisklimit Limit of risk constraintK Number of contingencies consideredλ Failure rateµ Repair rate

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Chapter 1

Introduction

1.1 Background of Cascading Outages

In power systems, cascading outage is a sequence of one or more dependentcomponent outages which are initialized by one or more common distur-bance(s). Cascading outages can either terminate before interrupting theelectricity service or continue until causing a blackout. If cascading goesout of control, it can cause massive disruption to electricity service. Forinstance, in the blackout event of August 14, 2003, 30 million people in theU.S. Northeast and Southeastern Canada suffered from the power supplyinterruption. According to the data provided in [1], the number of black-outs that lead to the outages of 1000MW or more doubles every 10 years.Many of these blackouts were made worse by cascading outages.

There are many sources of a cascading outage, such as lighting or othernatural disasters, contact between conductors and human errors. After theinitial event, subsequent outages are propagating in the system through var-ious mechanisms, and particular subsystems must generally be disconnectedfrom the system via the tripping of circuit breakers by relay or through hu-man intervention.

The most common propagation mechanism is cascading overload [8]. Af-ter one transmission line is overloaded and automatically removed from thenetwork by a circuit breaker for its protection, the load it originally carried isadded to other lines, leading to more lines become overloaded. When thereis a drop of the voltage magnitudes at substations, protective devices in thesystem may observe the abnormal condition and trigger the circuit breakers,as a result part or all of the system is de-energized. Other causes of subse-quent trips include hidden failures in protection devices, insufficient reactivepower resources, operation errors, etc. It is possible for several failure mech-anisms to appear in one cascading outage. Figure 1.1 displays the annual

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CHAPTER 1. INTRODUCTION 2

frequency of large blackouts in North America, the events caused by ex-treme natural events (hurricanes, ice storms, etc.) and supply shortages areremoved [8]. Many of these events are initiated by very small disturbancesthen compounded by cascading outages to ultimately result in blackouts.

Figure 1.1: Relative blackout frequency and contributions to blackout riskfrom large North American blackouts in various size categories. Data fromNERC for 19842006.

1.2 Need for Risk Assessment of Cascading Out-ages

A highly interconnected power system can provide economic benefits anda greater level of service reliability. However, the increasing interconnec-tivity and complexity of the power system give rise to the uncertainty inthe grid and thus a higher probability of cascading events. On the otherhand, the competition in power market leads to increased pressure on theeconomic efficiency of grid operation, which forces the Transmission SystemOperators (TSOs) to operate the grid closer to its limits. With increasingoperating stress in the system, transmission grids become more vulnerableto cascading outages. To cope with these changes, a risk-based assessmenttool for directly assessing and mitigating large cascading failures is required.

Chapter II presents an overview of existing methods for avoiding cascad-ing outages in power systems. Two main categories are discussed: determin-istic approach and risk-based approach. In Chapter III, a novel risk measurefor modeling risk of cascading outages is proposed and is incorporated intoan Optimal Power Flow (OPF) formulation. The structure of a full cascade

CHAPTER 1. INTRODUCTION 3

simulation is covered in Chapter IV, with which the impact of cascading pro-cesses and strategies to deal with them can be evaluated. In Chapter V, theperformance of the Risk-based Optimal Power flow (RBOPF) is comparedwith the traditional Security Constrained Optimal Power Flow (SCOPF) inseveral case studies on the one-area Reliability Test System (RTS)-96 sys-tem. The performance criteria are the operating cost, risk of initiating acascade and the impact of a full cascade. The final chapter concludes thethesis.

Chapter 2

Review of Existing Methods

2.1 Deterministic Approach

Power systems are subject to unpredictable disturbances and unavoidableoutages. In order to limit the consequence of these events, power systems areusually operated with a security margin. Maintaining a security margin willresult in a higher cost than the economic optimum because the transmissioncapacity is not fully utilized. The additional cost is ultimately paid for bycustomers. Although this security margin is maintained, it is not enough toabsolutely guarantee the security of the system; system blackouts still occuraround the world.

The current standard for security margins is the ’N-1’ criterion. Thesystem is required to operate without violations of operating requirementswhen any of a predefined list of contingencies occurs. In this approach,cascading outages are avoided through limiting post-contingency flow level.However, this approach has drawbacks. The power system’s operating statecan only take on the the discrete states of secure or insecure, dependingon whether any operating constraints have been violated. Furthermore,there is no assessment of the contingency probability and extent of violation.Thus, the level of operational risk actually faced by the power system is notreflected in this approach.

2.1.1 Security Constrained Optimal Power Flow (SCOPF)

Based on the ’N-1’ criterion, a widely-used formulation of the optimizationproblem is the Security Constrained Optimal Power Flow (SCOPF) formula-tion. Compared to the alternative Optimal Power Flow (OPF) formulation,SCOPF also includes the security constraints, which treat the security ofpower systems in a deterministic way. The system is either secure or in-secure depending on whether the constraint that post-contingency flow is

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CHAPTER 2. REVIEW OF EXISTING METHODS 5

lower than the line limit or not. The optimization aims at minimizing theoperating cost of the system, while satisfying the constraints of:

• power balance during normal state;

• hourly generation bids;

• maximum and minimum operating limits of generators;

• transmission line flow limits at normal state;

• limit on the post-contingency flow for each contingency and each af-fected circuit.

Taking under consideration the contribution of power generated by renew-able energy source (RES), the form of DC SCOPF is expressed as:

min CTG · PG (2.1.1)

subject to:

PG + Pres −Bθ = PL (2.1.2)

Pres = P fcres (2.1.3)

PminG ≤ PG ≤ PmaxG (2.1.4)

−Pmaxmn ≤1

xmn(θm,k − θn,k) ≤ Pmaxmn

for all line mn, k = 0, 1...N

(2.1.5)

θslack = 0 (2.1.6)

In the objective function, CG is the bid and PG is the power generation ofeach generator. The DC power flow equations are given by (2.1.2), whereB is the bus admittance matrix. The details of DC power flow model is in-troduced in AppendixB. The power in-feed from renewable energy source isassumed equal to the forecasted value in (2.1.3). Constraint (2.1.4) containsthe operating limit of generators. There are constraints for both pre- andpost-contingency power flow on each circuit. In (2.1.5), k represents for thestate of the system, when k = 0 the system is in normal state, when k > 0the system is in post-contingency state. The voltage angle on slack bus isset to be zero in (2.1.6).

In the formulation of SCOPF, the constraints of overload are made hardconstraints, but in practice this is not the case. According to the experienceof most control center operators, it is possible to operate the power systemwith a certain extent of overload. The use of hard constraints can result in


unnecessarily high energy cost. Furthermore, even when the flow is operatedunder its maximum limit and there is no problem with under voltage, thesystem is still not necessarily safe. For instance, a fast change of the lineflow with post-contingency flow lower than the line limit may still lead tothe inadvertent tripping of relay.

Another drawback of SCOPF is its inability to distinguish between vi-olations that occurred from the contingencies with different probabilities.In addition, different severities of violations are not distinguished in thismodel. In the next chapter, a risk index which can capture both likelihoodand severity of contingencies will be proposed.

2.2 Risk-based Approaches

Risk-based approach is the technique that is explicitly based on the conceptof risk. Unlike the past reliability indices, such as Loss of Load Probability(LOLP), which reflects the likelihood but not the severity, in this approach,the risk index includes both the probability and the impact of undesirableevents. In addition, most of the past reliability indices were measures ofthe system’s ability to avoid failure, while in risk-based approach, risk indexmeasures the system’s exposure to failure. The risk index is generally inthe form which weights the severity of a contingency with its occurrenceprobability :

Risk = Pr · Sev

Where Pr is the contingency probability and Sev is the severity resultedfrom the corresponding contingency. Risk index can be formulated in dif-ferent ways,Section 2.2.1 provides an overview of the risk indices developedin past publications. Depending on the application, there are risk measuresfor on-line or off-line risk assessment. In Section 2.2.2, the method for es-timating contingency probability is introduced. Then, the approaches forseverity modeling is summarized in Section 2.2.3.

2.2.1 Overview of Risk Indices

Over the last few years, there is a comprehensive series of papers exploringthe formulation of risk indices. McCalley and his colleagues have developedrisk indices for various types of security assessment. In [5], a risk index withthe following formulation is proposed:


Risk(Im|Xt) = E(Im(Xt+1)|Xt)

=

∫xt+1

∫Ei

Pr(Ei, Xt+1|Xt)×Risk(Im|Ei, Xt+1)dEidXt+1

In this measure of risk only a predefined set of contingencies is considered.The risk associated with the pre-contingency condition Xt is measured as theexpected value of the impact of the post-contingency condition Xt+1 giventhe current operating condition, i.e. E(Im(Xt+1)|Xt). The probability ofselected contingencies and uncertainty for the future operating condition areall included in the risk index. One distinctive feature of this index is thatthe impact of a specific contingency for a certain post-contingency conditionis considered to be uncertain, i.e. Risk(Im|Ei, Xt+1). For instance, thereis a probability for an overloaded line to trip, and this probability dependson the wind speed, ambient temperature,etc. Based on this formulation,they proposed a new measure for online Risk-based Security Assessment(RBSA) [6]. Online RBSA is capable of performing security assessment fora near-future condition, therefore it can provide operators with informationabout the consequences of their decisions.

Kirschen and his associates argued that the level of risk can be measuredas an expected outage cost, where the cost is calculated as the product of Ex-pected Energy Not Served (EENS) and Value of Lost Load (VoLL) [4]. Thecomputation of expected outage cost is integrated in a Monte Carlo simula-tion. Instead of using a list of predefined contingencies, random outages aregenerated to initiate the cascade. The consequences of random contingen-cies are measured in terms of load not served. In order to reach acceptableaccuracy, the simulation is repeated until the value of cost reaches sufficientprecision. In this approach, large amount of computation may be involvedwhich put high requirements on the computing resource.

In the research work of Zima [12], a sum of highest probabilities forthe post contingency line trips is used as the overall system risk. It isassumed that the probability of line failure is proportional to the extent ofits overloading. Therefore, by minimizing the slack variable, which measuresthe extent of overload, it is possible to reduce the cascading outage risk.

2.2.2 Model of Contingency Probability

There are different methods for estimating contingency probability. In tradi-tional contingency probability estimation performed for planning purposes,Pr is the probability for a component to be out of service during a longtime interval. Markov Model can be utilized to calculate Pr, as shown in


the Figure 2.1.

Figure 2.1: Single component Markov Model

In the model, λ and µ represent for the failure rate and repair rate in thelong term time frame. λ is the reciprocal of Mean Time To Fail (MTTF) andµ is the reciprocal of Mean Time To Repair (MTTR). Based on these twoparameters, the outage probability of a single component can be obtainedas: λ/(λ+ µ).

Another model can be used to estimate the occurrence probability ofa certain contingency in the next time interval, such as one day. In themodel, the contingency is assumed to satisfy a Poisson distribution. Theprobability of the ith contingency is:

Pri = (1− e−λi) ∗ exp(−∑j 6=i

λj), (2.2.1)

where λi and λj are the failure rates of contingencies i and j in the planningtime horizon, respectively.

2.2.3 Model of Severity

The approaches for modeling severity of cascading events can be summa-rized in the following two categories:

The first category models the effect of overall cascading process, and theparameter such as the cost of Expected Energy Not Served (EENS) is used toevaluate the severity, as discussed in the risk index proposed by Kirschen [4].The severity shows the outcome if a cascade is initiated in the system witha given dispatch and directly reflects the impact of cascading outages onthe customers in the system. The evaluation of such severity involves largeamount of computation and it is not yet suitable for optimization.

In the second category, the severity is modeled based on the system stateafter the initial contingency, such as extent of line overload. This type ofseverity allows system operators to better understand the potential of cas-cading outages for a given dispatch and to take risk reduction actions. The


severity is generally formulated to reduce the probability for an affected cir-cuit to have a secondary trip after the initial contingency. In many researchworks, this probability is assumed to be proportional to the extent of lineoverloading. For instance, in the risk model proposed by Zima [12], theybelieve that the probability will follow a curve as shown in Figure 2.2. It isassumed that the probability for line trip is zero when line flow |F | < Fmax,and 1 if |F | ≥ k · Fmax. Their consideration is that due to the temperatureincrease on an overloaded line, the sag of the line may be so severe to re-sult in a flashover towards the ground or trees and lead to the line trip. Aslack variable s is used to model the extent of overload. Zima proposes aweighting factor Wk corresponding to the slope of the characteristics shownin the figure, which is specific for each line. Therefore probability of postcontingency trip of subsequent line is Wk · sk. The objective function isformulated so as to minimize a sum of highest probabilities for the possiblepost contingency trips of subsequent lines.

Figure 2.2: Probability of the line trip as a function of its loading[12]

An alternative probabilistic model for cascading outage is proposed in [11].It is suggested in the paper that the probability of a secondary trip followsa initial disturbance is a function of: 1) The flow in the device after the dis-turbance and 2) the change of the flow caused by initial disturbance. Thefunction is assumed to be highly nonlinear:

p =eα+β1x1+β2x2

1 + eα+β1x1+β2x2,


where

x1 =|f2|f0

and x2 =|f2 − f1|

f0.

f0 is the line’s flow limit, f1 and f2 are the pre-contingency and post-contingency flow, respectively. Parameters x1 and x2 in this model canbe calibrated by statistical regression. In this paper, disturbance reportsand network models collected from industry or public website are used. Af-ter each disturbance, the line flows are collected and x1, x2 are calculatedaccording to the model. All the line flow change events and associated x1

and x2 are plotted in a diagram with x1 and x2 being the x-axis and y-axis. Those events that caused undesired cascading line trips are markedout. Then the data in this diagram is analyzed by logistic regression rou-tine, which gives the estimation for α and β. One short-coming of heuristicmethods is that the calibration of the model strongly depends on the casesconsidered - it remains a question whether the calibrated model can predictthe cascading event in other systems or situations. For cascading outages,various of cases can appear, therefore the model should be applicable to allthese cases. The methods that involve the estimation for parameters of themodel may not be suitable to be used to monitor the cascading failure inOPF.

In [9], a simple model is proposed to estimate the likelihood of line tripafter initial contingency. It is based on the assumption that the probabilityof a cascading trip of circuit j increases with 1) the post-contingency flowon the line 2) the power flow change on this line. The probability for circuitj to trip after the initial contingency i is given as

Prij =

1, PL,j ≥ 1.25 · Pmax,j

(PL,j−P0,j)(Ptrip,j−P0,j) , 0.9 · Pmax,j ≤ PL,j ≤ 1.25 · Pmax,j0, PL,j ≤ 0.9 · Pmax,j

(2.2.2)

where

P0,j pre-contingency flow on circuit j ;PL,j post-contingency flow on circuit j ;Ptrip,j assumed to be 125% of Pmax,j , the circuit will for sure to trip

if its load level is higher than this value;Pmax,j maximum limit of circuit j.

This formulation of severity is applicable to different cases, and wouldbe promising to be included into an OPF.

Chapter 3

Risk-based Optimal PowerFlow

In this chapter, we propose a risk-based measure to quantify the risk ofcascading outages in power systems. In Section 3.1, the assumptions andphysical motivations for the risk measure is presented. Section 3.2 describeshow the risk-based measure is included into an OPF formulation so as toreduce the risk of cascading outages. Section 3.3 discusses how the choiceof risk limit can affect the security and cost. An iterative algorithm usedto solve the optimization problem is introduced in Section 3.4. At the end,Section 3.5 includes an alternative on the formulation of risk measure.

3.1 Risk Modeling

It is not easy to include a risk measure into OPF algorithms. One of thechallenges is ensuring the requirement on computational power remains low,so that online application of the algorithm remains feasible. Moreover, to in-tegrate the constraint into OPF problem and solve via linear programming,the constraint must be formulated as a linear constraint - not a simple task.

Considering all above reasons, the following risk index is proposed in thisthesis for evaluating the risk of cascading process due to line overload. Itweights the severity of a contingency with the probability that it will occur,and it is contingency- and line-specific:

Riskij = Pri · Sev(Ei) (3.1.1)

where

i No. of contingency;j No. of line;Ei post-contingency state of ith contingency;

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CHAPTER 3. RISK-BASED OPTIMAL POWER FLOW 12

Pri the probability of ith contingency;Sev(Ei) the severity of ith contingency.

The main idea behind this equation is that for the contingency with higherprobability, less severity can be accepted, and for the infrequent contingency,we can tolerate higher severity. In deterministic approach, such as SCOPF,overloads are strictly avoided for all the selected contingencies. However,since most of these contingencies are associated with very low occurrenceprobability, the dispatch strategy generated from this assessment tends tobe overly conservative. As a result, the capacity of existing facilities arenot fully utilized and after a certain point, a high price is paid for relativelysmall gain in security.

The proposed risk index is intended to facilitate dispatch decision mak-ing. Therefore in the index, Pri represents for the occurrence probabilityof ith contingency in the next time interval, such as one day. Comparedto Markov model, Poisson model is more effective in predicting the failurethat will occur in the incoming time period [10]. It doesn’t provide goodrepresentation for the extended future, because the events that may occurin the distant future are not included in the model, such as repair. However,the transition for a single or more component from up state to down stateheavily effect the dispatch decision making, and the transition from downstate to up state by repairing is of little interest. Accordingly, it is assumedthat contingency events satisfy Poisson distribution and the Poisson modelis adopted for the contingency probability evaluation.

Similar to N-1 criterion, our method also focus on avoiding cascading out-ages by minimizing the probability to trip in secondary stage. The severityis modeled as the probability of a secondary trip, which is defined as theprobability that an initial event will lead to another dependent outage andthus start a cascade. By multiplying the probability of initial contingencywith the secondary tripping probability, the risk measure reflects the proba-bility that a cascade will be initiated. Following model is chosen to evaluatethe secondary tripping probability. It has been introduced in Section 2.2.3.Recalling the formulation of Prij :

Prij =

1, PL,j ≥ 1.25 · Pmax,j(PL,j−P0,j)

(Ptrip,j−P0,j) , 0.9Pmax,j ≤ PL,j ≤ 1.25 · Pmax,j0, PL,j ≤ 0.9 · Pmax,j

(3.1.2)

The engineering perspective supporting Prij is that: for a higher loadinglevel, more heat will be generated on the line, which increases the possibilityof conductor sag that can lead to a secondary trip. In addition, a large and


Figure 3.1: Plot of Prij with PL,j for different P0,j

fast change of loading level on the circuit reflects a large transient which maycause inadvertent tripping of relay. The most common practice to protecttransmission lines is to equip them with distance relays. Distance relaysare designed to respond when there is change in current, voltage, or thephase angle between the measured current and voltage. This operation isdone by comparing a relay’s apparent impedance to its pre-defined thresholdvalue. So when there is a large transient on the circuit, the relay may see animpedance change in a very short time period, which activates the distancerelay.

The value of Prij is plotted in Figure 3.1 with varying P0,j , where eachline represents for one pre-contingency flow. It can be seen that for thesame pre-contingency flow, there is a higher possibility of secondary trip ifthe post-contingency flow is larger. The point where each line crosses theaxis is the pre-contingency flow. For the same level of post-contingency flow,Prij is higher if there is a large change of loading level. This is based on theassumption that the relay is more likely to trip if it got high current levelas a result of a large jump than it had the same large current but no jump.Prij is set to be 0 when PL,j is lower than 0.9 · Pmax,j because assumptionis made that the relay will not trip if there is a large jump but its current issmall.

Another interpretation of Prij is that it is the ratio between the actualflow change and the maximum flow change allowed, where Ptrip,j −P0,j rep-resents the maximum acceptable flow change and PL,j − P0,j represents the


actual line flow change. Therefore the larger P0,j is, the smaller the ’jump’towards the higher level is accepted, which means that an additional loadadded to a circuit with its pre-contingency flow close to its limit is verylikely to trip.

Figure 3.2: Four considered cases of line flow change

Taking into account the signs of P0,j and PL,j − P0,j , four conditions ofline flow change are considered, as shown in Figure 3.2. Two different for-mulations of Prij are introduced in order to estimate the cascading outageprobability for these four conditions, each equation has the larger value oftwo equations when its conditions are met:

Prij =

(PL,j−P0,j)(Ptrip,j−P0,j) , P0,j ≥ 0 andPL,j − P0,j ≥ 0

P0,j ≤ 0 andPL,j − P0,j ≥ 0−(PL,j−P0,j)(Ptrip,j+P0,j) , P0,j ≤ 0 andPL,j − P0,j ≤ 0

P0,j ≥ 0 andPL,j − P0,j ≤ 0

(3.1.3)

To better illustrate the difference in identifying severity of initial contin-gency between N-1 criterion and Prij , following diagram 3.3 is presented.This diagram shows the post-contingency state of each circuit, where x axisis the normalized post-contingency flow and y axis is the normalized line flowchange. It can be observed from the diagram that the risk-based methodallows post-contingency flow to be higher than the circuit’s maximum limit.


For instance, some post-contingency states with normal N-1 constraint vio-lations are not considered as risky in the risk-based method. On the otherhand, for the states with no violations of N-1 constraint, they could beconsidered as dangerous if a large transient is induced after the initial con-tingency.

Figure 3.3: Normalized line flow change with normalized post-contingencyflow. Points are for post-contingency state of circuits. Stars are for statesfor normal N-1 constraint violations and circles are for the states with Prijlarger than 0.7

3.2 Incorporate Risk Measure into an OPF

This section illustrates how the proposed risk measurement is incorporatedinto a standard DC OPF formulation. In risk-based OPF (RBOPF), themeasure of risk can be included into OPF either as a part of the objec-tive function or as inequality constraints. For the proposed risk measure,Riskij = Pri · Prij , it works as a linear constraint which sets a limitationon the risk associated with each contingency and each affected line. Insteadof using traditional N-1 security constraints to limit the post-contingencyflow, we include constraints to limit the probability of secondary tripping.The formulation of RBOPF for cascading events is expressed as:

min CTG · PG


subject to:

PG + Pres −Bθ = PL

Pres = P fcres

PminG ≤ PG ≤ PmaxG

− Pmaxmn ≤1

xmn(θm − θn) ≤ Pmaxmn for all line mn

θslack = 0

Pri · Prij ≤ Risklimit (3.2.1)

One advantage associated with this formulation is that the risk is assignable.The risk is computed for each contingency and each affected line, as a result,it is easy to identify which contingency can cause high risk on remaininglines. The optimization variables are the power delivered by generators, thepower generated by renewable energy source and the voltage angle on eachbus:

[PG Pres θ]T

All these variables are for the system in normal state. The formulation ofrisk constraint is:

Pri ·PL,j − P0,j

Ptrip,j − P0,j≤ Risklimit (3.2.2)

To compute the post-contingency flow PL,j , linear sensitivity factors areused. When there is a single-fault or a multi-fault that occurs in the sys-tem, a certain amount of power will be shifted to the adjacent transmissionlines. Line Outage Distribution Factor (LODF) is a sensitivity factor whichcalculates the approximate change in the line flows when there is a lineoutage. The new line flow can be computed with LODF as follows:

PL,j = P0,j + LODFij · P0,i (3.2.3)

Risk constraint (3.2.2) can be rewritten as

Pri · LODFij · P0,i ≤ Risklimit · (Ptrip,j − P0,j)

⇒ PriRisklimit

· LODFij · P0,i + P0,j ≤ Ptrip,j (3.2.4)

Similarly, for generator outage, the post-contingency flow on remaininglines are computed with Generalized Generation Shift Distribution Factor(GGDF), which estimates the approximate change in the line flows whenthere is a generator outage

PL,j = P0,j +GGDFij · PG,i (3.2.5)


where PG,i is the output power of outage generator i. Accordingly, riskconstraint (3.2.2) can be rewritten as

Pri ·GGDFij · PG,i ≤ Risklimit · (Ptrip,j − P0,j)

⇒ PriRisklimit

·GGDFij · PG,i + P0,j ≤ Ptrip,j (3.2.6)

As mentioned in the equation (3.1.3), there is a symmetric formulation ofPrij for the condition that the line flow change is in the negative direction.The corresponding risk measure is included into OPF in a similar way. Itcan be observed that the constraints (3.2.4) and (3.2.6) are very similar tothe normal N-1 constraints, but with different weighting. Furthermore, dueto the simple formulation of risk constraints, they can easily be rewritten aslinear inequality equations, so that the optimization can be solved by linearprogramming directly.

Assuming that ith line is from bus m to bus n, and jth line is from busu to bus v, according to the DC power flow model, there is

P0,i =1

xmn(θm − θn) (3.2.7)

P0,j =1

xuv(θu − θv) (3.2.8)

If replace P0,i and P0,j in (3.2.4) with (3.2.7) and (3.2.8), following equationcan be obtained

PriRisklimit

· LODFij ·1

xmn︸︷︷︸a

(θm − θn) +1

xuv︸︷︷︸b

·(θu − θv) ≤ Ptrip,j . (3.2.9)

Construct this constraint into inequality matrix ismth nth uth vth

0 · · · 0 · · · a · · · −a · · · b · · · −b...

. . ....

...

0 · · · 0...

PGPresθ

≤ Ptrip,j

...Ptrip,N

Similarly, if replace P0,j in constraint (3.2.6) with (3.2.8), we get followingequation

PriRisklimit

·GGDFij︸︷︷︸a′

·PG,i +1

xuv︸︷︷︸b

·(θu − θv) ≤ Ptrip,j (3.2.10)


where b is the same as before, but coefficient a changes to be a′, aboveconstraint can be included into inequality matrix as

ith uth vth0 a′ 0 · · · b · · · −b · · ·

.... . .

......

0 · · · 0...

PGPresθ

≤ Ptrip,j

...Ptrip,N

3.3 Choice of Risk Limit

The goal is to uniformly reduce risk levels throughout the network, thereforethe same limit is set for all the risk constraints. From the risk constraint(3.2.1), following relationship can be obtained:

Prij ≤ Risklimit/Pri (3.3.1)

Depending on contingency probability, different limit will be set on sec-ondary tripping probability. For low probability contingency, a higher sever-ity can be accepted and vise versa.

Because it is necessary for Prij to have a range of [0, 1], and contingencyprobability can vary by 1 or 2 orders of magnitude, it is necessary to nor-malize the contingency probability so that a suitable upper bound is set onPrij . It is desired that Prij , the secondary tripping probability for RBOPFsolution, not exceed the maximum secondary tripping probability of SCOPFsolution, which is denoted Prij,max,scopf . The smallest contingency probabil-

ity is normalized as RisklimitPrij,max,scopf

so that Prij is bounded by Prij,max,scopf .

By also normalizing the largest contingency probability to 1 and linearlynormalizing all probabilities that lie in between, the contingency probabili-

ties are consequently all normalized into the range of[

RisklimitPrij,max,scopf

, 1]

and

the upper bounds of Prij vary in the range of [Risklimit, P rij,max,scopf ].Further, according to the formulation of risk constraint (3.2.2), there is

PL,j − P0,j ≤RisklimitPri

· (Ptrip,j − P0,j),

by grouping different variables, we get

PL,j ≤RisklimitPri

· Ptrip,j +

(1− Risklimit

Pri

)· P0,j .

Since the pre-contingency flow P0,j is limited below the maximum limit, andPtrip,j = 1.25 · Pmax,j , there is

PL,j ≤RisklimitPri

· 1.25 · Pmax,j +

(1− Risklimit

Pri

)· Pmax,j


⇒ PL,j ≤(

1 + 0.25 · RisklimitPri

)· Pmax,j

It can be observed that when a more conservative Risklimit is chosen, areduced value of maximum post-contingency flow can be allowed. Therefore,the choice of Risklimit involves the trade-off between the system securityand economic benefits. The flexibility of choosing Risklimit allows differentoperators a choice of strategy: highly secure, economic-secure or highlyeconomic.

3.4 Iterative Optimization

According to the formulation of severity Prij , it is a piece-wise functionwhich has non zero value only if the post-contingency flow on the circuitis higher than 90% of its maximum limit. However, for the formulationintroduced in the previous section, all the line flow solutions that violatethe risk constraints are optimized without checking if the post-contingencyflows are larger than 0.9 · Pmax or not. In order to incorporate the precon-dition of PL,j ≥ 0.9 · Pmax,j into the optimization, an iterative optimizationalgorithm is utilized. The main idea is that after each optimization, thepost-contingency flow of the affected circuit with risk constraint being ac-tive is checked. If the post-contingency flow is lower than 0.9 · Pmax,j , itmeans that the risk constraint is too conservative. In next optimization, therisk constraint is replaced by the constraint of PL,j ≤ 0.9 · Pmax,j , whichlimits the post-contingency flow in the safe range. Iterations will terminateif all the events with active risk level constraint satisfy the pre-condition ofPL,j ≥ 0.9 ·Pmax,j . Details of how the iterative optimization is performed isdrawn in the flow chart 3.4.

The geometric interpretation of the problem is as shown in Figure 3.5.Area P is the feasible set, and the objective function cTx is linear, whichdisplayed as several level curves. Hence, constraint replacement is permittedbecause the minimization of a linear optimization problem will always be onthe boundary of the feasible set. Considering risk level constraint for lineoutage:

PriRisklimit

· LODFij · P0,i + P0,j ≤ Ptrip,j (3.4.1)

and Post-contingency constraint:

PL,j ≤ 0.9 · Pmax,j (3.4.2)

For an specific event, if the first constraint (3.4.1) is active, the optimalsolution is on the boundary formed with this constraint. However, if thesolution doesn’t violate second constraint (3.4.2), it means the first con-straint is tighter than the second constraint, and hence too conservative.


Figure 3.4: Flow chart for iterative optimization algorithm

If the first constraint is then replaced with the second constraint for thisevent, the feasible set is enlarged, possibly further reducing cost. Sincethe post-contingency flow of this event is still limited within 0.9 · Pmax,j ,which is associated with zero risk, it ensures that the cost is not reduced atthe expense of increased system risk. Moreover, it could be proved that itonly needs several iterations to reach the termination condition of iterativeoptimization. Constraint (3.4.2) is equivalent to

P0,j + LODFij · P0,i ≤ 0.9 · Pmax,j (3.4.3)

By adding some terms and multiplicative factor of PriRisklimit

to both sides,the above equation can be rewritten as

PriRisklimit

· LODFij · P0,i + P0,j︸︷︷︸left

+

(Pri

Risklimit− 1

)· P0,j ≤

1.25 · Pmax,j︸︷︷︸right

+

(0.9 · Pri

Risklimit− 1.25

)· Pmax,j . (3.4.4)


Figure 3.5: Geometric interpretation of an Linear Programming[2]

It can be seen that the parts in the brackets in above inequality equationare the same with constraint (3.4.1), therefore above constraint is tighterthan constraint (3.4.1) if(

PriRisklimit

− 1

)· P0,j >

(0.9 · Pri

Risklimit− 1.25

)· Pmax,j

⇒ P0,j >0.9 · Pri − 1.25 ·Risklimit

Pri −Risklimit· Pmax,j := Pth (3.4.5)

where Pth is the power threshold. As introduced before, Pri is normalized

in a range of[

RisklimitPij,max,scopf

, 1]. Assume Pij,max,scopf as 1, Pri

Risklimitis changing

in the range of[1, 1

Risklimit

]. If plot Pth with varying value of Risklimit,

following diagram can be obtained Figure 3.6. Each line in the diagramrepresents for one value of Risklimit, and all the points on the same linerepresent for different value of Pri. It can be observed that Pth is mainlydistributed below 0.3 ·Pmax,j . However, for most of lines in the system, theirpower flow are larger than 30% of the max limit, which means for most oftime, constraint (3.4.2) is tighter than constraint (3.4.1). When risk levelconstraint is violated, it is very likely that the condition of PL,j > 0.9·Pmax,jis met. Therefore, the events for which constraints need to be replaced arevery rare and the optimization takes limited iterations.


Figure 3.6: Power threshold for different value of Pri and Risklimit

3.5 Alternate Form

Another formulation of Prij can be obtained by replacing the pre-contingencyflow P0, j in the denominator of Prij with the post-contingency flow PL,j

Pr′ij =

1, PL,j ≥ 1.25 · Pmax,j(PL,j−P0,j)

(Ptrip,j−PL,j), 0.9 · Pmax,j ≤ PL,j ≤ 1.25 · Pmax,j

0, PL,j ≤ 0.9 · Pmax,j

(3.5.1)

This formulation is more straightforward regarding the two engineeringexplanations discussed before. For a higher post-contingency flow PL,j , thedenominator of Pr′ij is smaller, which leads to a larger Pr′ij . For the samelevel of post-contingency flow PL,j , Pr

′ij will be higher if the line flow change

PL,j − P0,j is large.

If plot Pr′ij with increasing PL,j , Figure 3.7 will be obtained, with dif-ferent lines representing different P0,j . It can be seen that the plot of Pr′ijhas a profile of inverse proportional function. For a fixed value of P0,j , func-tion value Pr′ij increase slowly with small magnitude of PL,j . As magnitudeof PL,j getting closer to Ptrip,j , there is a steep increasing of Pr′ij , whichmeans the severity of cascading overload will increase rapidly when post-contingency flow is near to the operating limit. It can also be observed thatfor the same value of PL,j , if circuit flow jumps from a lower P0,j , whichmeans there is a large change of circuit flow, a higher likelihood Pr′ij for thecircuit to trip will be resulted.


Figure 3.7: Plot of new P ′ij with PL,j for different P0,j

One difficulty associated with the second formulation of Pr′ij is that itrequires a proper way to normalize P ′ij in order to have its value in therange of [0 1]. Due to time limitation, no further investigation of thesecond formulation of Pr′ij was pursued in this thesis work.

Chapter 4

Probabilistic CascadeSimulation

The risk-based constraint of RBOPF aims at limiting the probability thata cascade is initiated. To measure the consequence of the whole cascadeprocess, a cascade simulation is built to see the full details of cascade pro-cess. The goal is to show that when a system is subjected to the samedisturbance under the same condition, a more secure dispatch will lead toless losses than a less secure dispatch. Section 4.1 presents the structure ofthe cascade simulation model and briefly introduces the general idea behindthe simulation. In Section 4.2, 4.3 and 4.4, each part of the simulation areexplained in details, including the assumptions made and the functions usedin the simulation. Section 4.5 introduces the indices used to measure theimpact of cascade, which can work as a reference for comparing the securityof RBOPF dispatch with SCOPF dispatch.

4.1 Overview of the Method

Due to large number of components existing in power system, there aremany possible combinations of outages. An efficient way to consider all thepossible combinations is by utilizing a simulation model. Since the powersystem is subjected to stochastic outages, computing their cost requires aprobabilistic approach. The probabilistic measure of severity of secondarytrip introduced before is utilized here to decide how probable an affectedline will be disconnected from the system. Random sampling is used to triplines with non zero secondary trip probabilities, which ensures that all thepossible combinations of outages are considered.

The algorithm of the simulation is shown in the diagram 4.1. The sim-ulation consists of two parts. First part is for the initial outage simulationand second part is for cascading outages simulation. The secondary trip-

24

CHAPTER 4. PROBABILISTIC CASCADE SIMULATION 25

ping probability determines whether the cascade will be initiated. For theinitial stage, the state of the system results from a given dispatch and only asingle contingency is taken into account. After the initial outage, the post-contingency power flow is computed.

Based on the pre-and post-contingency flow, secondary tripping proba-bility can be estimated. If there is a circuit with non zero tripping prob-ability, the system goes into the cascading outages simulation. Cascadingoutages are modeled based on the Monte Carlo sample simulation. At thebeginning of each Monte Carlo run, random outages of circuits with nonzero tripping probabilities are generated. After these lines are removed, afunction is used to detect if there is island operation in the system. Theoutput of the function is a matrix that indicates which busses are includedin the island and the total number of islands.

Then for each island it needs to restore the generation/load balance. Inpower balance restoration, the corrective actions such as power redispatch,load shedding are modeled. After that, overload simulation is run for theisland to see if more lines get overloaded. The overload simulation will ter-minate if new islands are generated or there is no more overloaded lines. Incase of new islands generation, the matrix which contains the information ofthe islands is updated. Then repeat the same procedure for next island untilall the islands are checked. Monte Carlo simulation will stop if the numberof samples is enough for the convergence of the losses. In following sections,the details of each block included in the flow chart of cascade simulation aredescribed.


Initial Contingency

Cascade Outages

Run DC power flow

Generate random number to trip lines, check number

of islands

Overload simulation

New island generated?

Update island system

Restore load generation balance for the island

Blackout?yes

no

yes

no

Compute Losses

All islands are checked?

no

yes

Converge?

Ouput

no

Initialize system state

Calculate Pr_ij

For each island

yes

Simulate single outage

Cascade Simulation

Figure 4.1: Flow chart for Cascade Simulation


4.2 Initial Contingency Simulation

4.2.1 Initial System State

At the beginning, all the device in the system are in service and loads equalto the forecast. Power units generate active power according to a givendispatch plan.

4.2.2 Simulate Single Outage

The initial contingency taken into account is single line outage or singlegenerator outage. If a single line is tripped, the corresponding branch isremoved from the system. In case of generator outage, power generation ofother generators in the system is adjusted to compensate the power losses.It is assumed that there is 25% of operating reserve on each generator.

4.2.3 Post-contingency Flow Calculation

Since there is only single outage, it is possible to use linear sensitivity factorsLODF and GGDF to compute the post-contingency flow. These factors aresystem topology dependent, and are calculated based on the topology of thesystem before the initial outage.

4.3 Calculate Prij

This part of simulation determines if the outages will be stopped at thisstage or may continue propagating in the system. The secondary trippingprobability of each remaining line in the system is evaluated. If the post-contingency flow is higher than the tripping limit, the secondary trippingprobability of this circuit is assigned as 1, which means this line will definitelybe tripped in the following simulation. If the post-contingency flow lies in therange of 0.9 to 1 of the maximum limit, the secondary tripping probabilityof this circuit is calculated with equation(3.1.3). In case that the post-contingency flow is under 90% of the maximum limit, the cascading overloadis considered as not severe enough to lead to the circuit trip.

4.4 Monte Carlo Simulation

The cascading outages are modeled with Monte Carlo sample simulation.It consists of two important parts. The first part uses a random numbergenerator to determine if any other lines with nonzero probability of trippingdo in fact get tripped. The second part identifies any remaining circuitsthat get overloaded in the following stages. The indices used to measure theimpact of the cascading outages are evaluated in the simulation.


4.4.1 Generation of Random Outages

In order to simulate random outage, a vector of random numbers in therange of [0, 1] is sampled from a uniform probability distribution, each num-ber is for one remaining line in the circuit. If the random number is smallerthan the corresponding Prij of this circuit, this line is deemed to be failed.It is obvious that for the line with higher Prij , there is a higher possibilityto generate a number smaller than Prij . If there are any lines tripped inthis stage, the simulation continues with island detection and overload sim-ulation. Otherwise another new random numbers are sampled.

4.4.2 Number of Monte Carlo Runs

The sampling will repeat until the computed indices reach the predefinedaccuracy, which means that the variance of the estimated indices’ value issmaller than a predetermined acceptable estimate variance. The numberof samples required for the convergence is caculated with Central LimitedTheory (CLT), the samples’ mean is assumed to be normally distributed.The maximum variance depends on the confidence limits and degree of con-fidence chosen by the user. Suppose that X is the mean of estimated indicesvalue of n independent samples, confidence degree γ is the probability thatthe true population mean µ lies in the interval of [X − L, X + L], as

γ = P (X − L ≤ µ ≤ X + L),

where the confidence limit L is estimated as L = tα/2 × σX , and γ = 1− α.Standard values of α are 0.01, 0.05 and 0.1. The smaller the value of α,the higher the confidence that the true mean µ will lie in the confidenceinterval. When α is specified, the degree of freedom tα/2 can be found fromthe t-distribution table with n − 1 degrees of freedom. σX is the standarddeviation of the indices value estimated from n trials, σ2

X = σ2/n. σ isthe variance of the population distribution, since it is not known, it can bereplaced by the sample variance s. Given the confidence limit L and confi-dence degree γ, the number of samples required to have the convergence ofestimate indices value is calculated from

n = (s2t2α/2)/L2. (4.4.1)

Before starting the cascade simulation, the number of samples requiredfor the random number generation has to be estimated. First a small numberof ”trial runs” is simulated, such as 50 times. For each trial simulation, thenumber of sampling in Monte Carlo simulation is set as 1. Thereby yieldinga sequence of results X1, ..., Xn0 for the indices. Mean value and standard


deviation of each index are evaluated according to

an0 =1

n0(X0 + ...+Xn0) and sn0 =

√√√√ 1

n0 − 1

n0∑i=1

(Xi − an0)2.

The degree of confidence γ is chosen as 0.95, tα/2 for α equals to 0.05 is 1.96.The confidence interval L is chosen as 0.05 ·an0, so that there is 95% level ofconfidence that the true mean is lying within interval of an0±L. The numberof samples required for this precision is calculated from equation (4.4.1). Foreach index, the minimum number of samples required is obtained, and thelargest one of them is used in the cascade simulation. Once the sufficientnumber of Monte Carlo simulation is performed, the expected consequenceof cascade outages can be evaluated by dividing the accumulated value bythe number of samples.

4.4.3 Island Detection

After the contingency, the topology of the system is checked to see if thesystem breaks into islands. If the system is operating in islands, first itneeds to determine which bus belongs to which island. A detect functionis used here, the input for the function is the bus admittance matrix andthe output of the function is a matrix contains the information of the totalnumber of islands in the system and which buses are included in each island.Following is a example of the matrix for island system(

1 0 1 10 1 0 0

),

where the number of rows indicates the number of islands. For ith row, ifthe number of jth column if 1, it means jth bus belongs to ith island.

4.4.4 Restore Load-Generation Balance

If the system breaks into several islands, the balance between the powergeneration and power consumption in each island may not be maintained.The imbalance is measured as the difference between the change of load andthe change of generation.

Power Redispatch

The imbalance will be compensated by adjusting the power generation inall the other generators connected to the system. For the case that there ismore generation than load in the island, generators reduce the power deliveryproportional to their droop or maximum operating limits. Or more econom-ically, first reduce the power generation in most expensive generators and


then reduce the power generation in other generators proportionally. Forthe extreme case that there is no load connected to the island, all the gen-erators are shut down. After generators adjust their outputs, the conditionof generation tripping is examined. If any generator(s) gets overloaded afterthe power redispatch, the excess power will be shifted to other generatorsaccording updated participation factors. The updated participation factorsare computed based on the available capacity of remaining generators. Dueto the mechanical limits, generators can only operate in excess of 110%,otherwise there will be problem of power mismatch which leads to the de-celeration of rotor and droop of power frequency [4]. If the island loses toomuch active power generation and the total capacity of the remaining gen-erators in the island is still less than the load, then part of load has to beshed.

Load Shedding

After power redispatch, if the power generation and consumption in the is-land is still not balanced, there is a risk of large-scale collapse of the system.Experienced operators will choose to disconnect part of consumers from thesystem to avoid a system-wide blackout. The time needed to restore thepower supply after system blackout is much longer than that of a controlledload shedding [4]. The load is reduced proportional to the power consump-tion.

Blackout

An island is considered to be blackout when all the loads are shed. Thisoccurs only in the extreme case when there are no generation units connectedto the island.

4.4.5 Overload Simulation

After load-generation balance is restored for the system or the island, over-load simulation will be run for the system.

Overloaded circuit identification

When an element gets overloaded and trips, some of the loading will betransferred to other components, which may lead to outages of more com-ponents. In contingency analysis, the linear sensitivity factors LODF andGGDF are applied for the case with less or equal to three outages. Whilefor cascading outage analysis, which may involve four or more outages, itis computational costly to re-compute these sensitivity factors from time totime. Moreover, these sensitivity factors are only applicable to single ele-ment contingency, it is impossible to model the simultaneous outage of more


than one elements. Therefore after load generation balance is restored, analternative approach of solving DC power flow is used to obtain the newpower flows. The DC power flow calculation is non-iterative and it ensuresthe convergence of a solution.

In DC power flow equations, it is assumed that the voltage angle devia-tions between buses are small and voltage magnitudes are constant, whichmeans the system is able to maintain voltage levels. Therefore the event ofvoltage collapse is not considered in this modeling.

After load-generation balance is restored, DC power flow is resolved withthe new load condition and new dispatch plan. The overloaded lines areidentified with deterministic criteria (PL,j > Pmax,j). After the overloadedlines are tripped, the system topology is updated and a check is performedagain to see if more lines get overloaded.

Termination criteria

The overload simulation continues until the cascade overloading terminates.If there are new islands generated in this process, the matrix for island sys-tem has to be updated by adding the rows for the new islands at the bottomof the matrix. Each Monte Carlo run terminates if overload simulation isperformed for all the islands.

4.5 Severity Evaluation

In power system, unplanned load shed or system blackouts are usually causedby a combination of component outages due to component overload, protec-tion malfunction or operation errors. The likelihood of these contingenciesare small (in the magnitude of 10−3 or 10−4), therefore the occurrence prob-ability of a combination of failures is very low. However, the cost associatedwith catastrophic events is very high, due to the cost for power redispatchand the value that customers placed on the disruption of power supply.Consequently, it is valuable to weight the losses of severe cascading outageswith the probability that they will happen. We use the following equationto evaluate the impact of a full cascade

K∑i=1

(Pri · Sevi), (4.5.1)

where Pri is the probability of the initial contingency and Sevi is the sever-ity of the whole cascade process initiated by ith contingency.


For the severity of cascade process, except from the amount of power re-dispatch and load shed in the load-generation balance restoration, anothertwo parameters are considered. One is the number of islands blackout inthe cascade, the other is the number of lines tripped before termination ofcascade.

The number of lines tripped is evaluated using catastrophic outcome as-sessment, which is introduced in [9]. In the assessment the number of linestripped in each stage and the number of cascade stages reached in cascadingevents are analyzed. An metric named as catastrophic expectation index(CEI) is used to measure the severity of the outcome. The formulation ofCEI for cascading overload is

CEI =K∑i=1

Mi∑j=1

PrijNline,ij , (4.5.2)

where

K number of initial contingencies considered;Mi number of transmission lines with nonzero trip probability

following outage of ith contingency ;Prij the probability of secondary trip;Nline,ij total number of lines tripped in the cascade after jth line

is disconnected .

After simulation of initial contingency, at which we call secondary stageof the process, Prij is used to determine which of the remaining highlyloaded circuits will be tripped. Each of these circuits are then tripped.From tertiary stage of the process, all the remaining overloaded circuit (withPL,j > Pmax,j) are identified and disconnected. Repeat this process untilcascade terminates. If the process exceeds 5 stages, stop the simulationand assign Sevij as 100. The severity in the equation is the total number oftripped circuits before the termination of cascade. In the cascade simulationmodel built in this thesis work, it is possible to track the process of cascadeand record the number of lines tripped in each stage.

Accordingly, the indices used to measure to impact of full cascade isexpressed as

n∑i

(Pri · Predispatch,i|PLoadshed,i|Nblackout,i|CEIi). (4.5.3)

These indices summarizes the metrics by which the security of the systemwith a given dispatch is evaluated.

Chapter 5

Case Studies

In this chapter, case studies are performed with a test system to comparethe system performance when the system is dispatched using RBOPF withthat when the system is dispatched using SCOPF. First, under 90% of peakload condition of the test system, the security of dispatches generated fromSCOPF and RBOPF are compared. Section 5.2 presents the results of thecomparison and cascade simulations. In Section 5.3, different risk limitwere tested in RBOPF to see the variation of the security and cost. Moreextensive test were conducted in Section 5.4 to investigate the performanceof RBOPF under different load conditions. Since it is possible to have post-contingency flow higher than maximum limit in RBOPF solution, in Section5.5, the likelihood of this violation is evaluated.

5.1 Description of Test System

IEEE RTS 96 24-bus system (one area) is used as the test system. It isdeveloped by the Application of Probability Method Subcommittee. Thevalue of the test system is that it allows comparative and benchmark stud-ies to be performed on methods of power systems reliability analysis.

The system contains 38 branches and 32 generators. Figure 5.1 gives thesystem configuration. In the system, the total generation capacity is 3405MW and the total load under peak load condition is 2850 MW. The ap-pendix A at the end of this document contains the data for buses, branchesand generators. Failures rates and repair rates of transmission lines andgenerators for this system are available, which are included also in the ap-pendix A. Based on these reliability data, the line contingency and generatorcontingency probability can be estimated with equation (2.2.1).

33

CHAPTER 5. CASE STUDIES 34

Figure 5.1: IEEE 24-bus Reliability Test System [7]

5.2 Risk Assessment with Fixed Power Load andRisk Limit in RBOPF

In this section, the system performance of SCOPF and RBOPF are com-pared based on a case study of RTS 96 system with 90% of peak load. Therisk limit used in the RBOPF is fixed. First, the difference in identifyingseverity of a certain contingency between these two methods is illustrated.Then based on the criteria of secondary tripping probability, risk of cascad-ing outages, post-contingency flow level and line flow change, the security ofthe dispatches generated from these two methods are compared. Additionalcomparison is conducted by using the losses of cascade process as metrics.


5.2.1 Evaluate Risk of Cascading Events for SCOPF Solution

In this section, the proposed risk index is used to evaluate the risk of cascad-ing outages for the security constrained dispatch generated by SCOPF. Lineflow generated from the dispatch is used as initial flow P0,j . The contingen-cies taken into account are the outage of a single line or a single generatorin the system.

We first consider the line contingencies, the simulation of the single lineoutages are performed subsequently for each line. After each contingency,the post-contingency flow calculation is performed for all the remaining linesby using equation (3.2.3). Based on P0,j and the calculated PL,j for eachith contingency, the value of Prij can be evaluated. Similarly, the singlegenerator outages are simulated, and the post-contingency flow PL,j andPrij are also computed. In Figure 5.2, the risk for each line contingencyand the maximum secondary trip probability Prij under each contingencyare plotted together with the contingency probability.

Figure 5.2: Result for line outage contingency in SCOPF: the sequence isthe total risk, maximum Prij and probability of each contingency

It can be seen that the outages of the 12th, 23th and 28th line are amongthe most severe cases of Prij , and the details of these contingencies and thepre-and post-contingency flow are listed in Table 5.1. These three cases cor-responding to following situations in the configuration of 24-bus system: a)


the load connected to bus 8 is fed by the power transmitted on the 11th,12th and 13th line. When the 12th line is failed, the transmission load onthe 12th line is added to the 13th line, which largely rises the stress on the13th line and increase the possibility for the 13th line to trip. b) 23th line isin charged of the power transmission from bus 14 to bus 16, when this lineis in outage, the excess power generated by the generator connected to bus14 has to be transmitted through another route to bus 16 (bus 14-11-9-3-24-15-16), which increases the burden on the 6th line (bus 3-9). c) Power istransmitted from bus 16 to bus 17 through the 28th line, when it is failed,the power is transmitted to the upper area through the 25th line (bus 15-21).

All the affected circuit listed are characterized with high post contin-gency flow (> 0.9 · Pmax,j) and relatively large flow change. The outage ofthe 12th line will lead to a large flow change of 0.6 · Pmax,j , and the post-contingency flow on the 13th line reaches the maximum limit of the circuit.The outage of 23th and 28th line will increase the stress on the 6th and 25thline respectively. However, the post-contingency flow and line flow changein these two cases are not as large as the outage of 12th line, the resultedPrijs are lower in these two cases. In this load condition, except from thelisted case of 12th line outage, the outage of the 25th or 26th line will causethe overload of the 28th line. However, the pre-and post-contingency flowchange in these two cases are relatively low (0.24 · Pmax,j), the associatedPrijs are lower compared to the outage of 6th and 25th line.

No. of outage line 12th line 23th line 28th lineConnection bus 8-9 bus 14-16 bus 16-17

Overloaded Circuit 13th line 6th line 25th lineConnection bus 8-10 bus 3-9 bus 15-21

Pij 0.71 0.64 0.56P0,i (MVA) 84.22 300.64 301.84P0,j (MVA) 55.77 -75.90 229.24PL,j (MVA) 140 -139.45 380.16Pmax,j (MVA) 140 140 4004P/Pmax,j 0.60 0.45 0.38PL,j/Pmax,j 1 0.99 0.95

Table 5.1: Three most severe cases of Prij for line outages

In the solution of SCOPF, there is only the outage of 32th generatorthat causes post-contingency flow higher than 0.9 · Pmax, the resulted Prijis not very severe due to the small line flow change. Therefore the result isnot discussed in detail here.


The evaluation result shows the difference in identifying contingencyseverity between the traditional N-1 criterion and the risk-based measure.The N-1 criterion in SCOPF only inhibit the circuit flow when post-contingencyflow violates the maximum limit. Therefore the severity is either 0 or 1 in N-1 criterion. In risk-based measure, the severity is changing continuously andit is proportional to the post-contingency flow and pre-to post-contingencyflow change. Consequently, the post-contingency states without N-1 con-straint violations can be considered as high risk if there is a large flowchange. In addition, all the circuits with high flow (> 0.9 · Pmax) have non-zero severity and would be reduced in proportion to the change of line flowand contingency probability.

5.2.2 Risk Assessment up to Secondary Cascade Stage

Under the same loading condition, we ran RBOPF for the test system withRisklimit=0.61. In the formulation of RBOPF introduced before, the up-per bound of secondary trip probability Prij is set to be in the range of[Risklimit, P rij,max,scopf ], where Prij,max,scopf is estimated based on the re-sult of SCOPF. It could be observed from the Figure 5.2 that, in case ofline outage contingencies, the maximum Prij in SCOPF is 0.703, which isthe event that the 13th line got overloaded after the outage of the 12thline. Therefore Prij,max,scopf is set as 0.703. Non zero Prij for each lineoutage contingency and each affected line of RBOPF solution is plotted inthe diagram 5.3 (right). As a comparison, the Prij of SCOPF solution isalso plotted (left).

Figure 5.3: Distribution of Prij for line outages in SCOPF (left) and RBOPF(right). Warmer colors indicate s higher probability of secondary tripping.As shown, there is a marked improvement in security for all but a few of thenodes in the RBOPF case.


In RBOPF, there are less post-contingency states with non zero Prijthan SCOPF. For instance, after the outage of 12th and 13th line, the post-contingency flows in RBOPF are lower than 90% of maximum limit, whilein SCOPF, high values of Prij are resulted (represented by red points). Onthe other hand, the contingency of 25th, 26th and 28th line can lead tomore severe Prij with the dispatch of RBOPF, as shown in the diagram,the color of these points are warmer. Value of risk can be obtained byweighting the severity of a contingency with its probability. Diagram 5.4shows the risk distribution. For SCOPF (left), It can be observed that thepost-contingency states resulted from 12th, 13th and 21th line outages arevery risky, and these risks are eliminated in the dispatch of RBOPF. In bothdispatches, there is no distinct difference in the risk levels of 25th, 26th and28th line contingency.

Figure 5.4: Distribution of Risk for line outages in SCOPF (left) andRBOPF (right)

To better illustrate the change of risk level from SCOPF to RBOPF,the maximum secondary tripping probability and the overall risk for eachcontingency are plotted in Figures 5.5 and 5.6. In SCOPF, it optimizes thesolution irrespective of the contingency probability, so high values of Prijare resulted at the contingencies with high probability, which significantlycontribute to the risk. In RBOPF, although there are higher risks due tocontingency of 25th, 26th and 28th line, the increase are not significantcompared to the risk addition caused by outage of 12th, 13th and 21th linein SCOPF. The solution of RBOPF shows an apparent risk reduction fromSCOPF, as shown in Figure 5.6.

According to the formulation of risk index, the risk constraint of RBOPFaffects the solution when the post-contingency flow exceeds 90% of maximum


Figure 5.5: Maximum Pij for each line contingency

Figure 5.6: Risk for each line contingency

limit, and the flow is reduced proportional to the contingency probability.This is reflected in the Figure 5.7. For all the contingencies with post-contingency flow higher than 0.9 · Pmax,j , there is a lower post-contingencyflow in the contingencies with higher probability in RBOPF, such as theoutages of 12th, 13th, 22th and 23th line. At the same time, for the contin-gencies of 25th, 26th and 28th line with relatively lower probabilities, thepost-contingency flow are allowed to be higher than maximum limit. Thisprovides the potential for the cost reduction. High flows are allowed whenthe contingency is low probability, which makes it possible for less expen-sive generators to increase their power delivery. Since Prij is controlled forhigh probability contingency, the cost can be reduced without significantlyincreasing the risk.

The risk constraint of RBOPF limits the line flow change on the circuitafter the contingency so as to ensure the transient stability of the system. InFigure 5.8, the maximum line flow change under each contingency is plotted.It can be observed that the large ’jumps’ that appear in the solution ofSCOPF are all cut down in RBOPF.


Figure 5.7: Maximum post-contingency flow for each line contingency

Figure 5.8: Maximum line flow change for each line contingency

5.2.3 Risk Assessment of Full Cascade

In last section, the solutions of SCOPF and RBOPF are compared in termsof post-contingency flow, line flow change, risk level and etc. Althoughthese assessments provide a good indication of system’s exposure to cascad-ing outages, it is possible that these contingencies with non zero probabilityof secondary trip may result in a catastrophic outcome, such as cascadingoverload, transient instability or system blackout. To look at the full cas-cade depth when a cascade is initiated, cascade simulations are performedfor the dispatches of SCOPF and RBOPF (Risklimit = 0.61).

For the RTS 96 system, the Monte Carlo simulation requires 1881 sam-ples to have convergence of result. The averaged amount of power redispatchin the cascading process in case of line outage is plotted in 5.9. There areless contingencies can result in power redispatch in RBOPF. For instance,it requires no power dispatch for the outages of 12th, 13th or 21th line. Thereason is that these contingencies will not initiate cascading outages withthe dispatch strategy of RBOPF.

The amount of power redispatch caused by cascading outage dependsnot only on the probability of secondary trip but also on the power imbal-ance caused by the line disconnection. It has been shown that the Prij for


contingency of 12th and 13th line are among the highest probability. Theoutage of 12th line may result in cascading trip of 13th line and vise versa.However, after outages of these two lines, only a small island consists oftwo buses (bus 7 and bus 8) is produced and the power imbalance in theisland is not so large. The resulted averaged power redispatch is not sig-nificant even if secondary trip probability is large. For the same severity ofpower imbalance, the averaged amount of power redispatch is proportionalto the secondary tripping probability. The cascade initiated by contingencyof 25th, 26th or 28th line will all finally lead to an isolated island consistsof bus 17, 18, 21 and 22. However, the secondary trip probability of 28thline contingency is higher than the other two contingencies, it requires moreaveraged power redispatch to restore the power balance.

Figure 5.9: Power redispatch for each line outage

Figure 5.10: Probabilistically-weighted power redispatch for each line outage

In Figure 5.10, the probability for initial contingency is included intothe result, which shows a significantly different conclusion if the contin-gency probability is taken into account. It is more obvious that the dispatchof SCOPF requires more power redispatch than the dispatch of RBOPF ifcascade is initiated. In SCOPF, cascading outages are avoided without con-sidering the contingency probability, therefore it may require large amountof power redispatch at the contingencies with high occurrence probability,such as 21th line contingency. In RBOPF, the risk of cascading outages


for high probability contingencies are eliminated, consequently less powerredispatch is required.

The amount of load shed and the number of islands have blackouts aredisplayed in Figure 5.11 and 5.13, respectively. In the dispatch of SCOPF,both the initial contingencies of 21th and 22th line can lead to load shed ofmore than 100 MW. Moreover, there are islands blackout in the cascadingprocess initiated by these two contingencies. In the simulation result withRBOPF solution, the outage of 21th line does not contribute to the loadshed and the number of blackouts. The reason is that the risk of cascadingoutages of 12th line is eliminated by the risk constraint of RBOPF.

Figure 5.11: Load shed for each line outage

Figure 5.12: Probabilistically-weighted load shed for each line outage

Figure 5.13: Blackouts number for each line outage


Figure 5.14: Probabilistically-weighted blackouts number for each line out-age

By the end of cascade simulation, it is possible to perform catastrophicoutcome assessment. The propagation of cascading outages in the system areshown in following figures, where Figure 5.15 and 5.16 are for the dispatchof SCOPF and Figure 5.17 is for the dispatch of RBOPF. By comparing thefigures of both dispatches, it can be seen that in the solution of RBOPF, thepotential of cascading outages for 12th, 13th and 21th line contingency areeliminated. Only the additional cascading outage for circuit 38 contingencyappears in RBOPF, which terminates in the secondary stage.

7

23

29

12

13

13

12

21

22 23

6

1

22

22 23

236

2 3 4

5 6 7 8

9 24 27

2 3 6

7 24 27

1 2 3 4

5 6 7 8

9 24 27

22

23

6 21

1 2 3 4

5 6 7 8

9 24 27

Primary Event

Level 1

Level 2

Level 3

Level 4

Cascad

ing Se

qu

en

ce

Level 1Probability

0.3952 0.7064 0.614 0.4949 0.3112 0.154 0.3806

Severity 1 0 0 13 13 6 13

Figure 5.15: CEI evaluation for SCOPF(Part I)


27

23

29

23

6

25

28 26

28 30

26 28

26

Primary Event

Level 1

Level 2

Level 3

Level 4

Cascad

ing Se

qu

en

ce

Level 1Probability

Severity

26

28 25

28 30

25 28

25

28

26 25

26

25 26

25

0.641 0.3952 0.4841 0.4841 0.55740.4953 0.2398 0.4953 0.2398 0.31070.5574

0 1 1 2 12 0 1 0 1 0

Figure 5.16: CEI evaluation for SCOPF(Part II)

27

23

25

28 26

28 30

26 28

26

Primary Event

Level 1

Level 2

Level 3

Level 4

Cascad

ing Se

qu

en

ce

Level 1Probability

Severity

26

28 25

28 30

25 28

25

28

26 25

26

2526

25

0.3638 0.61 0.3639 0.5684 0.610.568 0.51 0.284 0.51 0.610.284

7

23

29

1 2 3 4

5 6 7 8

9 24 27

1 2 3 4

5 6 7 8

9 24 27

22 38

28

6 21

23

29

0.3721 0.2609

1 0 1 1 11 2 0 2 10 0 0

0.3568

13

23

6

Figure 5.17: CEI evaluation for RBOPF

The value of CEI evaluated for the solutions of SCOPF and RBOPF(Risklimit = 0.61) are summarized in Table 5.2. SCOPF solution causes ahigh CEI: 21.19. The main cause is the 21th line contingency which leadsto blackouts in the system. After contingency probability is considered,the extent of CEI reduction is more significant. Given the assumption that


Dispatch Strategy CEI for cascading Pri · CEISCOPF 21.19 0.027

RBOPF 9.76 0.011

% of reduction 53.94% 59.26%

Table 5.2: Value of CEI for dispatches of SCOPF and RBOPF

Prij is a good measure of secondary outage probability, CEI assessmentprovides evidence that RBOPF can generate more secure operating pointsthan SCOPF.

5.3 Risk Assessment with Fixed Power Load andVarying Risk Limit in RBOPF

Under the 90% of peak load condition, more risk limits were tested inRBOPF to investigate the variation of security and cost with different risklimits.

5.3.1 Effect of Risk Limit on Generator cost

The generation cost is plotted with risk limit in the diagram 5.18. The ten-dency shows the cost is reduced with less conservative risk limit, and theamount of cost reduction due to iterative optimization is larger for lower risklimit. The reason is that there is a higher possibility for the risk constraintwith lower risk limit to be tighter than the constraint of PL,j ≤ 0.9 ·Pmax,j .Consequently, more risk constraints which are too conservative have to bereplaced by the constraint of PL,j ≤ 0.9 ·Pmax,j . The feasible set is enlargedafter the constraint replacement, which provides the opportunity to furtherreduce cost. In addition, for the listed risk limits, the cost of RBOPF solu-tion is always lower than the SCOPF solution (170.36 $/hour).

To better understand the cost reduction from SCOPF to RBOPF, perunit generation cost of each generator and the power delivered by each ofthem in the solutions of SCOPF and RBOPF (Risklimit = 0.61) are pre-sented in Figure 5.19. In the dispatch of RBOPF, there is less generationon the 3th (↓ 25.91MW ) and 21th generator (↓ 22.48MW ) with per unitcost of 10.239 $/hour and 9.537 $/hour respectively. On the other hand,the 22th generator increase its generation significantly (↑ 40MW ), the gen-eration cost of this generator is relatively low (5.23 $/hour). As a result,the overall generation cost is reduced in RBOPF.


Figure 5.18: Cost reduction through iteration optimization, where the blueline represents the cost of first iteration and the red line represents for thecost of last iteration.

Figure 5.19: Per unit generation cost and power delivered by each generator


5.3.2 Risk Assessment up to Secondary Cascade Stage

The total risk of both line outage contingency and generator outage contin-gency are evaluated for the solutions of SCOPF and RBOPF with differentrisk limits. The equations utilized for computing the total risk are

Risktot,line =

K∑i=1

Nl−1∑j=1

(Pri · Sevij) (5.3.1)

Risktot,gen =K∑i=1

Nl∑j=1

(Pri · Sevij) (5.3.2)

Risktot = Risktot,line +Risktot,gen (5.3.3)

where all the possible initial contingencies and post-contingency states areconsidered. In the equation, K is the number of initial contingencies, itequals the number of lines in case of line outage and equals the numberof generators in case of generator outage. Nl is the number of lines in thesystem.

Figure 5.20: Generation cost for a given total risk in case of line or generatorcontingency: the numbers marked out are risk limit in RBOPF

Figure 5.20 plots the generator cost against the total risk. In the bluearea, the solutions of RBOPF are better than the SCOPF solution in termsof lower total risk level and less generation cost. There is less cost because


in RBOPF the post-contingency flows are allowed to exceed their maxi-mum limit, so that less expensive generators can work at a higher output.Although there is no explicit limitation on the total risk level of all con-tingencies in RBOPF, lower total risk level can be achieved in RBOPF byuniformly risk control throughout the system. The result proves that in thiscase, the cost reduction in RBOPF is not at the expense of a higher risk.

5.3.3 Risk Assessment of Full Cascade

The dispatches of SCOPF and RBOPF with different Risklimit have beenevaluated with a full cascade simulation. Certain scenarios cause island op-eration, making a power redispatch or load shed necessary, or a blackoutunavoidable. The amount of power redispatch, load shed and the numberof blackout islands for the solutions of RBOPF and SCOPF are plotted in,respectively, Figure 5.21, 5.22 and 5.23.

Figure 5.21: Generator cost for a given total probabilistically-weightedpower redispatch


Figure 5.22: Generator cost for a given total probabilistically-weighted loadshed

Figure 5.23: Generator cost for a given total probabilistically-weightedblackouts number

These figures show that when Risklimit is set below 0.63, the four criteriaof generator cost, probabilistically-weighted power redispatched, probabilistically-weighted load shed, and probabilistically-weighted blackout number all favorRBOPF over SCOPF. The advantage of the solutions of RBOPF is moreapparent in the total load shed and total blackouts number reduction, where


a risk limit below 0.64 is enough to show superiority of the RBOPF. If therisk limit in RBOPF is set below or equal to 0.63 to ensure RBOPF outper-forms SCOPF in all four criteria, the load shed and the number of islandsblackout in cascading process are significantly reduced (by 50%) from thatof the SCOPF.

5.4 Risk Assessment for Varying Risk Limit andPower Load

To further examine that the solutions of RBOPF outperform the solutionof SCOPF. Extensive test is performed with different load conditions. Thepercentage of total risk reduction from that of SCOPF is presented in Figure5.24.

Figure 5.24: Total risk reduction from SCOPF in different load conditions

For peak load condition, there is a large extent of reduction, which isaround 60%. While for 80% and 90% of peak load condition, there is around10% to 20% of reduction, respectively. SCOPF affects the solution only ifthe post-contingency violates the circuit’s limit, by clamping the flow at thelimit. In RBOPF, it controls the post-contingency flow that higher than90% by reducing it proportional to the probability, so it tends to controlhigh flows throughout the system and not just those that contribute the cir-cuit’s limit violation. When there is high stress in the system, the solution ofSCOPF is associated with more high post-contingency flow (> 90%·Pmax,j),and more contingencies with non zero risk of cascading outages.


In the future, system operators are expected to operate the grid closerto its limit. Given that the proposed risk index properly measures the riskof cascading outages, the RBOPF with proposed risk-based measure couldbe a promising tool for reducing the operation risk for heavy load conditionas it effectively reduce the high flow throughout the system.

The dispatch strategies for different load conditions are tested with cas-cade simulation. Results of cascade losses are displayed in Figure 5.25, 5.26and 5.27. Risk limits used in RBOPF are labeled above each column. Itcould be observed that when appropriate risk limit is chosen, there is lesscascade losses for the solutions of RBOPF irrespective of the load condition,which proves the capability of RBOPF solution in cascading prevention.

Figure 5.25: Total power redispatch in different load conditions

Figure 5.26: Total load shed in different load conditions

Figure 5.27: Total blackout number in different load conditions


5.5 Probability of N-1 Constraint Violation

Since the usual N-1 constraint is not necessarily fulfilled in RBOPF, it is ofinterest to investigate the likelihood of RBOPF solution to violate the nor-mal N-1 constraint. A method for calculating the probability that RBOPFsolution violates the SCOPF hard constraints is introduced in [3] is usedhere. The probability of violation is considered as the average probabilityfor the circuit to have the post-contingency flow larger than its power rating,and it is expressed as

Pvio =1

nc

nc∑k=1

K∑i=1

(Pri ∗Nvio,k(Ei)) (5.5.1)

where

Pvio the probability of violations;nc the number of cases taken into account;Nvio,k(Ei) the number of N-1 constraint violations at state Ei

in the kth case.

3 snapshots (80%, 90% and 100% of peak load condition) are consideredin the probability estimation. 5 different risk limits are used in RBOPFfor each snapshot. There are 38 initial line contingencies and 32 initialgenerator contingencies, which result in 37 post-contingency states and 38post-contingency states, respectively. Therefore, there are in total 39330post-contingency states, among which there are 32 times of N-1 constraintviolations. Taken into account the probability of these contingencies thatcan cause violations, the average probability of violation for the consideredcases is 0.23%. More cases can be used for this calculation by taking moresnapshots of the system. It can be concluded from the result that althoughN-1 constraint is not necessarily to be fulfilled in RBOPF, the probabilityfor the RBOPF solution to violate the hard constraint of SCOPF is still low.

Chapter 6

Conclusions and FutureWork

This thesis provides a novel risk-based measure for evaluating the risk ofcascading events. The risk of cascading events is reflected in a risk index,which accounts for both the probability and severity of the initial contin-gency. The severity is modeled as the probability of a secondary trip.

The proposed risk index is included into an OPF formulation as riskconstraints, through which the risk controlled dispatch strategies are iden-tified. A uniform risk limit is set for the risk constraints in order to controlthe probability of secondary tripping Prij in proportion to the likelihoodof the initial contingency. The maximum secondary tripping probabilityof SCOPF solution can be a good reference for choosing the upper boundof Prij . The flexibility for choosing the Risklimit allows a better tradeoffbetween security and economy. The discontinuity in the risk formulationis captured by solving the optimization iteratively. The potential of costreduction by using iterative optimization was shown in the result. In addi-tion, solving the optimization requires very short time because the problemis linear and only a few iterations are needed.

The dispatch strategy generated by SCOPF is evaluated with the pro-posed risk index. Result shows the difference between the deterministic andrisk-based criterion in identifying severity and risk of a certain contingency.In deterministic criterion, the severity of initial contingency rises from 0 to1 as the post-contingency flow reaches the circuit limit. On the contrary, inrisk-based criterion, the severity is changing continuously, and it takes intoaccount the post-contingency flow and the change of line flow from pre-topost-contingency state. Consequently, the event with large flow change andpost-contingency flow lower than the circuit limit may have a higher sever-ity than the event which violates the circuit limit bus with small flow change.

53

CHAPTER 6. CONCLUSIONS AND FUTURE WORK 54

The solution of RBOPF shows an apparent reduction in total risk levelof all the contingencies from SCOPF solution. This is achieved by havingmore uniform risk control in RBOPF. The improvement in risk doesn’t comeat the expense of a higher cost. The risk constraint in RBOPF allows highflows when the contingency is low probability, which provides the oppor-tunity for less expensive generators to increase power delivery. Analysis ofpost-contingency flow and change of line flow from pre-to post-contingencystate proved that the risk constraint can effectively limit the high post-contingency flow for the high probability contingencies and reduce large lineflow change.

The comparison of risk of cascading outages up to secondary stage pro-vides a good indication of system stress and reflects the probability thata cascade will be initiated. To show the superiority of RBOPF in reduc-ing the impact of a full cascade over SCOPF, the dispatch is checked witha full cascade simulation. According to the simulation result, when risklimit in RBOPF is low enough, the reduction of power redispatch, load shedand blackouts number from the SCOPF to RBOPF can be observed. Fur-ther comparison is based on catastrophic outcome assessment for cascadingevents. Solution of SCOPF has a much higher value of CEI comparing withthe solution of RBOPF with similar cost, as the RBOPF solution is lessexposed to occurrence of cascading outages initiated by some contingencies.

To further examine the performance of RBOPF and SCOPF, the studyfor different load conditions is conducted. The observations indicate thatRBOPF is more effective in risk reduction for heavy load condition, whichmeets the expectation. Moreover, lower impact of cascade can be ensured invarious load condition when the appropriate risk limit is chosen. Althoughadditional comparison could be performed based on other criteria, e.g. volt-age instability, oscillatory instability, such comparisons are not expected toresult in different conclusions since RBOPF uniformly reduces the high flowand operating stress in the system. The future research based on the pro-posed risk-based OPF algorithm might include the following aspects:

• The conclusions drawn in this thesis is based on the precondition thatsecondary tripping probability is properly measured. Therefore, it is impor-tant to validate the assumptions of secondary tripping probability.

• In the proposed formulation of RBOPF, there is no explicit limitationon the total risk level of all the contingencies. Accordingly, it is expected toexplore the method to include the constraint of total risk level into OPF.

• It is suggested to use the maximum secondary tripping probability of

CHAPTER 6. CONCLUSIONS AND FUTURE WORK 55

solution of SCOPF as the upper boundary for Prij in RBOPF, other way ofchoosing upper bound can be explored to see how the performance varies.

• Since the goal of improving the security assessment approach is tocope with the uncertainty induced by increasing penetration of fluctuatingrenewable energy sources, it would be of interest to additionally include therisk of wind fluctuation into RBOPF model.

• Before the adoption of new risk-based measure in power system, carefuland extensive testing are required. Therefore large system and more snap-shots should be tested, whereas the motivation and direction for performingsuch testing are provided in this thesis.

Appendix A

Data of RTS 96

Table A.1: Branch data

Branch No. From Bus To Bus X Limit Failure Rate Repair Rate(Ω) (MW) (1/year) (1/year)

1 1 2 0.014 140 0.24 547.5

2 1 3 0.211 140 0.51 876

3 1 5 0.085 140 0.33 876

4 2 4 0.127 140 0.39 876

5 2 6 0.192 140 0.48 876

6 3 9 0.119 140 0.38 876

7 3 24 0.084 320 0.02 876

8 4 9 0.104 140 0.36 876

9 5 10 0.088 140 0.34 876

10 6 10 0.061 140 0.33 250.29

11 7 8 0.061 140 0.3 876

12 8 9 0.165 140 0.44 876

13 8 10 0.165 140 0.44 876

14 9 11 0.084 320 0.02 876

15 9 12 0.084 320 0.02 876

16 10 11 0.084 320 0.02 876

17 10 12 0.084 320 0.02 876

18 11 13 0.048 400 0.4 250.29

19 11 14 0.042 400 0.39 876

20 12 13 0.048 400 0.4 876

56

APPENDIX A. DATA OF RTS 96 57

Branch No. From Bus To Bus X Limit Failure Rate Repair Rate(Ω) (MW) (1/year) (1/year)

21 12 23 0.097 400 0.52 876

22 13 23 0.087 400 0.49 876

23 14 16 0.059 400 0.38 876

24 15 16 0.017 400 0.33 876

25 15 21 0.049 400 0.41 876

26 15 21 0.049 400 0.41 796.3

27 15 24 0.052 400 0.41 796.3

28 16 17 0.026 400 0.35 796.3

29 16 19 0.023 400 0.34 796.3

30 17 18 0.014 400 0.32 796.3

31 17 22 0.105 400 0.54 796.3

32 18 21 0.026 400 0.35 796.3

33 18 21 0.026 400 0.35 796.3

34 19 20 0.04 400 0.38 796.3

35 19 20 0.04 400 0.38 796.3

36 20 23 0.022 400 0.34 796.3

37 20 23 0.022 400 0.34 796.3

38 21 22 0.068 400 0.45 796.3

Table A.2: Generator data

Generator No. Bus No. Limit(MW) Cost($/hour) MTTF(h) MTTR(h)

1 1 20 24.842 450 50

2 1 20 24.842 450 50

3 1 76 10.239 1960 40

4 1 76 10.239 1960 40

5 2 20 24.842 450 50

6 2 20 24.842 450 50

7 2 76 10.239 1960 40

8 2 76 10.239 1960 40

9 7 100 17.974 1200 50


Generator No. Bus No. Limit(MW) Cost($/hour) MTTF(h) MTTR(h)

10 7 100 17.974 1200 50

11 7 100 17.974 1200 50

12 13 197 18.47 950 50

13 13 197 18.47 950 50

14 13 197 18.47 950 50

15 15 12 21.227 2940 60

16 15 12 21.227 2940 60

17 15 12 21.227 2940 60

18 15 12 21.227 2940 60

19 15 12 21.227 2940 60

20 15 155 9.537 960 40

21 16 155 9.537 960 40

22 18 400 5.23 1100 150

23 21 400 5.23 1100 150

24 22 50 1 1980 20

25 22 50 1 1980 20

26 22 50 1 1980 20

27 22 50 1 1980 20

28 22 50 1 1980 20

29 22 50 1 1980 20

30 23 155 9.537 960 40

31 23 155 9.537 960 40

32 23 350 9.537 1150 100


Table A.3: Load data

Load No. Bus No. Load(MW)

1 1 108

2 2 97

3 3 180

4 4 74

5 5 71

6 6 136

7 7 125

8 8 171

9 9 175

10 10 195

11 13 265

12 14 194

13 15 317

14 16 100

15 18 333

16 19 181

17 20 128

Appendix B

DC Power Flow

DC power flow is a simplified, linear form of modelling AC system. Thecalculation is non-iterative and it ensures the convergence of solution. Thismethod is less accurate than the non linear AC power flow. However, whenthe approximation is acceptable, this method is faster in computation andsimpler compared to AC power flow.

In DC power flow calculation, following assumptions are made: 1) branchesare considered to be lossless; 2) All bus voltage magnitudes are close to 1p.u.; 3) Voltage angle differences across branches are small enough. In thedc model, the active power flow in a branch is given by:

Pij = x−1ij ∗ (θi − θj) (B.0.1)

Active power injection at bus i is

Pinj,i = (∑j∈Ωi

x−1ij )θi +

∑j∈Ωi

(−x−1ij θj) j = 1, 2, ..., N, (B.0.2)

N is number of buses in the system. If write above equation in matrix form:

Pinj = B′θ (B.0.3)

wherePinj is the vector of the net injections Pinj,i = Pg,i − Pl,iB′ is the nodal admittance matrix with

B′ij = −x−1ij (B.0.4)

B′ii =∑j∈Ωi

x−1ij (B.0.5)

θ is the vector of voltage angles θkThe equation relates the the power flows on each line and power injected on

60

APPENDIX B. DC POWER FLOW 61

each bus is:Pline = bAB−1Pinj (B.0.6)

where Pline is the vector for active power flows on each branch; b is diagonalmatrix of the line susceptances, diag(1/xi); A is line-bus incidence matrixwith two non zero entries in each row (aki = 1 and akj = −1 for line kleaving from bus i and arriving to bus j).

Bibliography

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[2] Stephen Poythress Boyd and Lieven Vandenberghe. Convex optimiza-tion. Cambridge university press, 2004.

[3] RC Dai, H Pham, Y Wang, and JD McCalley. Long-term benefitsof online risk-based direct-current optimal power flow. Proceedings ofthe Institution of Mechanical Engineers, Part O: Journal of Risk andReliability, 226(1):65–74, 2012.

[4] DS Kirschen, KRW Bell, DP Nedic, D Jayaweera, and RN Allan. Com-puting the value of security. In Generation, Transmission and Distri-bution, IEE Proceedings-, volume 150, pages 673–678. IET, 2003.

[5] James D McCalley, Vijay Vittal, and Nicholas Abi-Samra. An overviewof risk based security assessment. In Power Engineering Society Sum-mer Meeting, 1999. IEEE, volume 1, pages 173–178. IEEE, 1999.

[6] Ming Ni, James D McCalley, Vijay Vittal, and Tayyib Tayyib. Onlinerisk-based security assessment. Power Systems, IEEE Transactions on,18(1):258–265, 2003.

[7] J Nikoukar, MR Haghifam, and A Parastar. Transmission cost alloca-tion based on the modified z-bus. International Journal of ElectricalPower & Energy Systems, 42(1):31–37, 2012.

[8] Marianna Vaiman, Keith Bell, Yousu Chen, Badrul Chowdhury, IanDobson, Paul Hines, Milorad Papic, Stephen Miller, and Pei Zhang.Risk assessment of cascading outages: Methodologies and challenges.Power Systems, IEEE Transactions on, 27(2):631–641, 2012.

[9] Fei Xiao and James D McCalley. Power system risk assessment and con-trol in a multiobjective framework. Power Systems, IEEE Transactionson, 24(1):78–85, 2009.

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[10] Fei Xiao, James D McCalley, Yan Ou, John Adams, and Steven My-ers. Contingency probability estimation using weather and geographicaldata for on-line security assessment. In Probabilistic Methods Appliedto Power Systems, 2006. PMAPS 2006. International Conference on,pages 1–7. IEEE, 2006.

[11] Jianfeng Zhang and Fernando L Alvarado. A heuristic model of cas-cading line trips. In Probabilistic Methods Applied to Power Systems,2004 International Conference on, pages 647–650. IEEE, 2004.

[12] Marek Zima and Goran Andersson. On security criteria in power sys-tems operation. In Power Engineering Society General Meeting, 2005.IEEE, pages 3089–3093. IEEE, 2005.

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