Var Planning Problem Considering Conditional

Value-at-Risk Assessment

Julio Cesar Lopez and J.R.S MantovaniDepartment of Electrical Engineering

Sao Paulo State University – UNESP

Ilha Solteira, SP – Brazil

Email: {julio.lopez, mant}@dee.feis.unesp.br

Javier Contreras SanzEscuela Tecnica Superior de Ingenieros Industriales

Universidad de Castilla - La Mancha, 13071

Ciudad Real, Spain

Email: Javier.Contreras@uclm.es

Abstract—This paper presents the reactive power plan-ning solution under risk assessment through the CVaR(Conditional-Value-at-Risk) using stochastic programming. Loaduncertainty is modeled by distribution function. Uncertaintyin the reactive power availability of existing and new reac-tive power sources is modeled through probabilistic constraintswith a ρ-quartile measure. The taps settings of the under-loadtap-changing transformers are modeled as discrete settings. Theproblem solution in this paper includes a reasonable number ofpossible future scenarios that calculate a set of solutions whichallow to find the best flexible planning and adapting to futurescenarios of power system operation such that the planning hasfound local optimum solution quality. The tradeoff between riskmitigation and cost minimization is analyzed. The efficacy of theproposed model is tested and justified by the simulation resultsusing the CIGRE-32 electric power system.

Index Terms—Conditional-Value-at-Risk, Chance-constrained,mixed-integer non-linear programming, reactive power planning,stochastic programming.

NOMENCLATURE

The following notation is used throughout the paper:

SetsB Set of buses.

L Set of transmission lines.

PQ Set of pq buses.

PV Set of pv buses.

SH Set of candidate buses to install new reactive

power sources.

SHE Set of buses with existing reactive power sources.

Slack Slack bus.

T Set of load tap changing transformers.

Ω Set of scenarios.

Indexese Index for buses with existing reactive power

sources.

i Index for buses.

j Index for {PV ∪ Slack} buses.

k Index for candidate buses to install new reactive

power sources.

l Index for PQ buses.

m Index for under-load tap-changing transformers.

n Index for transmission lines.

p Index for PV buses.

sl Index for Slack bus.

ω Index for scenarios.

ConstantsKFCk,KFRk Investment costs of capacitive and induc-

tive reactive power sources in bus k.

KV Ck,KV Rk Operating costs of capacitive and induc-

tive reactive power sources in bus k.

NSm Steps number of under-load tap-changing

transformers m.

PGp(ω) Active power produced of generator unit

p in scenario ω.

PDl(ω), QDl(ω) Active and reactive load in bus l in sce-

nario ω.

Qmin

Gj, Qmax

GjReactive lower and upper limits of gen-

erator unit j .

Qmax

Ck, Qmax

RkUpper limit of new capacitive and induc-

tive reactive power sources in bus k.

Qmin

SHe, Qmax

SHeUpper and lower limits of existing reac-

tive power sources in bus e.

RGm Transformation regulation of under-load

tap-changing transformers m.

Smax

n Upper total capacity limit of transmission

line n.

V min

i , V max

i Upper and lower voltage magnitudes in

bus i.

α Confidence level used for the CVaR cal-

culation.

β Risk weighting factor.

γupe , γdw

e Up and down expected reactive power

availability of existing reactive power

sources in bus e.

γck, γrk Expected reactive power availability of

new capacitive and inductive reactive

power sources in bus k.

π(ω) Weight of scenario ω.

ρupe , ρdwe Specified confidence level for the up-

per and lower limits of existing reactive

power sources in bus e.

ρck, ρrk Specified confidence level for the new

reactive capacitive and inductive sources

in bus k.

978-1-4799-3656-4/14/$31.00 ©2014 IEEE

ψupe , ψup

e Standardized normal value for up and

down limits of existing reactive sources in

bus e.

ψck, ψrk Standardized normal value for the new

reactive capacitive and inductive sources

in bus e.

Variablesntapm(ω) Tap position of under-load tap-changing

transformer m in scenario ω.

pi(ω), qi(ω) Active and reactive power injections in

bus i in scenario ω.

pn(ω), qn(ω) Active and reactive power flows by

branch n in scenario ω.

pGsl(ω) Active power generation in the slack bus

in scenario ω.

qGj(ω) Reactive power generation in bus j in

scenario ω.

qshe(ω) Reactive power generation of existing

sources in bus e in scenario ω.

qck(ω), qrk(ω) Capacitive and inductive reactive power

generation of new sources in bus k in

scenario ω.

tapm(ω) Tap setting of under-load tap-changing

transformer m in scenario ω.

vi(ω), δi(ω) Voltage magnitude and angle in bus i in

scenario ω.

uck, urk Binary decision variables: 1 if qc or qr

are built in bus k, 0 otherwise.

V aR Variable used to calculate the

Value-at-Risk.

η(ω) Auxiliary variable associated to scenario

ω and used to calculate the CVaR.

I. INTRODUCTION

Reactive power planning (RPP) plays an important role in

the operation of the transmission systems. The purpose of RPP

in a competitive market is to find the minimum cost installation

plan of new reactive power resources so that the voltage of

each node is maintained within an acceptable level for any

unexpected operating state caused by power wheeling or non-

utility generator operation, etc. Unfortunately, the risk of a

disastrous domino effect is growing in many countries due to

the current trend of operating power systems under the most

critical points of stability and capacity limits. A powerful rea-

son for this practice is economical. Several classical optimiza-

tion algorithms such as linear programming [1], [2], quadratic

programming [3] nonlinear programming [4], mixed integer

nonlinear programming [5], [6], decomposition methods [7],

[8], methods based on Successive Linear Programming [9],

[10], mathematical programming techniques such as gradient

method [11], newton method [12], have been also used to solve

deterministic reactive power optimization problems. An EPSO

(Evolutionary Particle Swarm Optimization) algorithm is used

in [13] to solve the RPP considering different contingencies

and load levels. In [14] it is applied the PSO (Particle Swarm

Optimization) including line flows and improving the system

voltage profile. A TS (Tabu Search) is used for solving a two-

stage RPP problem in [15].

In this paper we propose a RPP model incorporating risk

aversion through the CVaR based in stochastic optimization,

which is defined as the expected value of a cost smaller

than the (1 − α)-quantile of the costs distribution. CVaR(α,

x) is computed as the expected cost in the (1 − α)100%

worst scenarios. Load uncertainty is modeled as a distribution

function. Uncertainty in the availability equivalent capacity

reactive power sources is modeled by probabilistic constraints

with a ρ-quartile measure to determine the confidence level

to meet the load under contingencies on the existing and new

reactive power sources. The taps settings of the under-load

tap-changing transformers are modeled as discrete settings

taps, this is modeled considering take into account that a

transformer tap is a connection point along a transformer

winding that allows a certain number of turns to be selected.

This means that a transformer with a variable turn ratio is

produced, enabling voltage regulation of the output. The tap

selection is made with a tap changer mechanism, which is

categorized by a regulation range (RGm) and a step count

(NSm), e.g., ±10%, 32 steps. The tap range defines the

limits of the device’s regulating ability. The tap increment

defines the resolution of the device. These considerations

are used to calculate a set of solutions in order to find the

best planning that is adapted to future changes so that an

optimum solution is obtained. The objective is to minimize

the sum of investment costs (IC) and expected operation costs

(EOC) optimizing the sizes and locations of new reactive

power sources to ensure power system security. Naturally

the proposed RPP problem is a non-convex, mixed-integer

non-linear, large-scale optimization problem which requires

considerable computational burden to be solved.

This paper is organized as follows: Section II presents the

RPP under CVaR assessment formulation. Section III describes

the uncertainty model used for load scenario generation. In

Section IV a case study is presented to show the advantages

of proposed model. Section V presents some final conclusions.

II. VAR PLANNING CONSIDERING CVAR ASSESSMENT

FORMULATION

The proposed model is formulated as a stochastic

mixed-integer nonlinear programming problem, take into ac-

count the risk assessment, which is based on the CVaR and it

is incorporated in the objective function. Thus the constraints

in the developed model contain stochastic variables, the so-

called chance constraints. In the model we consider P(·) and

V(·) symbols as the Probability and Variance as follows:

min f =

[∑ω∈Ω

∑k∈SH

KFCkuck +∑ω∈Ω

∑k∈SH

KFRkurk

+∑ω∈Ω

π(ω)

( ∑k∈SH

KV Ckqck(ω) +∑

k∈SH

KV Rkqrk(ω)

)

+ β

(V aR+

1

1− α

∑ω∈Ω

π(ω)η(ω)

)](1)

s.t. :

PGp(ω) + pGsl(ω)− PDl(ω)− pi(ω) = 0 (2)

qGj(ω)−QDl(ω)− qk(ω) + qshe(ω)+

qck(ω)− qrk(ω) = 0 (3)

Qmin

Gj≤ qGj(ω) ≤ Qmax

Gj(4)

V min

i ≤ vi(ω) ≤ V max

i (5)

tapm(ω) = 1 + 2RGm

(ntapm(ω)

NSm

)(6)

−NSm

2≤ ntapm(ω) ≤

NSm

2(7)(

p2n(ω) + q2n(ω)

)≤ (Smax

n )2 (8)

P

(qshe(ω) ≤ γup

e Qmax

SHe

)≥ ρupe (9)

P

(qshe(ω) ≥ γdw

e Qmin

SHe

)≥ ρdwe (10)

P

(qck(ω) ≤ γc

upk Qmax

Ckuck

)≥ ρck (11)

P

(qrk(ω) ≤ γr

upk Qmax

Rkurk

)≥ ρrk (12)

η(ω) ≥

[ ∑k∈SH

KFCkuck(ω) +∑

k∈SH

KFRkurk(ω)

+∑

k∈SH

KV Ckqck(ω) +∑

k∈SH

KV Rkqrk(ω)− V aR

](13)

uck ∈ {0, 1} (14)

urk ∈ {0, 1} (15)

η(ω) ≥ 0 (16)

ntapm(ω) ∈ Integer (17)

qck(ω) ≥ 0, qrk(ω) ≥ 0 (18)

The objective function (1) consists of three terms. The

first corresponds to the investment costs in new reactive

power sources, the second is the expected operation costs

of new reactive power sources and the third term is equal

to the CVaR function multiplied by a weighting factor β,

this factor is used for modeling the tradeoff between the

expected cost and the risk, which is measured by the CVaR.

Equations (2) and (3) represent the nodal active and reactive

balance equations. Equation (4) represents the upper and

lower capacity limits of reactive power generation. Equation

(5) represents the upper and lower voltage limits. Equations

(6) and (7) represent the taps settings and taps positions of

under-load tap-changing transformers. Equation (8) represents

the power transfer capability by the branches. Equations (9)

and (10) represent the probabilistic equations for the up and

down capacity of the existing reactive power sources with a

ρ-quartile confidence level. Equations (11) and (12) represent

the probabilistic equations for the upper and lower capacity

limits of the new reactive power sources with a ρ-quartile

confidence level. Equation (13) is used to compute the CVaR

value.

A. The Transformed Deterministic Chance constraintsThe main goal in problems with probabilistic constraints is

to determine their deterministic equivalents. Exact calculation

for the normal distribution function can be performed for

random variables [16]. Probabilistic constraints of reactive

power sources take into account the fact that the capacity of

the existing and newly installed sources is affected by the

corresponding equivalent availability of each source. Because

of a component failure or a similar condition, the output of

a source may be reduced. This type of outages result in the

derating of the sources. The equivalent availability of a source

accounts for such derating outages [17]. With this background

the set of probabilistic constraints (9)–(12) are transformed

into a set of deterministic constraints as follows:

qshe(ω) ≤

{γupe Qmax

SHe− ψup

e

[V (γup

e )(Qmax

SHe

)2]1/2}(19)

qshe(ω) ≥

{γdwe Qmin

SHe+ ψdw

e

[V(γdwe

) (Qmin

SHe

)2]1/2}(20)

qck(ω) ≤

{γckQ

max

Ck− ψce

[V (γce)

(Qmax

Ck

)2]1/2}uck

(21)

qrk(ω) ≤

{γrkQ

max

Rk− ψre

[V (γre)

(Qmax

Rk

)2]1/2}urk

(22)

III. LOAD UNCERTAINTY MODELING

Many approaches are available for scenario generation [18],

[19], [20]. In this paper it is assumed that the normal proba-

bility density functions of the system loads are available at all

load-buses. Each node in the scenario tree has an individual

predecessor node, but presumably several successors. Each

path from the root node to the leaf nodes is defined as a

scenario. Uncertainty is modeled by a tree demand. Each load

scenario is associated with the probability extracted from an

annual load curve, usually considering its duration.

IV. CASE STUDY

To validate the proposed model, numerical results are pre-

sented for the CIGRE-32 power system [21]. The optimization

problem is solved using the optimization solver KNITRO 8

[22] in AMPL v2012 [23], in a Dell PowerEdge R910x64 PC

server, 128 GB of RAM and 1.87 GHz under the Windows

Server 2008 operating system.

In this paper the results are shown for three more significant

scenarios (low load, average load and peak load) of the thirty

demand scenarios analyzed.

The upper and lower voltage limits in all buses are within

[0.95− 1.05] p.u. The upper and lower tap limits of the

under-load tap-changing transformers are associated with the

position in the transformer winding with a transformation

regulation of 10% and step count of ±16%. All buses are

considered as candidates to install new reactive power sources.

The investment and operation costs of new reactive power

sources considered are 150 US$ and 15 US$/MVAr respec-

tively. The maximum capacity of new reactive capacitive and

inductive sources is fixed in 1000.00 MVAr. All reactive power

variables are considered as continuous.

Figure 1 shows the tap position of the under-load

tap-changing transformer for β = 0 and β = 7. This figure

shows that the taps are set according to the integer variable

position in the LTC winding. All settings are within the limits

established by the regulation (10% and ±16).

Table I shows the reactive power dispatch of new installed

reactive power sources for different values of β.

TABLE IREACTIVE POWER ALLOCATION AND DISPATCH FOR DIFFERENT VALUES

OF β

Scenario Bus Capacitive Inductive

β = 0

low demand4021 – 150.004032 – 407.93

average demand No allocation Sources

peak demand

4031 600.00

–

1022 400.001043 321.634022 950.004032 564.174043 600.00

41 326.05

β = 2

low demand4021 – 150.004032 – 407.93

average demand No allocation Sources

peak demand

4021 150.00

–

4031 600.001022 400.001043 242.944022 943.964032 237.63

42 400.0041 274.33

β = 7

low demand4021 – 150.004032 – 407.93

average demand No allocation Sources

peak demand

4021 150.00

–

4031 600.001022 400.004022 905.204032 238.994043 416.10

42 400.0041 288.35

Figure 2 shows the plot of the efficient frontier of the

CVaR versus the reactive power planning cost. These results

are significant since they provide relevant information to the

−16

−12

−8

−4

0

4

8

12

16

Tap

Posi

tion

β=0

β=7

(a) Low Demand

−15

−10

−5

0

5

10

15

Tap

Posi

tion

β=0

β=7

(b) Average Demand

−15

−10

−5

0

5

10

15

LTC Branch

Tap

Posi

tion

1044

−4044

1045

−4045

4042

−42

4041

−41

4047

−47

4043

−43

4046

−46

4051

−51

4061

−61

4062

−62

4063

−63

β=0

β=7

(c) Peak Demand

Fig. 1. Tap position of the under-load tap-changing transformer.

decision maker.

Typically, a solution with a low expected cost involves

a high risk of experiencing large costs in some scenarios.

Solving the problem for different values of β yields a set of

solutions with different values of expected cost and CVaR.

Consequently, it is possible to represent the expected cost

versus the CVaR or the cost standard deviation for different

values of β, yielding the so-called efficient frontier.

Figure 3 depicts the expected cost distributions for β = 0

and β = 7. It can be observed that the interval spanned by the

cost for β = 0 is wider than that resulting one for β = 7.

This fact indicates that the volatility of the procurement cost

1.654 1.655 1.656 1.657 1.658 1.659 1.66 1.661

x 104

7570

7580

7590

7600

7610

7620

7630

7640

CVaR [US$]

Rea

ctiv

e P

ow

er P

alnnin

g C

ost

[U

S$]

β = 0β = 1

β = 2

β = 3

β = 4

β = 5

β = 6

β = 7

Fig. 2. Reactive planning cost: efficient frontier.

is higher in the risk-neutral case (β = 0).

0 500 1000 1500 2000 2500 3000 3500

0.014

0.020

0.028

0.03

Operation Costs [US$]

β=0

β=7

Fig. 3. PDF of the reactive planning cost for different values of β.

V. CONCLUSIONS

In this paper, the RPP problem is formulated considering

CVaR assessment. Uncertainty in the availability equivalent

capacity reactive power sources is modeled by probabilistic

constraints with a ρ-quartile measure. The taps settings of

the under-load tap-changing transformers are modeled as dis-

crete settings taps. The proposed framework can quantify the

security risk created by uncertainties while minimizing the

operation and investment costs. This systematic approach can

help to transmission planner and operator system or decision

maker to carry out reactive planning considering risk and

uncertainties explicitly formulated in the proposed model.

An extension of the proposed problem can be formulated

considering a multi-stage analysis. In planning problems the

solution run time is negligible, therefore in this paper it is not

considered.

ACKNOWLEDGMENT

The authors would like to thank the economical support by

FAPESP grants. 2010/16728-5 and 2012/21570-7 and FEPISA

in Sao Paulo - Brazil.

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