# Application of Interval Arithmetic in the Evaluation of Transfer Capabilities by Considering the Sources of Uncertainty

Authors: Prabha Umapathy; C. Venkataseshaiah; M. Senthil Arumugam
Journal: Discrete Dynamics in Nature and Society
 (2009)

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## Abstract

Total transfer capability (TTC) is an important index in a power system with large volume of inter-area power exchanges. This paper proposes a novel technique to determine the TTC and its confidence intervals in the system by considering the uncertainties in the load and line parameters. The optimal power flow (OPF) method is used to obtain the TTC. Variations in the load and line parameters are incorporated using the interval arithmetic (IA) method. The IEEE 30 bus test system is used to illustrate the proposed methodology. Various uncertainties in the line, load and both line and load are incorporated in the evaluation of total transfer capability. From the results, it is observed that the solutions obtained through the proposed method provide much wider information in terms of closed interval form which is more useful in ensuring secured operation of the interconnected system in the presence of uncertainties in load and line parameters.


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## Body

## 1. Introduction

Due to the deregulation of the power industry, there
has been an increasing interest in quantifying the transmission transfer
capabilities of power systems. Interconnected systems are replacing isolated
systems for economic and reliability reasons, but the resulting looped networks
give rise to new technical concerns. Assessment of total transfer capability and
available transfer capability (ATC) are such issues in a power system. ATC can
be mathematically  defined as the total transfer capability (TTC) minus the
transmission reliability margin (TRM), minus the sum of existing commitments
and the capacity benefit margin (CBM) [1].In
the past few years, a number of methods have been proposed for ATC
determination. The literature of ATC calculation can be divided into two broad
categories: deterministic methods and
probabilistic methods.  The deterministic
methods can be further divided into the continuation power flow (CPF) method
[2], the repeated power flow (RPF) method [3], and the optimal power flow (OPF)
methods [4, 5].  Starting from a solved
base case, the CPF method obtains a series of power flow solutions by
increasing the transfer parameter by using a prediction-correction scheme.  RPF method starts from a base case and
repeatedly solves the power flow equations, each time increasing the power
transfer by a small increment until it reaches an operation limit.  OPF method formulates the TTC calculation as
an optimization problem, with equality constraints of power flow and inequality
constraints from operation limits.  The
objective function, obviously, is to maximize power flow on the specified
transmission route.  These methods seem
to be appropriate and efficient in management of the transmission system.  However, because it fails to consider the
probabilistic nature of the power system, the obtained ATC may be too
conservative and therefore lead to costly and inefficient use of system
resources.The
quantitative uncertainty is quantifiable in nature and is represented in
numerical terms by a mathematical function with deterministic parameters.  This could be achieved either by
probabilistic variables or by interval variables.  In the former, the uncertainty is defined by
probability density functions [6, 7]. The application of the interval
Arithmetic method for power flow analysis of transmission networks was first
proposed in [8].  This concept has been
applied for radial distribution system analysis [9, 10]. The basic idea in this
paper is to extend the application of interval arithmetic in order to consider
the probabilistic aspects of the system input parameters in the calculation of
total transfer capability. The problem is formulated as an optimization
problem, where the objective function is to maximize the power transfer between
a specific set of generator(s) and load(s) without violating any of the system
constraints.The remainder of this paper is
organized as follows: in Section2, the problem formulation of TTC is
presented; the functional procedure of the interval arithmetic is discussed in
Section 3 and calculation of base case power flow is briefly discussed in Section
4; the results for two important case studies on transfer capability are
discussed in Section 5 and finally the conclusions are summarized in Section 6.


## 2. Problem Formulation for Total Transfer Capabilty

The mathematical formulation of TTC can be expressed
as follows.The
objective function for the OPF reflects the maximum power transfer from one
bus/area to another bus/area.Objective function is(2.1)f(x)=Max(∑i∈SDΔPDi).The equality constraints of the OPF
reflect the power system. The power system is enforced through the power flow
equations which require that the net injection of real and reactive power at
each bus is equal to zero.Equality constraints are(2.2)PGi−PDi−∑j=1n|Vi||Vj|(Gijcosδij+Bijsinδij)=0,QGi−QDi−∑j=1n|Vi||Vj|(Gijcosδij+Bijsinδij)=0.The
inequality constraints of the OPF reflect the limits created to ensure system
security. The limiting constraints considered in this paper consist of three
types: generation capacity, transmission line capacity, and voltage level.The
generators’ maximum and minimum outputs for real and reactive powers are as follows:(2.3)|PGi|min≤|PGi|≤|PGi|max,|QGi|min≤|QGi|≤|QGi|max.For the maintenance of system
security, the transmission line MVA ratings are taken into account(2.4)|Sij|≤|Sij|max.To
maintain the quality of the electrical service and system security, bus voltage
limits are considered(2.5)|Vi|min≤|Vi|≤|Vi|max, whereΔPDi:
is active power increment of load bus;PGi, QGi:
are real
and reactive power generation at busi;PDi, QDi:
are real
and reactive load demand at busi;n:
is
bus number of the system;Vi, Vj:
are voltage magnitudes at busi,j;Sij:
is  line MVA limit.


## 3. Interval Arithmetic Method

Interval arithmetic is a
powerful tool to determine the effects of uncertain data.  It can deal with numbers that vary within a
range. The basic concepts of interval arithmetic are discussed as follows.An
interval numberX=[x1,x2] is the set of real numbers x such that x1≤x≤x2.  Here x1andx2 are known as the lower limit and upper limit
of the interval number, respectively.
Let X=[x1,x2] and Y=[y1,y2] be the two interval numbers.  The basic arithmetic operations of addition,
subtraction, multiplication, and division of these two interval numbers are
defined as follows  [11]: (3.1)X+Y=[x1+y1,x2+y2],X−Y=[x1−y1,x2−y2],X*Y=[min(x1*y1,x1*y2,x2*y1,x2*y2),max(x1*y1,x1*y2,x2*y1,x2*y2)],X÷Y=X*Y−1.However, for the purpose of
power flow analysis, calculations are based on complex numbers rather than real
numbers.  The basic relations involving
complex interval numbers are described as follows.   In
general, any complex numberZ=X+iY—where i is the complex operator—is
said to be a complex interval number if both its real and imaginary parts, X and
Y, respectively, are interval numbers.
Therefore X can be represented as X=[x1,x2] and Y can be represented as Y=[y1,y2].LetZ1=A1+iB1 and Z2=A2+iB2 be the two
complex interval numbers. The addition, subtraction, multiplication, and
division of these two complex numbers are defined as follows:(3.2)Z1+Z2=(A1+A2)+i(B1+B2),Z1−Z2=(A1−A2)+i(B1−B2),Z1*Z2=(A1*A2−B1*B2)+i(A1*B2−A2*B1),Z1÷Z2=C+iD,where, (3.3)C=(A1*A2+B1*B2)÷(A22+B22),D=(A2*B1+A1*B2)÷(A22+B22).


## 3.1. Interval Power Flow Analysis

The power flow analysis method used in this work is
fast-decoupled load flow algorithm.
However, to account for the uncertainty of the input load parameters,
the real and reactive power loads are treated as interval numbers rather than
fixed numbers.  Similarly, to account for
the uncertainty of the input line parameters, the resistance and reactance of
the line are treated as interval numbers. Consequently, the complex arithmetic
has been replaced by complex interval arithmetic(3.4)P(K)=[P(k)ℓ,P(k)u],Q(k)=[Q(k)ℓ,Q(k)u],R(jj)=[R(jj)ℓ,R(jj)u],X(jj)=[X(k)ℓ,X(jj)u],V(k)=[V(k)ℓ,V(k)u],
where,
P(k) is real power load of kth bus;Q(k) is reactive power load of kth bus;V(k) is voltage magnitude at kth bus;R(jj)
is resistance of the branch—jj;X(jj)
is reactance of the branch—jj;P(k)ℓ and P(k)u are lower and upper limits of real power load of kth
bus;Q(k)ℓ and Q(k)u
are lower and upper limits of reactive power load of kth bus;R(jj)ℓ and R(jj)u
are lower and upper limits of resistance of a particular branch—jj;X(jj)ℓ and X(jj)u are lower and upper limits of reactance of a particular branch—jj.


## 3.2. Uncertainty Parameters in Load and Line Variation Modeling

The system is assumed to operate under normal
conditions but line and load parameters vary within a certain range.  In this paper,±10% variations in load
parameters and ±3% for line parameters are considered from their rated nominal
value(3.5)P(k)ℓ=0.90P(k),P(k)u=1.10P(k),Q(k)ℓ=0.90P(k),Q(k)u=1.10Q(k),R(jj)ℓ=0.97R(jj),R(k)u=1.03R(jj),X(jj)ℓ=0.97P(jj),X(k)u=1.03X(jj).
Various case studies have been conducted and the results
of the following cases are presented:(i)
the system under normal operating conditions,(ii)
uncertainties due to load parameters only,(iii)
uncertainties due to line parameters only,(iv)
uncertainties due to both load and line parameters.
The basic steps used for computing the TTC using interval
arithmetic for each transaction are given in Figure 1.Figure 1
Flow diagram for the computation of ATC.


## 4. Basecase Power Flow Calculation

The IEEE 30 bus system shown in Figure2 is adopted to
illustrate the proposed method.  The
system is divided into 3 areas.  The
system has six generators, with two generators in each area.  We assume each area as a utility.  The utility in a certain area wants to import
power from another area.  Hence TTC is
evaluated between areas. The base case load flow for fixed values of input is
given in Table 1.Table 1
Base case load flow.
Bus numberArea numberV (p.u)Angle (deg)GenerationLoadPQPQ111025.970.9900211−0.41549160.9731.9921.712.7310.983138−1.522074——2.41.2410.980093−1.794728——7.61.6510.982406−1.863823——00610.973184−2.266957——00710.967355−2.651837——22.810.9810.960624−2.725769——3030910.980506−2.996933——001030.984404−3.374936——5.821110.980506−2.996933——001220.985468−1.536912——11.27.513211.4761633711.36001420.976677−2.308035——6.21.61520.980229−2.311835——8.22.51620.977396−2.644486——3.51.81720.976865−3.392339——95.81820.96844−3.478388——3.20.91920.965287−3.958205——9.53.42020.969166−3.871024——2.20.72130.993383−3.488393——17.511.22231−3.39272921.5939.56002321−1.58922819.27.953.21.62430.988566−2.631461——8.76.72530.990215−1.689989——002630.972194−2.139346——3.52.32731−0.82843926.9110.54002810.974715−2.265929——002930.979597−2.128498——2.40.93030.967883−3.041524——10.61.9Figure 2
IEEE 30 bus test system.


## 4.1. Base Case Power Flow Incorporating Various Uncertainties Using Interval Arithmetic

The
transfer capability is computed from a base case constructed from the system
information available at a given time. Usually, there is some uncertainty or
inaccuracy in this computation because the transfer capability computed at the
base case does not reflect evolving system conditions.  These uncertainties include inaccurate or
incorrect network parameters, effects neglected in the data and approximations
in transfer capability computations.   As
already discussed in Section3, the uncertainties in load and line parameters
have been taken into account by assuming that they vary over a range.  When the load demands in the system vary
within some intervals, the bus voltages and other quantities such as line flows
and line losses also vary within certain intervals.  The intervals of variation of bus voltages for
the system under consideration have been calculated for all the three uncertainties
and are presented in Table 2.  In this
table, the symbols Vℓℓ and Vuℓ denote the lower limit and
upper limit of the voltage, respectively.Table 2
Bus voltage magnitudes with different uncertainties.
Bus numberFixed
valuesLoad uncertaintyLine
uncertaintyLoad &
line uncertaintiesLower limitUpper limitLower limitUpper limitLower limitUpper limitVℓℓ(p.u)Vuℓ(p.u)Vℓℓ(p.u)Vuℓ(p.u)Vℓℓ(p.u)Vuℓ(p.u)111111112111111130.9831380.9813060.9865370.9824880.9845350.9806830.98697240.9800930.97730.9837610.9799930.9812570.9765480.98428550.9824060.9796750.9849890.9818240.9829770.9789980.98547560.9731840.969030.9772160.9723690.9741020.9680130.97794570.9673550.9632140.9723140.9668320.96890.9620150.97319680.9606240.954010.9651980.9584740.9610550.9525160.96630490.9805060.9776270.9822250.9793530.9806340.97690.982787100.9844040.9822940.9848970.9831130.9841470.9817280.98537110.9805060.9776270.9822250.9793530.9806340.97690.982787120.9854680.9836730.9869490.9848670.9858370.9831290.98738131111111140.9766770.9740630.9789310.9757810.9772890.9732240.979605150.9802290.9780560.9822410.9795420.9808190.9773440.98281160.9773960.9742980.978830.9758530.9773550.9734670.979506170.9768650.9735390.9778690.9749760.9765080.9726960.978563180.968440.9651170.9714160.9672970.9693150.9639980.972322190.9652870.9612350.9680890.9635860.9658220.9599980.969093200.9691660.9655310.9714850.9675540.9695440.9644310.972382210.9933830.9921670.9927870.9922530.9927180.9919220.993009221111111231111111240.9885660.9872470.9898030.9881730.9888810.9868530.990118250.9902150.9891560.9912790.9899140.9905270.9888150.991552260.9721940.9690920.9749680.971170.9729080.9681280.975744271111111280.9747150.9703290.9786860.9737980.9754480.9693530.979372290.9795970.9774830.9817870.9790010.9802920.9767630.98236300.9678830.9644670.971250.9668590.9678590.9633330.972154Figures3 and 4 show the voltage profile for load
uncertainty and line uncertainty for fixed, lower, and upper limit
voltages.  One can see that the load
uncertainty has a considerable effect on the voltage profile compared with the
line uncertainty.  However, when both
load and line uncertainties are taken into account, the effect is much more and
is reflected in the voltage profile shown in Figure 5.Figure 3
Voltage profile for load uncertainties.Figure 4
Voltage profile for line
uncertainties.Figure 5
Voltage profile for
load and line uncertainties.


## 5. Results and Discussions

Several cases on the transfer capability are studied
and from them the results of two important case studies are presented and
discussed below.


## 5.1. Case 1: The Transfer Capability from Area 2 to 1

Using
the proposed interval arithmetic method, TTC is evaluated for both fixed input
parameters and for various uncertainties.
The loads are modeled as constant power factor loads.  The active loading (in MW) of area 1 before
and after the transaction is shown in Table3.Table 3
Active loading of area 1.
Bus number23478Before21.72.47.622.830.0After23.55.111.225.832.4The
generation of area 2 increases from 56.2 to 70.0 MW and the load at area 1 from
84.5 to 98 MW and the limiting condition was the overloading of lines6–8. The sum of the loads at the sink area is
taken as the TTC.  Table 4 gives the
total load of area 1 and total real and reactive losses in interval parameters.  One may observe that the range of the total
loss is higher for load uncertainty compared to line uncertainty.  Obviously, the losses increase when both load
and line uncertainties are taken into account.Table 4
Base case for Cases 1 and  2.
Varying parametersReal
loadReal
loadReactive
loadReactive
loadTotal
real lossTotal
real lossTotal
reactive lossTotal
reactive lossPℓℓ (MW)Puℓ (MW)Qℓℓ (MVAr)Quℓ (MVAr)Pℓℓ (MW)Puℓ (MW)Qℓℓ
(MVAr)Quℓ
(MVAr)Fixed
line and load84.584.556.456.42.442.448.998.99Load
uncertainty76.0592.9550.8662.141.963.2157.3411.47Line
uncertainty84.584.556.456.42.3822.5428.729.30Load
and line uncertainty76.0592.9550.8662.141.8453.3227.1011.85Table5 shows the values of different operating parameters after the transaction has
been carried out.  The TTC values are
presented in interval form for all the categories of uncertainties. It is
observed that the interval arithmetic method indicates a wider range of
intervals.Table 5
Interval values for the operating parameters in case 1.
Varying parametersTotal transferReactiveTotal realTotal reactiveTotalcapabilityloadlosslosslossLower limit (MW)Upper limit (MW)Qℓℓ (MVAr)Quℓ (MVAr)Pℓℓ (MW)Puℓ (MW)Qℓℓ (MVAr)Quℓ (MVAr)Sℓℓ (MVA)Suℓ (MVA)Fixed line and load989863.3763.372.682.6811.5211.5211.8311.83Load Uncertainty88.89108.6559.4572.642.223.8748.78149.0614.53Line Uncertainty90.49104.2359.973.42.8082.9910.2713.4410.6513.77Load and Line Uncertainty86.45106.4356.0265.992.653.310.7514.8611.0715.22


## 5.2. Case 2: The Transfer Capability from Area 3 to 1

The
generation of area 3 is increased proportionally when the loads in area 1 are
increased. The generation at area 3 is increased from 84.5 to 94 MW.  The load active power vector of area 1 in bus
number sequence is [0.0, 27.7, 3.06, 9.70, 0.0, 0.0, 29.1, 38.29, 0.0, 0.0].
The limiting conditions are overloading in lines6–8 and 21-22.  It is observed that the TTC lower limit and
upper limit vary over a wide range when both load and line uncertainties are
taken into account.  Table 6 shows the
values of TTC and other system parameters in interval form after the
transaction has been carried out.Table 6
Interval values for the operating
parameters in case 2.
Varying parametersTotal TransferReactiveTotal realTotal reactiveTotalcapabilityloadlosslosslossLower limit (MW)Upper limit (MW)Qℓℓ (MVAr)Quℓ (MVAr)Pℓℓ (MW)Puℓ (MW)Qℓℓ (MVAr)Quℓ (MVAr)Sℓℓ (MVA)Suℓ (MVA)Fixed line and load107.85107.8571.9871.982.62.610.0410.0410.3710.37Load uncertainty97.058118.6364.979.312.053.398.812.29.0412.66Line uncertainty102.35114.5266.3380.412.512.699.7110.3610.0310.70Load and line uncertainty95.03116.2865.8373.731.983.58.5212.638.7513.11


## 6. Conclusion

The need for efficiency in electrical power
deregulation has increased the need for improved calculations of TTC by
incorporating the uncertainties in transmission.  This paper proposes a probabilistic approach
using interval arithmetic to estimate the TTC. The solution obtained from the
interval arithmetic method is found to be more informative in qualitative terms
about the system analysis when compared to the conventional deterministic
approach.  It is conjectured that power
planners and operators may benefit not only from the expected TTC evaluation,
but also from an evaluation of confidence intervals of the calculation.  The transfer capability from area 2 to 1 and
3 to 1 of the IEEE 30 bus test system is computed with fixed line and load, with
line uncertainty, with load uncertainty, and with both line and load
uncertainties. The total loss for the lower limit and upper limit is computed
for all four types and compared. The results clearly elucidate that the total
loss occurring is minimum when the bus is operated with line uncertainty. The
obtained results are useful to ensure a secured operation of the interconnected
network when dealing with extensively large uncertainties in load and line
parameters.



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*Source: 527385-2009-06-01.xml*