IEEE TRANSACTIONS ON POWER SYSTEMS 1 Interval Power Flow

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IEEE TRANSACTIONS ON POWER SYSTEMS 1

Interval Power Flow Analysis UsingLinear Relaxation and Optimality-BasedBounds Tightening (OBBT) Methods

Tao Ding, Student Member, IEEE, Rui Bo, Senior Member, IEEE, Fangxing Li, Senior Member, IEEE,Qinglai Guo, Member, IEEE, Hongbin Sun, Senior Member, IEEE, Wei Gu, Member, IEEE, and

Gan Zhou, Member, IEEE

Abstract—With increasingly large scale of intermittent and non-dispatchable resources being integrated into power systems, thepower flow problem presents greater uncertainty. In order to ob-tain the upper and lower bounds of power flow solutions includingvoltage magnitudes, voltage angles and line flows, Cartesian coor-dinates-based power flow is utilized in this paper. A quadraticallyconstrained quadratic programming (QCQP) model is then estab-lished to formulate the interval power flow problem. This non-convex QCQP model is relaxed to linear programming problemby introducing convex and concave enclosures of the original fea-sible region. To improve the solutions bounds while still encom-passing the true interval solution, optimality-based bounds tight-ening (OBBT) method is employed to find a better outer hull of thefeasible region. Numerical results on IEEE 9-bus, 30-bus, 57-bus,and 118-bus test systems validate the effectiveness of the proposedmethod.

Index Terms—Convex/concave envelopes, interval power flow,linear relaxation, optimality-based bounds tightening (OBBT),quadratically constrained quadratic programming (QCQP),uncertainty.

NOMENCLATURE

, Real and imaginary part of voltage magnitude ofbus .

, Real and imaginary part of th element of busadmittance matrix .

Manuscript received July 20, 2013; revised November 22, 2013 and January19, 2014; accepted February 20, 2014. This work was supported in part by Na-tional Key Basic Research Program of China (973 Program) (2013CB228203),National Science Foundation of China (51277105) and National Science Fundfor Distinguished Young Scholars of China (51025725). The work of F. Liwas supported in part by CURENT, a US NSF/DOE Engineering ResearchCenter under NSF Award Number EEC-1041877. Paper no. TPWRS-00942-2013. (Corresponding author: R. Bo).T. Ding is with the Department of Electrical Engineering, Tsinghua Univer-

sity, Beijing, China, and also with the Department of Electrical Engineering andComputer Science, The University of Tennessee, Knoxville, TN 37996 USA.R. Bo is with the Midcontinent Independent Transmission System Operator

(MISO), Eagan, MN 55121 USA (e-mail: [email protected]).Q. Guo and H. Sun are with the Department of Electrical Engineering,

Tsinghua University, Beijing, China.F. Li is with the Department of Electrical Engineering and Computer Science,

The University of Tennessee, Knoxville, TN 37996 USA.W. Gu and G. Zhou are with the School of Electrical Engineering, Southeast

University, Nanjing, China.Color versions of one or more of the figures in this paper are available online

at http://ieeexplore.ieee.org.Digital Object Identifier 10.1109/TPWRS.2014.2316271

Total number of buses.

Set of PQ and PV buses.

Set of PQ buses.

Set of PV buses.

Reference bus.

, Lower and upper bound of .

, Active and reactive power injection at bus .

, Lower and upper bound of .

Total number of branches.

, Voltage magnitude and angle of bus .

I. INTRODUCTION

R ENEWABLE energy, especially wind, is regarded asan important alternative to traditional power generating

sources due to its non-exhausted nature and benign environmenteffects [1]. However, the main challenge is associated with itsintermittency and volatility. In particular, the wind speed ishighly dependent on the weather conditions, the geographicalregion, and seasons of the year [2]. To ensure system reliability,the forecasting uncertainty must be considered into generationscheduling, and interval power flow provides promising ap-proach to achieve the boundary information of system statusesunder uncertainties.The load flow problem is typically formulated as a set of non-

linear equations as functions of bus voltages. Some algorithmsspecific to load flow problem with uncertain demand have beenpresented in [3]–[14], where stochastic, fuzzy, and probabilityprogramming techniques were introduced in order to model theuncertainties. Such approaches incorporate model uncertaintiesexpressed as fuzzy membership functions [4], [5] and proba-bility density functions [6]–[10] into its solution procedure.Probabilistic method was used in [6]–[8] where equations

were linearized for further convolution. In [9], an approach in-volves semi-volatile variable was proposed to improve calcu-lation effectiveness, which transformed convolution operationinto simple addition operation. In [10], Monte Carlo methodwith Latin super-cube sampling is introduced to study proba-bilistic load flow, but it requires several models that sample var-ious combinations of input values. So the computation requiredfor these types of studies could be expensive.

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2 IEEE TRANSACTIONS ON POWER SYSTEMS

However, the aforementioned methods are typically depen-dent on pre-defined probability distribution function or mem-bership function of uncertainty. It may be difficult to identifyaccurate probability distribution functions due to data avail-ability and stochastic nature of the uncertainty. In addition, thesemethods provide probabilistic solution to the system reliability[11]–[13]. With the consideration of the above limitations, in-terval analysis has been frequently referred to as an effectivealternative to resolve such uncertainties [14]–[17].Interval arithmetic (IA) characterizes load and other parame-

ters by a range of values rather than a single number. In [14] and[15], power flow model with interval variables is proposed andcalculated with Krawczyk-Moore interval iterative method. IAis useful for considering the effect of power injection uncertain-ties, and solving the set of nonlinear equations. However, onelimitation of IA is that it may result in very conservative andenlarged interval solution. Therefore, affine arithmetic (AA),which is introduced in [16] and [17] as an improvement of IA,has been proposed to take into account the correlation betweenuncertain operands and sub-formulas involved in the intervalcalculation.AA method linearizes the nonlinear equations at the oper-

ating point using Taylor series or Horner’s rule and uses Cheby-shev’s approximation theory to represent the remainder of thelinearization. However, approximation errors are unmeasure-able. In addition, it is difficult for AA method to deal withsine and cosine function in energy balance equations formu-lated in polar coordinates due to the strong nonlinear character-istic. This paper therefore utilizes the energy balance equationin Cartesian coordinates to formulate the interval power flowproblem as a quadratically constrained quadratic programming(QCQP) problem. Due to the non-convex nature of the QCQPproblem, firstly a linear relaxation method is employed to relaxthe QCQP problem to linear programming (LP) problem to getrelaxed bound solution. Then, optimality-based bounds tight-ening (OBBT) method is introduced to tighten the feasible re-gion of the relaxed problem to find the optimal outer hull of theQCQP problem. The relaxed linear programming model is thenevaluated under the tightened feasible region. The interval solu-tion of voltage magnitudes, angle and branch flow can be solvedhighly efficiently for all buses and branches.The rest of the paper is organized as follows: Section II

presents the interval power flow and its QCQP form in Carte-sian Coordinates. Section III proposes using linear relaxationmethod and OBBT method to solve the problem. In addition,a sparse matrix technique is introduced to reformulate theproblem with the consideration of sparsity of power networks.In Section IV, numeric results and comparisons on standardIEEE test systems are presented to illustrate the effectivenessof the proposed method in obtaining interval values of the loadflow. Finally, conclusions are drawn in Section V.

II. INTERVAL POWER FLOW PROBLEM AND ITS QCQP FORM

The energy balance equations in polar coordinates includingactive and reactive power flow are represented as (1)–(2):

(1)

(2)

In power flow problems, uncertainties exist widely, amongwhich the active and reactive power supply and demand areoften of the most interest, and therefore are considered in the in-terval power flow model. Other parameters of the network suchas the resistances and reactances of transmission lines and trans-formers are regarded as constants.Note that the energy balance equations contain strong non-

linear functions such as and , which are difficult todeal with when obtaining the AA form of the injected powers[16]. Usually, the linear polynomials using Taylor series orHorner’s rule are introduced to linearize the system of equa-tions at the operating point with respect to the variables, wherethe omitted remainders in the approximation might cause addi-tional level of inaccuracy. An iterative method was proposedin [18] that eliminates the effect of remainders with an initialguess of deviation so as to narrow the solution’s bounds.In order to overcome the approximation errors caused by

strong nonlinear energy balance equations in polar coordinates,we transform the original problem into Cartesian coordinatesform in (3)–(5), which only contain quadratic polynomials:

(3)

(4)

(5)

In interval power flow analysis, power injections are volatileand can be represented using intervals, such thatand . However, the maximum (or minimum)voltage magnitude cannot be directly obtained due to the non-linear characteristics of the power flow equations.We solve the interval solution for one bus at a time. Take

voltage magnitudes for example (and the branch flow will bepresented later): the interval power flow (IPF) problem can beformulated as a non-convex and nonlinear programming (NP)in (6)–(10) with quadratic objective and constraints, which is astandard QCQP problem and is labeled IPF-QCQP in this paper:

(6)

subject to

(7)

(8)

(9)

(10)


DING et al.: INTERVAL POWER FLOW ANALYSIS USING LINEAR RELAXATION 3

It can be deduced that the total number of decision variablesis . Traditional interior point method cannot pro-

duce the global optimal solution of QCQP problem becauseof its non-convexity. On the other hand, this type of problemhas the special quadratic structure, which enables the utilizationof a linear relaxation method proposed in [19] and an OBBTmethod proposed in [20]. The next section will discuss howthese methods can be employed to achieve the optimal solutionbound.

III. SOLUTION TECHNIQUES USING LINEAR RELAXATIONAND OPTIMALITY-BASED BOUNDS TIGHTENING

A. Linear Relaxation Model for Quadratically ConstrainedQuadratic Programming (QCQP) Problem

First, we consider the following form of QCQP problemswithout the loss of generality:

(11)

(12)

(13)

where denotes the number of quadratic constraints, () are indefinite matrices, ( )

are -dimensional vectors, the set is assumed tobe nonempty and bounded.

is called bilinear function if the functions andare both linear functions. For example is a

bilinear function.Let a mapping : be a bilinear function, then

is the convex envelope of over , if it is thepoint-wise supremum of convex underestimation of over .Similarly, is the concave envelope of the over ,if it is the point-wise infimum of concave overestimation ofover .Theorem 1: The convex and concave envelope of the bilinear

function over a rectangular region [21]

(14)

are given by the expressions(15) and (16), and then we can ar-rive at :

(15)

(16)

Let dummy variable and substitutefor in QCQP model, the objective and constraints becomebilinear. Furthermore, the convex envelopes and concaveenvelopes as defined in (15)–(16) are introduced to obtainthe tractable linear relaxation (LR) model of the non-convexproblem QCQP, formulated as follows:

(17)

(18)

(19)

(20)

(21)

(22)

(23)

where the operator “ ” can be defined as, for .

For minimization problem of the original QCQP, theQCQP-LR is a relaxed linear programming model and there-fore provides a lower bound of the original QCQP. Likewise,the QCQP-LR model provides an upper bound for the maxi-mization problem of the original QCQP.

B. Sparse Matrix Technique for IPF-LR Model

Based on the above linear relaxation method, the QCQPproblem can be relaxed into linear programming problemby adding dummy variables and approximating the originalconstraints with concave and convex envelopes.Intuitively, one dummy variable needs to be created for

each pair of any two nodes , and the resulting matrixwill contain elements. It presents challenge when ap-plied directly to large scale power systems. Fortunately, due tothe sparse nature of power networks, variable is not neededfor any two nodes that have no direct connection. Utilizing thesparsity can greatly reduce the size and complexity of the LRoptimization model. Therefore, this subsection will introducean approach to deal with sparse matrix and present the IPF-LRmodel in reduced form.Let be a finite set and denote the pairs of by ( ) in

(24):

(24)

For a given power network graph with[20], where the elements of are the vertices of representingall the buses, and the elements of are the edges of repre-senting transmission lines and transformers. According to theenergy balance functions in (7)–(8), the quadratic terms inand have the same structure except for the constant , andthe operator sign. Note that only up to six additional dummyvariables , , , , and are needed foreach edge , and bilinear terms are not neededfor . Meanwhile, in order to reduce the number ofadditional dummy variables, the bus admittance matrix

, describing the bus-bus relationship, should be con-verted into bus-branch relationship. An sparse connectionmatrix is defined such that its element is 1 if branchis connected to bus and 0 otherwise.We firstly define and as the susceptance vector and

conductance vector of the branch admittance, respectively, anddefine and as the susceptance vector and conductancevector of the bus self-admittance. Then, utilizing the connectionmatrix , we convert those vectors into four matrices

, , and , asformulated in (25)–(28). Fig. 1 illustrates the data structure ofthe sparse susceptance and conductance vectors:



Fig. 1. Data structure of sparse susceptance and conductance vector.

(25)

(26)

(27)

(28)

Further, we define , , ,, , , . Then dummy

variable vectors are created, , ,, , , .

With the four susceptance and conductance matrices , ,and , and the dummy variable vectors, (7)–(8) can be

converted into (30)–(31). Based on the linear relaxation methodfor QCQP model introduced in Section III-A, the IPF-QCQPcan be converted into IPF-LR expressed in (29)–(56).Through this sparse matrix technique, the number of added

matrix variable is reduced from elements to ,which is particularly important for applications to large-scalesystems:

(29)

subject to

(30)

(31)

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

(44)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

(56)

where is the scheduled voltage magnitude for PV bus ;

and are the given real and imaginary parts of reference busvoltage.It should be noted that (29) formulates upper bound and lower

bound of the interval voltage magnitude of each bus. Intervalsof other system statuses including voltage angle, branch reac-tive and active power of power flow, can be established throughchanging the model objective function as follows and addingadditional constraints.1) Voltage Angle: The voltage angle interval can be obtained

by solving the original IPF-QCQP problemwith replaced objec-tive function .Note that voltage angles are normally between

, in which tangent function is monotonous.Therefore, we can firstly solve an IPF-QCQP problem with(57) as objective function to get the interval of a new dummyvector , where , and then calculate the voltage angleinterval using . After reformulatingas a quadratic constraint , linear relaxation methodin (15) and (16) is used again for to generate (58)–(61).Combining (58)–(61) with the original constraints (30)–(56),together with the new objective function (57), we can obtainthe following model for obtaining interval solution of :

(57)

(58)

(59)

(60)

(61)

(62)



(63)

(64)

(65)

(66)

(67)

(68)

(69)

(70)

(71)

(72)

(73)

(74)

(75)

(76)

(77)

(78)

(79)

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

2) Branch Active and Reactive Power: The original formof branch active and reactive power is formulated in (89) and(90). Using the aforementioned susceptance and conductancematrices as well as the dummy variable vectors, (89) and (90)can be expressed in linear representation as in (91) and (92)respectively:

(89)

(90)

(91)

(92)

3) Generator’s Reactive Power: As for generator’s reactivepower, the original form is formulated in (93). Similarly, (93)can be expressed in linear representation as in (94) according tothe aforementioned method. Under power injection variations,generators’ reactive power output may exceed its limit, giventhe scheduled voltage. In this situation, the PV bus will be con-verted to PQ bus with reactive power assigned to the limitingvalue, and the solution needs to be recalculated:

(93)

(94)

C. OBBT Method for Optimal Interval Solution

It is evident that the relaxed IPF-LR model is dependent onthe hyper-rectangular variable domain such as (56). The initialdomain can be so large that leads to a very conservative intervalsolution, and so it is desirable to find a tighter hyper-rectangledomain to achieve the optimal interval solution. For this reason,the OBBT method [20] is employed in this section to solve theIPF-QCQPmodel, which will tighten the relaxed model IPF-LRbased on linear optimality by cycling through each participatingvariable until the volume of the domain stops to shrink.Define to represent the variable set in

the th iteration. Let denote the constraints ofIPF-LR including (30)–(56) and (58)–(61), and let bethe (IPF-LR) solver that maps the hyper-rectangular variabledomain to a real value that repre-sents the optimal objective value of (IPF-LR) model. Then,the hyper-rectangular variable domain can be tightened or‘squeezed’ through an iterative linear programming solver,which is illustrated in the following three steps.1) An initial estimation of the solution interval is obtainedas follows by linearizing power flow at a given operatingpoint with the Jacobi matrix calculated by

(96) and applying an offset ( , )to ensure optimal solution bound is within the initialestimation:

(95)

(96)



Then, the initial variable domain is set aswhere

is a big number. Set .

2) Solve

for . Set .3) . If , stop; otherwise,go to 2).

Note that the above process is only needed once for the wholeinterval power flow calculation both for “min” and “max”modelsimultaneously.The (IPF-LR) model is a linear programming model over a

bounded and closed set (or compact set) , and the objectivefunction can be both “min” and “max” models. Based onOBBT method, the variable domain satisfies[20], and as such the domain sequence ,obtained from steps 2) and 3), is monotonously decreasingand geometrically a nested sequence of hyper-rectangles.Therefore, the corresponding convex hull satisfies

and forms a monotonously decreasing se-quence [19]. As a result, the sequenceis a monotonously increasing sequence for the “min” modeland a monotonously decreasing sequence for the “max” model.At the same time, the sequence isalso a bounded sequence (detailed proof is presented in theAppendix). According to the convergence theorem [23], thesequence is a Cauchy sequence andhas a limit, and therefore the proposed OBBT-based algorithmcan converge after finite number of iterations.The gradually tightened hyper-rectangular variable domain

and the corresponding convex hull as well as the feasible regionof the original problem are illustrated in Fig. 2, where the solidred area represents the feasible region of the original QCQPproblem.After the process stops (and OBBT algorithm converges), the

optimally tightened convex hull over hyper-rectangularvariable domain and the corresponding interval solution areobtained. Because the dummy vector is a component of , theinterval solution of bus angle is immediately available. Linearprogramming models in (97)–(100) are solved for interval solu-tion of bus voltage, branch active and reactive power and gener-ator’s reactive power respectively. Linear programming modelsinstead of interval operation using (29), (91) and (92) are uti-lized to avoid conservative operations of IA due to the correla-tion among dummy variables , , , , and :

(97)

(98)

(99)

(100)

Fig. 2. Gradually tightened variable domain and convex hull using OBBT-based method.

Fig. 3. Flowchart of the proposed method.

It should be noted that (97)–(100) represent the IPF-LRmodels over the improved and tighter outer hull of originalfeasible space. Besides, as the interval solutions of bus voltageand branch active and reactive power can be solved indepen-dently, parallel computing can be utilized to further improvethe computing efficiency. The flowchart of proposed method isshown in Fig. 3.

IV. NUMERICAL RESULTS

A. 9-Bus System and Comparison With Monte CarloSimulation

A 9-bus test system available in [24] was analyzed using theproposed methodology, with a variation on load and gen-eration. The computation was performed using MATPOWER,YALMIP and CPLEX 12.4 on a personal computer with Intel®Core™ i5 Duo Processor T420 (2.50 GHz) and 8 GB RAM.The interval solutions obtained by solving for the maximum andminimum of voltage magnitudes and angles for load bus (a.k.a.



TABLE IINTERVAL SOLUTIONS OF VOLTAGE MAGNITUDES AND ANGLES OBTAINED FROM PROPOSED METHOD AND MONTE CARLO SIMULATION METHOD

TABLE IIINTERVAL SOLUTIONS OF ACTIVE AND REACTIVE LINE FLOWS OBTAINED FROM PROPOSED METHOD AND MONTE CARLO SIMULATION METHOD

PQ buses) and generator bus (a.k.a. PV buses), as well as lineflows, were listed in Tables I and II. Results for transmissionbuses were not presented for simplicity. The results were ver-ified by comparing with Monte Carlo (MC) stochastic simula-tion results, which randomly samples power injection for 5000trials to get maximum/minimum voltage magnitude and angleof each bus. Here, we assume Monte Carlo simulation (MC)with sufficient number of samples can yield the “correct” in-terval solutions.From the comparison between proposed method and Monte

Carlo simulation, it is seen that the proposed method can es-timate the upper/lower bound of voltage magnitude with max-imum error no more than 0.5%. Furthermore, the ratio betweenthe length of intervals obtained from the MC simulation andthose obtained from the proposed method is presented in thelast column of Tables I and II. The ratio is mostly in the rangeof 60% to 90%, indicating the estimated interval length is rea-sonably enlarged compared to that from MC method.It should also be noted that, when one variable happens to

sit at the interval lower or upper bound at a given scenario, theother variables don’t necessarily sit at their respective intervalbounds.

B. 57-Bus System and Comparison With Affine ArithmeticMethod and Monte Carlo Simulation Method

The proposed method was also verified on the IEEE 57-bussystem, with a variation on load and generator powers.

The results were compared with those of MC simulation with5000 sample size. Figs. 4 and 5 depict the interval lower andupper bounds for voltage magnitudes and angles; Figs. 6 and7 show the interval results of branch active and reactive powerflows. The comparison between QCQP and MC simulation il-lustrates the effectiveness of the proposed method, and the max-imum error of voltage magnitude is no more than 3% and av-erage error is about 1%. The error of active/reactive power flowintervals is also very small, which demonstrates the effective-ness of the proposed method.Furthermore, in comparison to Affine Arithmetic method

[15], we can see that the proposed method produces lowerbounds and upper bounds that are significantly closer to the truelower and upper bounds obtained from Monte Carlo simulationmethod. Particularly, estimated intervals for some buses such asbus 23, 31, and 37 using the proposed method are much betterthan those using AA method. The performance improvementof the proposed method over AA method has been observedin both voltage magnitude bounds and voltage angle bounds,which is because the proposed method only involves verylimited interval arithmetic operations.It should also be pointed out that the proposed method arrives

at a better solution than AA method at the cost of more compu-tation time. For example, it took the proposed method about 19 s(8% of simulation time by MC simulation) to obtain the intervalsolution whereas it only took AA method 7 s (3% of simulationtime by MC simulation).



Fig. 4. Bounds of voltage magnitudes under 20% variation.

Fig. 5. Bounds of voltage angles under 20% variation.

Fig. 6. Bounds of active power flows under 20% variation.

C. Reactive Power Limit

Table III lists the generator reactive power limit and sched-uled voltage, the initial interval analysis results for reactive

Fig. 7. Bounds of reactive power flows under 20% variation.

power and voltage magnitude before bus type conversion, andthe final interval analysis results after bus type conversion. The“Initial solution” column represents the case without consid-ering reactive power limit, and “Final solution” column showsthe case with reactive power limit considered. Table III alsoshows reactive power outputs of generator 3, 9, and 12 are outof limit, as highlighted in red and bold font. Therefore, the bustypes for those generator buses are changed from PV to PQ andthe solution is recalculated.Taking generator 9 as an example, the interval results of the

proposed method are compared with those of MC simulation inFigs. 8 and 9. With the violation of reactive power limit, gener-ation 9’s bus type is switched to PQ bus, and the generator busvoltage magnitude will not hold constant any more. Rather, itbecomes an interval [0.977, 0.984] as shown in Fig. 9, whereasthe true interval of voltage magnitude is [0.9775, 0.9821]. Fig. 8illustrates that estimated reactive power interval for generatorbus with bus type conversion is also very close to the true in-terval. Furthermore, Figs. 10 and 11 show the impact of dif-ferent uncertainty level on voltage magnitude without/with re-active power limit respectively. It is observed that more gener-ators may reach to their limits with the increase of power injec-tion uncertainty, and the voltage magnitude intervals of all busesbecome wider, especially for bus 3 to bus 15 which are close tothe generator buses that are converted to PQ buses.

D. Accuracy and Computation Time Analysis

The proposed method has been tested on different test sys-tems including the IEEE 9-bus, IEEE 30-bus, IEEE 57-bus, andIEEE 118-bus systems, to analyze the accuracy and computa-tion time. In addition, parallel computing can be adopted tosolve for multiple or all buses simultaneously. Table IV showsthe maximum and average errors under different level of un-certainties, as well as computation time for interval solutions ofall buses. Considering 5000-sample MC method may not givesufficiently accurate results t, 100 000-sample MC method wasused instead and its results are treated as true solutions. The re-sults of 5000-sample MC method is also provided to demon-strate the fact that the accuracy of MC method is dependent onthe sample size.



TABLE IIIINTERVAL SOLUTION OF GENERATOR REACTIVE POWER AND VOLTAGE MAGNITUDE WITH REACTIVE POWER LIMIT ENFORCED

Fig. 8. Generator 9’s reactive power from proposed method and Monte Carlosimulation method.

Fig. 9. Generator 9’s voltage magnitude from proposed method and MonteCarlo simulation method.

Table IV shows that the maximum and average errors becomelarger with increased uncertainty level. The computation timeincreases when bus type conversion occurs. For instance, forIEEE 9-bus system, there is no reactive power limit violation,and therefore computation time is nearly the same for differentuncertainty levels. On the other hand, for IEEE 118-bus system,there is no generator reactive power limit violation when un-certainty level is 10%. When the uncertainty level is increasedto 20%, there is 1 violation in generator reactive power limit

Fig. 10. Impact of variation level on voltage magnitude without reactive powerlimit.

Fig. 11. Impact of variation level on voltage magnitude with reactive powerlimit.

which causes almost doubled computation time, and at 30% un-certainty there are 2 violations and the computation time almosttripled.Table IV shows the proposed method delivers reasonably ac-

curate results when compared to MC method, which often takesmuch longer time to run with larger sample size or loses accu-racy with smaller sample size.Lastly, it should be pointed out the interval solution fromMC

method by its nature is a subset of the true interval, while the



TABLE IVACCURACY AND COMPUTATION TIME OF VOLTAGE MAGNITUDES WITH REACTIVE POWER LIMIT ENFORCED

Fig. 12. Illustration of interval solutions from different methods.

solutions from the proposed method and AA method are super-sets of the true interval, as illustrated in Fig. 12. In other words,MCmethod gives an underestimated interval length and the pro-posedmethod produces an overestimated interval length. There-fore, the proposed method is suitable for application where thefull range of the true interval needs to be captured.

V. CONCLUSIONS

This paper aims to study the interval power flow problemwith uncertain power injection. In the Cartesian coordinates, theenergy equation function contains only quadratic polynomials,and therefore the interval power flow problem can be formu-lated as standard QCQP optimization model. A linear relaxationmethod is utilized and applied for each bilinear term to obtainan interval solution that is guaranteed to include the true intervalsolution. To minimize the gap between the relaxed interval so-lution and the true one, an optimality-based bounds tightening(OBBT) method is employed which improves the outer hullof the feasible region of the original QCQP problem. Further-more, utilizing the sparse nature of the power flow problem, thecomplexity of relaxation model is reduced through utilizing thesparse matrix technique. More importantly, as the interval solu-tion of each bus and branch can be calculated independently,parallel computing technology can be utilized to further ex-pedite the solution process. The results of several IEEE testsystems have demonstrated the effectiveness of the proposedmethod, which outperforms the methods reported in literatures.

With the proposed method, interval solutions of all bus voltageand all branch flow under various uncertain conditions can be ef-ficiently solved, which provides great insights in operating andplanning for the uncertainties in power systems.

APPENDIXSHOW IS A BOUNDED SEQUENCE

Without loss of generality, we use the “min” modelas an example to prove the boundedness of sequence

. The proof can be easily adaptedfor the “max” model.First, the linear programming model is the relaxed model of

original QCQP, and therefore we obtain

(A1)

According to the lemma from [19], we have

(A2)

where is the matrix with the entries .It then yields

(A3)

We know is bounded because , and therefore wehave

(A4)From (A3) and (A4), we derive

(A5)



From (A3) and (A4), we also derive

(A6)

By (A5) and (A6), we have shownis therefore a bounded sequence.

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Tao Ding (S’13) received the B.Eng. and M.S. de-grees in electric power engineering from SoutheastUniversity, China, in 2009 and 2012, respectively.Presently, he is pursuing the Ph.D. degree at TsinghuaUniversity, China.He is now a visiting scholar with the Department

of Electrical Engineering and Computer Science,The University of Tennessee, Knoxville, TN, USA.His current research interests include power systemeconomics and optimization methods, power systemvoltage stability, and control.

Rui Bo (S’02–M’09–SM’10) received the B.S. andM.S. degrees in electric power engineering fromSoutheast University, China, in 2000 and 2003, re-spectively, and the Ph.D. degree from the Universityof Tennessee, Knoxville, TN, USA, in 2009.He worked at ZTE Corporation and Shenzhen

Cermate Inc. from 2003 to 2005. He has been em-ployed at Midcontinent Independent TransmissionSystem Operator (MISO) since 2009. His interestsinclude power system operation and planning, powersystem economics, and computational methods.

Fangxing Li (M’01–SM’05) received the B.S.E.E.and M.S.E.E. degrees from Southeast University,China, in 1994 and 1997, respectively, and the Ph.D.degree from Virginia Tech, Blacksburg, VA, USA,in 2001.He is presently an Associate Professor at The Uni-

versity of Tennessee, Knoxville, TN, USA (UTK).Prior to joining UTK, he had worked at ABB Con-sulting as a senior engineer and then a principal en-gineer from 2001 to 2005. His current interests in-clude renewable energy integration, power markets,

distributed energy resources, and smart grid.Dr. Li is a registered Professional Engineer in North Carolina, an Editor of

the IEEE TRANSACTIONS ON SUSTAINABLE ENERGY, and a Fellow of IET.

Qinglai Guo (M’09) was born in Jilin City, China,on March 6, 1979. He received the B.S. degree fromthe Department of Electrical Engineering, TsinghuaUniversity, Beijing, China, in 2000 and the Ph.D. de-gree from Tsinghua University in 2005.He is now an Associate Professor at Tsinghua

University. He is a member of CIGRE C2.13 TaskForce on Voltage/Var support in System Operations.His special fields of interest include the EMSadvanced applications, especially the automaticvoltage control.



Hongbin Sun (SM’12) received the double B.S.degrees and Ph.D. degree from Tsinghua University,Beijing, China, in 1992 and 1997, respectively.He is now Changjiang Chair Professor of Educa-

tion Ministry of China, full professor of electricalengineering in Tsinghua University, and assistantdirector of State Key Laboratory of Power Systemsin China. From September 2007 to September 2008,he was a visiting professor with the School ofElectrical Engineering and Computer Science at theWashington State University, Pullman, WA, USA.

In the past 15 years, he has developed a commercial system-wide automaticvoltage control systems which has been applied to over 30 large-scale powergrids in China. He published more than 300 academic papers. He held over50 patents in China. His interests include smart grid, renewable energy andelectrical vehicle integration, power system operation, and control.Prof. Sun is an IET Fellow.

Wei Gu (M’06) received the B.Eng. degree and thePh.D. degree in electrical engineering from SoutheastUniversity, China, in 2001 and 2006, respectively.He is now an Associate Professor in the School of

Electrical Engineering, Southeast University. His re-search interests include power system stability andcontrol, smart grid, renewable energy technology andpower quality.

Gan Zhou (M’09) received the B.Sc. (Eng.) degree in automation from theNanjing University of Science and Technology, Nanjing, China, in 2000, andthe M.Sc. (Eng.) and Ph.D. degrees in electrical engineering from SoutheastUniversity, Nanjing, China, in 2003 and 2009, respectively.He is now an Associate Professor in the School of Electrical Engineering,

Southeast University. His current research interests include power system sta-bility and control, and high performance parallel computing technique for powergrid.

Documents

IEEE TRANSACTIONS ON POWER SYSTEMS 1 Interval Power Flow