An Advanced STATCOM Model for Optimal

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

IEEE TRANSACTIONS ON POWER SYSTEMS 1

An Advanced STATCOM Model for OptimalPower Flows Using Newtons MethodBehzad Kazemtabrizi, Member, IEEE, and Enrique Acha, Senior Member, IEEE

AbstractThis paper presents the optimal power flow (OPF)formulation of a recent power flow STATCOMmodel [1]. The newmodel puts forward an alternative, insightful interpretation of thefundamental frequency operation of the PWM-controlled voltagesource converter (VSC), in an optimal fashion. The new modelmakes provisions for the explicit representation of the convertersinternal ohmic and switching losses which in the context of an OPFformulation, yields an optimum operating point at which thesepower losses are at a minimum. The STATCOM model possessesunparalleled control capabilities in the operational parameters ofboth the AC and DC sides of the converter. Such control mod-eling flexibility is at its best when expressed in the context of anOPF solution using Newtons method. The STATCOM equationsare incorporated into the OPF formulation using Lagrangianfunctions in quite a natural manner for efficient optimal solutionsusing a single frame-of-reference. The inequality constraint set ofvariables is handled equally well using the multipliers method.The prowess of the new model is demonstrated using two samplesystems.

Index TermsFACTS, Newtons method, optimal power flows,STATCOM, voltage source converter.

I. INTRODUCTION

L IKE the static VAR compensator (SVC), the primaryfunction of the STATCOM is to provide flexible reactivepower support at key points of the transmission system but ata faster speed of response and with an enhanced performance[2]. The STATCOM may take the form of one of the manypossible converter topologies available today, made up offully controllable power electronic valves and driven by PWM(or equivalent) control [3]. The most popular switched-modeconverter topologies fulfilling the requirements of providingfast voltage support are the two-level and the three-levelPWM-driven VSCs, together with the newer modular multi-level converter (MMC) VSCs. They are normally connectedto a point of the power grid using a step-up transformer withtap-changing facilities [4][7]. The fundamental frequencyoperational behavior of the VSC, as seen from its AC side,resembles that of a controllable voltage source. Such a char-acteristic has been exploited to good effect in power systemstudies to represent the STATCOM as a controllable voltage

Manuscript received January 29, 2012; revised February 13, 2013, May 26,2013, and September 16, 2013; accepted October 23, 2013. Paper no. TPWRS-00086-2012.B. Kazemtabrizi is with the School of Engineering and Computing Sciences,

Durham University, Durham, U.K. (e-mail: [email protected]).E. Acha is with the Department of Electrical Engineering, Tampere Univer-

sity of Technology (TUT), Tampere, Finland (e-mail: [email protected]).Digital Object Identifier 10.1109/TPWRS.2013.2287914

source behind coupling impedance [8], [9]. This is not dissim-ilar to the way in which synchronous condensers are representedin power flow studies. Such a simple concept represents wellthe fact that at the fundamental frequency, the STATCOMconverters output voltage may be adjusted against the ACsystems voltage in the converter to achieve very tight controltargets, a capability afforded by the switched-mode convertertechnology [1][9]. Nevertheless, for all its attractiveness thisconcept fails to explain the operation of the STATCOM fromits DC side. Some of the most obvious shortcomings of theSTATCOM model based on the equivalent voltage sourceconcept are: 1) there is no easy way to ascertain whether ornot the converters operation is within the linear region ofoperation [10]; 2) switching losses tend to be neglected; 3) theinternal ohmic losses of the converter along with the effectsof the converters magnetics are normally lumped togetherwith those of the interfacing transformer which, more oftenthan not, is a tap changer. This has provided the motivation todevelop a more realistic STATCOM model for fundamentalfrequency operation [1]; one which overcomes the limitationsof the equivalent voltage source representation and is suitablefor assessing the impact of both conventional multi-level andmodular multi-level converters (MMC) [11], [12], on largepower networks and in an optimal manner.This paper may be considered a companion paper of [1]

where the conventional power flow solution of the STATCOMmodel has been put forward. In the OPF problemwhich isthe subject matter of this papera chosen system objectivefunction (or a group of functions) is solved towards its op-timum operating point subject to systems realistic operatingboundaries.In the optimal power flow (OPF) formulation presented in

this paper, the system objective function is chosen to be thecost of generators active power dispatch [13]. It should benoted that the set of results obtained from an OPF solution maynot necessarily agree with those obtained from a conventionalpower flow solution even when applied to the same system.In an OPF solution, the solution space is shaped by the actionof different controllers in the system that set the boundarieson control state variables and functions (i.e., nodal active andreactive power flows) [2]. Adhering to the necessary optimalitycriteria will eventually result in convergence towards a dif-ferent operating point (optimum) than the one obtained by theconventional power flow calculation. The OPF formulation re-quires creating a Lagrangian function with appropriate penaltyfunctions to keep the system operating conditions within theiracceptable boundaries whilst adhering to the necessary opti-mality criteria. The reason is that the key part of the optimality

0885-8950 2013 IEEE


2 IEEE TRANSACTIONS ON POWER SYSTEMS

Fig. 1. (a) STATCOM schematic representation. (b) Voltage source converterequivalent circuit. (c) On-load tap-changing transformer equivalent circuit.

criteria found in OPF formulations is not incorporated in theconventional power flow formulation. For instance, and asexemplified by the OPF simulations presented in this paper,the converters internal switching losses are reduced whencompared to those obtained with a conventional STATCOMpower flow algorithm.Optimal solutions of the new STATCOM model yield con-

siderable reductions in power system losses and in the con-verters internal power losses, when compared to the solutionsfurnished by the STATCOM model solved using conventionalpower flows [1]. Furthermore, optimal solutions with the newSTATCOMmodel will also yield improved solutions comparedto the optimal solutions provided by the voltage source repre-sentation of the STATCOM, and with less computational com-plexity.

II. STATCOM NEW MODEL

The equivalent electric circuit for the STATCOM model isshown in Fig. 1.The STATCOM consists of two main componentsa voltage

source converter (VSC) and a tap-changing coupling trans-former (LTC), as illustrated in Fig. 1(a). The VSC is modeled asan ideal complex tap-changing transformer, shown in Fig. 1(b).The reason for using a complex tap changer to model the VSC

operation stems from the following fundamental relationshipapplicable to the PWM controlled operation of the VSC:

(1)

where tap magnitude of the ideal complex tap-changingtransformer corresponds to the amplitude modulation coeffi-cient of an actual two-level, three-phase VSC, defined as

, in which the PWM-controlled VSC operates in thelinear range with [5]. The phase angle is thephase angle of the complex voltage relative to the systemphase reference.It should be noted that such aggregated relationships are also

applicable to represent the fundamental frequency operation ofthree-level, three-phase VSCs driven by PWM control since inthis application the interest is in the relationship betweenand through and . This would be regardless of thenumber of switches and converter levels.On the other hand, modular multilevel converters (MMC)

have a different construction design and operating principlesthan PWM-driven converters. They comprise several small DCchoppers with bi-directional switches, making up sub-modulesof each leg of the three-phase converter. Assuming that theoutput DC voltage of each sub-module is controlled to main-tain an average value of then the constant input DC voltagein each leg of a three-phase MMC-VSC with N sub-moduleswould be [11], [12]. It follows that the numberof active sub-modules in the multi-level converter dictates thevalue of the voltage magnitude on the AC side of the converter.It turns out that (1) also represents very well the aggregated af-fects of this operation if one thinks of as a discrete tap asopposed to the continuous tap associated with the PWM-drivenVSC converters. For numerical efficiency within the power flowor the OPF solution a continuous tap is assumed and at the end ofthe convergent solution, the nearest physical tap is selected andone further iteration is carried out to fine tune the overall powerflow solution. This would not be different to schemes adoptedelsewhere for the tap selection of LTC transformers where dis-crete taps are considered as opposed to continuous ones [14].As shown in Fig. 1(b), the complex tap-changing transformer

represents the internal operation of the converter under PWMcontrol. The converters input DC voltage, is provided bythe capacitor bank which is connected in parallel witha resistor (conductance) of value representing the con-verters internal switching losses at a constant DC input voltage.The reactive power control feature of the VSC is, on the otherhand, represented in the valve set modeled by a notional vari-able shunt susceptance in the AC side of the ideal, complex, tap-changing transformer. The VSC model is completed by addinga series impedance to the AC side of the complex-tap trans-former in which the series resistor is associated with theohmic losses which are proportional to the AC terminal currentsquared and the series inductance represents the convertersinterface magnetics.The converters switching losses are modeled by the fol-

lowing quadratic expression [1]:

(2)


KAZEMTABRIZI AND ACHA: AN ADVANCED STATCOM MODEL FOR OPTIMAL POWER FLOWS USING NEWTONS METHOD 3

where is the converter switching losses under constant DCvoltage and nominal load current conditions. To incorporate theeffects of actual converter operation this term is corrected by thesquared ration of actual-to-nominal currentwith the quadraticexponent chosen to reflect the power behavior of the switchingresistance (conductance).The reactive power property of the converter is modeled

using a variable shunt branch susceptance to account for thecalculated reactive power (either generation or absorption) inthe converter depending on its control requirements, whichmay be set to either direct nodal voltage regulation or reactivepower control [1].The VSCs operation at fundamental frequency is defined by

the following nodal admittance matrix which is developed inmore detail in Appendix A. See equation (3) at the bottom ofthe page.Furthermore, the coupling transformer is taken to be a con-

ventional tap-changing transformer with discrete tap steps, asshown in Fig. 1(c). The nodal matrix representation of the clas-sical tap-changing transformer, represented by the equivalentcircuit of Fig. 1(c), is [2]

(4)

Notice that in (4) the tap is real as opposed to complex andthat it is taken to be on the transformers primary side. It shouldbe noted that in the course of the OPF solution process the OLTCtap is treated as a continuous variable but in practice this is adiscrete variable. Therefore, at the end of each internal iterativeloop, is rounded off to its nearest integer.

A. STATCOM Nodal Power Equations

The nodal power equations of the full STATCOM modelwithin the OPF is calculated by combining the nodal powerequations of the VSC and the OLTC modules.1) VSC Module: The VSC nodal power injections are de-

rived from the product of its nodal voltages and current in-jections, in complex conjugate forms. See equation (5) at thebottom of the page.

Carrying out straightforward complex algebra, the nodal ac-tive and reactive power equations for the VSC model are de-rived:

(6)

2) OLTC Module: Similarly, the nodal power injections ofthe OLTC module are derived from the nodal voltage and cur-rent relationships at both ends of the OLTC:

(7)

This expression yields the following explicit nodal power in-jections for the OLTC model:

(8)

where .

(3)

(5)



Suitable combination of the two set of equations, (6) and (8),yields the required nodal power injections at the three nodes ofinterest, namely, , and 0:

(9)

(10)

(11)

where and in the summation symbol in (9)(10) are thecontributions of all branches connected to node other than theOLTC transformer. These calculated nodal powers are requiredby the optimal power flow formulation.

B. Practical Implications1) STATCOM Design Requirements: As illustrated in

Fig. 1(b), the VSC is assumed to be connected between asending bus, , and a receiving bus, 0, with the former takento be the VSCs AC bus and the latter taken to be the VSCsDC bus. The voltage input at the DC bus is provided by theDC capacitor bank, of value , and kept constant at avalue . The voltage magnitude is regulated withinsystem-dependent maximum and minimum values afforded bythe following basic relationships:

(12)

The PWM-controlled VSC is taken to operate within thelinear region [6]. Hence, the PWM amplitude modulation coef-ficient is within the bounds: and for a two-level,three-phase VSC, .2) Simplifying Assumptions: It should be noted that the

ideal phase shifter decouples, angle-wise, the circuits to the leftand to the right of the ideal transformer, i.e., the phase anglevalue at node 0 is independent of circuit parameters or networkcomplexity to the left of the ideal phase shifting transformer.From the numerical perspective, the phase angle voltage at bus0 keeps its value given at the point of initialization, which inthis STATCOM application will be taken to be zerowhenlooked at it from the vantage of rectangular coordinates, itsimaginary part does not exist [1]. In the course of the OPFsolution process the variations of the DC bus angle are kept tozero by penalizing this angle throughout the solution process,hence, . Alternatively, the entries corresponding tothis state variable in the OPF formulation may be removedaltogether, resulting in a more compact formulation that wouldyield identical results.

III. STATCOM NEW MODEL FORMULATION IN OPTIMALPOWER FLOW (OPF) USING NEWTONS METHOD

A. Augmented Lagrangian FunctionsThe constrained OPF problem is formulated using the La-

grangian function given in (13) for the STATCOMmodel by ap-plying explicit multipliers to system equality constraints given

in (9)(11) and penalizing the resultant Lagrangian function forany state variable violations [2]:

(13)

where corresponds to the set of functional equalityconstraints for the system including the STATCOM device.

corresponds to the summation of the values of theproblems objective functions which are taken to be the gener-ators quadratic cost functions as given in [13]. And,is an explicit quadratic penalty function for penalizing theLagrangian function for any state variable violations.The explicit state variables pertaining to the STATCOM new

model comprise the variables for both the converter and theOLTC modules. This is shown in (14):

(14)

It should be noted that apart from these explicit expressions,solving the OPF requires defining a Lagrangian function for thewhole systemwhich would include the nodal voltages and phaseangles of all buses (except for the Slack bus for which onlythe nodal voltage magnitude is required), transformer tap ratios,as well as any other variables associated with a given powercontrolling equipment such as the STATCOM. As mentioned inSection I this is done by combining the effects of all the systemLagrangian functions.The OPF problem to be solved in this paper is on the cost of

generators active power dispatch, each possessing a quadraticcost function with an expression similar to the one presented in[11].

in (13) then corresponds to the generators scheduled ac-tive power dispatch which is subject to the systems operatingconditions [i.e., ]. In such circumstances the OPF problemis concerned with minimizing the overall cost of active powerdispatch subject to realistic operational conditions and controlsettings. The controls are set by the STATCOM explicit statevariables in the system. The problem constraints essentially rep-resent the network actual operating conditions. Voltage mag-nitudes and phase angles in buses, generators active powers,nodal power injections and mismatches in each bus are amongthe most important operating constraints in OPF-related studies.Applying Newtons method [15][18] to the Lagrangian func-tion (13) and assuming that no penalty function terms exist atthe start of the OPF iterative processthe system is assumed towork under normal operating conditions and all the variables areinitialized within their respective limitsthe linearized systemof equations for the OPF is defined as

(15)

where vector is the vector of primal-dualvariables (dual variables are the Lagrange multipliers for bothequality and inequality constraints, and , respectively)[15][18].The matrix of coefficients, , is a combination ofHes-

sian and Jacobian terms obtained from second order deriva-tives of the Lagrangian function in (13) with respect to the en-tries of vector . This results in a formulation which yields aquadratic rate of convergence. Commensurate with the power



flow Newton-Raphson application, the Jacobian sub-matrix in(15) keeps the same level of sparsity as the nodal admittancematrix and so does its Hessian sub-matrix. This contrasts withan earlier formulation based solely on the use of an alternativeHessian matrix [20], which contains little sparsity. The gradientvector, , which comprises the first order derivatives of theLagrangian function with respect to the entries of vector oughtto maintain a decreasing pace throughout the course of the itera-tive solution [2], [18][20], to ensure a reliable solution towardsthe optimum.The linearized system of equations may be written down

more explicitly as

- - - - - - - - - - - - - - (16)

It should be noted that the gradient term, , actually corre-sponds to the mismatch of nodal power calculations. It shouldalso be noted that the second order derivatives of the Lagrangianfunction with respect to the Lagrange multipliers are zero, i.e.,

.

B. STATCOM New Model Equality Constraints

For the STATCOMmodel in Fig. 1, the set of functional con-straints comprise the nodal active and reactive power mismatchequations at nodes: , and 0

(17)(18)(19)(20)(21)(22)

where and are the calculated injected powers atnode ; and are active and reactive powers generated atnode ; and are active and reactive powers consumedat node ; and are the calculated injected powersat node ; , , and will be zero for anypractical purpose; and are the calculated injectedpowers at node 0; , , and are the active and re-active power generated and consumed at node 0, respectively.The above expressions have to be satisfied for an optimum so-lution to be acceptable, otherwise it is said that the solution isinfeasible.

C. STATCOM New Model Control in OPF

Two control modes are available in the STATCOM newmodel introduced in this paper, namely, active power flowcontrol (if applicable) and nodal voltage regulationcontrolconstraints given in (23) and (24) are used to this end.1) Nodal Voltage RegulationConstraint on State Variables:

The voltage regulation constraint is a variable equality con-straint which is added to the OPF formulation by means of aquadratic penalty function of the form (23)[12]:

(23)

where is either or 0 and is the target nodal voltagemagnitude which must remain within operational limits andis a non-zero integer termed the penalty factor.Equation (23) is used by default to enforce the STATCOMs

nodal voltage regulation at the AC bus using the OLTC trans-former in Fig. 1(c). It should be noted that within the OPFformulation, the nodal voltages at both nodes (AC systemvoltage) and (VSC AC output voltage) may be controlled bythe combined action of the OLTC and VSC. However, the VSCAC system voltage is rather set free to vary within its permittedboundaries; as a result of this, nodal voltage regulation is notimposed on this node. The VSC DC input voltage is providedby the DC capacitor bank which is initialized as a PV-type busin the OPF solution process. The VSC explicit nodal voltagecontrol is therefore on the DC bus not on the AC bus. TheAC side voltage is regulated by the action of the OLTC trans-former, whereas the DC side voltage is determined by the DCcapacitors design requirementsas discussed in Section II-B.Hence, the DC voltage is set to a pre-determined level (corre-sponding to the VSC input DC voltage) throughout the solutionprocess.The value of the penalty factor, , dictates the hardness of

the voltage regulation boundaries. However, choosing the initialvalue of the penalty factor is a highly empirical exercise, whichis rooted in experience and trial and error [2]. Choosing too largea value may lead to inaccurate and unfeasible results whereassmall values may lead to a poor rate of convergence and possiblestagnation. For the test cases presented in this paper a value of

has been used for the penalty factor .2) Active Power Flow RegulationConstraint on Functions:

Active power flow through the VSC converter is controlled byvarying the phase shift that exists in the converters ideal trans-former model (i.e., the angle ).For explicit active power flow control inside the converter, an

additional functional equality constraint is introduced in form of(24):

(24)

where normally is the calculated nodal activepower at the DC bus, which is set to zero or , as detailedabove. For the purposes of modeling the VSCs DC bus, thisis a PV-type bus with active power set to either zero or to apre-specified value, say, .Notice that the latter option is only possible if any form of

energy storage is available in the STATCOMs DC bus. How-ever, reactive power in the DC node is always set to zero. Incontrast to the model of the VSC based on the concept of acontrollable voltage source, in the new STATCOM model theOPF algorithm modifies the phase angle in such a way thatthe amount of the active power flowing through the convertercorresponds to the target active power flow upon convergence.This is a distinguished feature of the new model which is com-pletely absent from the controllable voltage source model. The



explicit Jacobian and Hessian terms associated with the activepower flow control constraint in the STATCOMmodel are givenin Appendix B.

D. STATCOMs Explicit Lagrangian Function

The STATCOMs Lagrangian function is given by (25):

(25)

The Lagrangian in (25) contains no quadratic penalty func-tion for inequality constraints at the start of the OPF process, i.e.,these are not activated at the outset. A full iterative solution of(15) requires the convergence of an inner loop by solving (16) ina true Newton-Raphson fashion. Once the process converges toa specified tolerance in the inner loop then all the variable limitsare checked against their respective boundaries and the activebinding set is identified [17], [18]. Variables that have violatedtheir limits make up the active set; they are forced to their ceil-ings and incorporated in (15) using the penalty functionin which the term is defined as the Lagrangian multiplier forits corresponding active set. Inclusion of the active set uponconvergence of the first inner loop completes the first iterationof the outer loopreferred to as a one global iteration. Hence,a second outer loop is initiated, which now incorporates bothequality and inequality constraints [15], [18], and [21], [22]. Inprinciple, convergence for the local iterations is achieved in truequadratic fashiona hallmark of the Newton-Raphson method.However, the active set is updated outside the Newton-Raphsonsolution, a procedure that impairs the overall convergence ofthe Newton-Raphson OPF solution, which is termed NewtonsOPF method. Furthermore, experience has shown that the innerloops convergence may be better assured by employing a de-celerating factor at the point of updating the statevariables and Lagrange multipliers at the end of each local iter-ation. This is particularly the case during the local iterations ofthe first two global iterations [2]. The use of such decelerationfactors impairs further the quadratic convergence characteristicsof the Newton-Raphson method, i.e., the number of local itera-tions will increase. However, experience has shown that this isa very powerful resource owing to the highly non-linear natureof the problem at hand.The active power flow constrain at the converter, (23), is nor-

mally enforced by default at the start of the solution processitsLagrangian is included in (25). It is either set to zero DC poweror to a pre-specified DC positive/negative power injection if theSTATCOM is provided with any form of energy storage. Ofcourse, it is always possible in Newtons OPF solution not towish to enforce this constrain, something that is done by en-forcing its associated Lagrange multiplier to zero using a suit-able quadratic penalty function of the form given in (26):

(26)

where the term is the Lagrangian multiplier pertaining to theconverters active power flow constraint. It is noted that all themultipliers in (25) have been initialized at zero values.

E. STATCOM Linearized System of Equations

Application of Newtons method to the STATCOM La-grangian function (25), taking due account of the state variablevector (15) and the Lagrange multipliers for the active equalityconstraints set, is given in (27):

(27)

F. Inequality Constraints Set

The inequality constraint set includes the following limits: theconverters PWM amplitude modulation coefficient, , theOLTC transformers tap ratio, , and the nodal voltage magni-tudes. The control variables are the STATCOMs nodal voltagesat both nodes (AC and DC), the ideal transformers complex tap,

, which is allowed to vary between 0 and and theOLTCs tap ratio, which is allowed to vary between 0.6 and1.2. No limits are imposed in the phase angles of the nodal volt-ages or in the VSCs complex tap angle.

IV. TEST CASES

To assess the accuracy, flexibility and robustness of the pro-posed STATCOM model, two test cases are presented in thissection. The first case is a rather contrived system where theSTATCOM is fed from a synchronous generator through a trans-mission line, as shown in Fig. 2. The second case comprises amodified version of the IEEE 30-node system [23] in which aSTATCOM is assumed connected at node 24 tomaintain voltagemagnitude at 1 p.u. at that node. An existing OPF program usingNewtons method written in MATLAB [2] has been extended



Fig. 2. Fictitious three-node radial systemSTATCOM new model represen-tation.

TABLE IOPTIMAL POWER FLOW SOLUTION FOR THE FICTITIOUS

RADIAL THREE-NODE SYSTEMNEW MODEL

to implement the new STATCOM model and to carry out quitecomprehensive tests, two of which are presented below.

A. Radial SystemNew ModelThe three-node system in Fig. 2 comprises one generator,

one transmission line, one load and one STATCOM which isused to regulate voltage magnitude at its AC node at 1.02 p.u.whereas its DC bus voltage is kept at p.u. The followingparameters are used in the contrived test system1) transmis-sion line: and ; 2) LTCtransformer: and ; 3) VSC:

, , and; 4) Load: and

The generator node is taken to be the slack bus. The objectivefunction to be minimized is shown as follows:

(28)

The optimal power flow solution for the radial system is givenin Table I. The voltage at the slack bus is also penalized to keepit at 1 p.u. The VSC DC voltage is fixed at 1.4142 p.u. and thevoltage at bus 2 is controlled using the LTC transformer. Thefinal value of the variable tap changer at the optimum is 1.02which is rounded off to 1.0 for a discrete mechanical tap. Thenodes STATCOM-AC and STATCOM-DC in Table I, correspondto the AC and DC nodes of the STATCOM, respectively; Gencorresponds to the generator bus; andVSC-AC andVSC-DC cor-respond to the AC and DC nodes of the voltage source converteras given by the new model shown in Fig. 1. The final value ofthe objective function in (28) is calculated according to the gen-erators optimum active power dispatch, with values given inTable II. The values of the penalty factors for all the quadraticpenalty functions are initiated at .The OPF for the radial system in Fig. 2 converges in three

global iterations to a tolerance of . Table III gives the

TABLE IIOBJECTIVE FUNCTION VALUE AT THE OPTIMUM

TABLE IIIGLOBAL ITERATIONS AND LOCAL ITERATIONS

number of local iterations incurred at each global iteration, witha deceleration factor . The converters AC terminalvoltage is free to vary within its allowable boundaries andarrives at the final value of 1.05 p.u. with the angle of .The voltage angle at the DC bus is kept constant at the pointof initialization using a quadratic penalty function to nullifyits corresponding increments throughout the OPF process.The STATCOM consumes 0.0260 p.u of active power and theconverter switching losses are .The converter valve set generates 0.6120 p.u. of reac-

tive power to maintain the voltage at bus 2 at 1.02 p.u.Furthermore, minimum transmission line losses stand at

and The generatorsactive and reactive powers limits are set at:and , respectively.

B. Radial SystemControllable Voltage Source ModelFor the sake of completeness and in order to contrast the re-

sults provided by the new model with those provided by theSTATCOMmodel based on the controllable voltage source con-cept [2], [3], the contrived radial system of Fig. 2 is solvedagain but this time using the latter model. The voltage source isconnected behind a coupling impedance (representing the VSCinternal magnetic and ohmic losses). A shunt conductance ofvalue 1% is connected between the coupling impedance and thevoltage source in order to represent the ohmic losses inside theconverter.The three-node systemwith the STATCOMmodeled as a con-

trollable voltage source is shown in Fig. 3.The network parameters for this system remain very much the

same as in the test case of Fig. 2. The OPF solution convergesin 3 global iterations. These results are compared to those pro-duced by the new STATCOM model in Tables IV and V. Asexpected both models yield similar results. The STATCOM con-trollable voltage source model generates 0.6110 p.u. of reactivepower to maintain the voltage at node 2 at 1.02 p.u.The converter voltage stands at 1.0510 p.u. The con-

verter voltages behind the converter impedance for both modelsare compared in Tables IV and V.The powers calculated by both models are presented in

Table VI. The switching losses in this test case are modeled by



Fig. 3. Fictitious three-node radial systemSTATCOM controllable voltagesource model representation.

TABLE IVCONVERTER VOLTAGE MAGNITUDES FOR DIFFERENT STATCOM MODELS

TABLE VCALCULATED VOLTAGE PHASE ANGLES FOR DIFFERENT STATCOM MODELS

TABLE VIOPTIMAL POWER FLOW SOLUTION FOR DIFFERENT STATCOM MODELS

connecting a shunt conductance of 1% between the couplingimpedance and the variable voltage source. In fact, this shuntresistive branch should be connected at the DC bus of the VSCas opposed to its AC side but is precluded because the voltagesource converter model does not have such a bus and the resultsconcerning switching losses will be inaccurate and optimistic.By comparing the results given by the OPF solution of the

three-node radial system with the STATCOM modeled usingthe new model and a controllable voltage source model respec-tively, the following limitations are clearly observed: 1) lackof explicit DC bus representation which means that the con-verter voltage is represented by only one state variable per-taining to the controllable voltage source. Therefore there is nodirect means of controlling the AC output voltage of the con-verter by varying the DC input voltage. 2) In the controllable

Fig. 4. STATCOM supplying reactive power at node 24 of the modified IEEE30-node system to regulate voltage magnitude at 1.02 p.u.

voltage source model there is no way of limiting the opera-tion of the PWM modulation coefficient within the linear re-gion, therefore the results obtained from a controllable voltagesource model do not provide sufficient information to distin-guish the regions of operation of the converter. This may bedone by only introducing a new explicit state variable in theOPF formulation, further complicating the overall formulationof the problem, whereas, with the new model, this is alreadyincluded in form of the complex tap ratio of the transformermodeling the PWM-control of the VSC. 3) Lack of the capa-bility of appropriate modeling of energy storage in the DC sideof the converter due to the inability for explicit representationof the DC-side bus. This may be remedied by adding an addi-tional equality constraint in form of an active power flow, how-ever with the new model; this is included inherently within theconverter model. All is needed to add the energy storage is tochange the value of the converter active power flow control toa non-zero negative value. 4) Inaccurate and optimistic calcula-tion of the converters internal switching losses.Carrying out the OPF solution with this a priori detectedmod-

eling inaccuracy will yield a different optimum operating point(i.e., 157.07 $/h), which is optimistic.

C. Modified IEEE 30-Node SystemIn order to test the performance of the new STATCOM

model in a larger network, the IEEE 30-node system [23] isselected. The fixed bank of capacitors at node 24 is replacedwith a STATCOM, which is used to regulate voltage magnitudeat that node at 1.02 p.u. The modified portion of the 30-nodesystem is shown in Fig. 4. The nodal voltage magnitudes areallowed to vary between 0.9 and 1.1 p.u. at all 24 load busesand between 0.9 and 1.05 p.u. at all six generator buses. Node1 is taken to be the slack bus.All the STATCOM parameters and limits are taken to be the

same as in Test Case A except for the transformer leakage re-actance, which takes a value of 0.3690 p.u. (the tap changingtransformer is not shown in Fig. 4). The penalty factor foreach quadratic penalty function is initiated at .The generators fuel cost functions given in Table VII are

used for the six generators available in the 30-node test system.The Newtons OPF arrives to the solution in eight global

iterations. The slow convergence rate is the result of enforcinginequality constraints in voltage magnitudes for violatednodes. A summary of the most relevant results are shown



TABLE VIIFUEL COST FUNCTIONS OF THE SIX GENERATORS

IN THE IEEE 30-NODE TEST SYSTEM

TABLE VIIIACTIVE AND REACTIVE POWER INJECTIONSAT THE GENERATOR AND STATCOM NODES

TABLE IXCONVERTER OPERATING PARAMETERS AT THE OPTIMUM

in Tables VIIIX. The STATCOM consumes 0.0245 p.u. ofactive power of which 0.81% is for VSC internal switchinglosses, whilst 1.64% accounts for OLTC ohmic losses. TheSTATCOM generates 0.3432 p.u. of reactive power to maintainthe voltage magnitude at node 24 to 1.02 p.u. The OLTC finaltap is rounded off to 0.7. Notice that the powers shown are theoutput powers at the OLTC transformer terminals as opposed tothe VSC terminals. Table X shows the final values of generatorcost functions at the optimum after eight global iterations to atolerance of . The increased number of iterations was dueto a voltage limit violation in node 24 which has been fixedto its lower boundary by the action of its inequality constraintmultiplier. The total value of the objective function is the sumof all six generators costs obtained with the generators outputsat the optimum, which stands at 972.8759 $/h.

V. CONVERTER SWITCHING LOSSESOne of the advantages of the proposed new model is that it

gives provisions for explicit representation of the converters in-ternal switching losses under converters operating conditions.This is done by applying (2) to the converters otherwise con-stant switching losses under constant input DC voltage and ratedcurrent (i.e., 1 p.u.).

TABLE XOBJECTIVE FUNCTIONS VALUES AT THE OPTIMUM

TABLE XISWITCHING LOSSES AS GIVEN BY OPF AND CPFSOLUTIONSFICTITIOUS THREE-NODE SYSTEM

It is observed that applying the OPF formulation as outlinedin this paper will result in a further reduction of the convertersswitching losses even under similar operating conditions as forwhen a conventional power flow (CPF) algorithm is appliedto the same model. This is evident in Table XI when the newmodel is used in a fictitious three-node system similar to the oneshown in Fig. 1 and the STATCOM is tasked with maintainingthe voltage at node 2 to 1.05 p.u.It is seen that applying the OPF under the same operating

criteria will result in an approximate 30% reduction in the valuecalculated for the converters switching losses. The STATCOMcurrent magnitude in the case of OPF solution algorithm is0.7095 p.u. whereas in the case of applying CPF the STATCOMcurrent magnitude comes at 0.8421 p.u.

VI. CONCLUSIONSA new STATCOM model suitable for optimal power flow

solutions using Newtons method has been introduced in thispaper. The new model departs from the idealized controllablevoltage source concept that has been used so far for representingthe fundamental frequency operation of the STATCOM in OPFformulations. Instead, it treats the DC-to-AC converter of theSTATCOM as a transformer device with a variable complextapjust as DC-to-DC converters have been linked, conceptu-ally speaking, to step-up and step-down transformers [6]. ThePWM control of the VSC is modeled explicitly by means of thecomplex tap of the ideal transformer whose magnitude repre-sents the PWM amplitude modulation coefficient and its phaseangle corresponds to the phase shift that would exist between thefundamental frequency voltage and current wave forms. More-over, the phase angle of the complex tap in the new VSC modelcoincides with the phase angle of the conventional, equivalentvoltage source model of the VSC. The converters DC bus ismodeled as a type-PV bus with constant DC voltage magnitude



TABLE XIIEFFECTS OF ENERGY STORAGE ADDITION IN THE FICTITIOUS

THREE-NODE RADIAL SYSTEM

and zero phase angle, i.e., when expressed in rectangular coor-dinates, the imaginary part of this voltage does not exist. TheSTATCOM-OPF model is tested in a radial system configura-tion to showcase the regulating properties of the new model. Alarger system comprising several generators has been selectedto show that the new STATCOM model performs equally wellwithin Newtons OPF solution.

APPENDIX AVSC MODEL DESCRIPTION

The VSC is modeled as an ideal complex tap-changing trans-former, as shown in Fig. 1(b). The following relationship definesthe complex tap ratio in the ideal complex tap-changing trans-former:

(A1)

The nodal matrix equation is quite straightforwardly derived byperforming basic nodal analysis on the VSCs equivalent cir-cuit model in Fig. 1(b). The current through the admittance con-nected to nodes and 1 is defined as

(A2)

where .At node the following relationship applies to the current

flowing in this node:

(A3)

Combining (A2) and (A3) will ultimately yield the VSCsnodal admittance matrix shown in (3).

APPENDIX BEXPLICIT JACOBIAN AND HESSIAN TERMSFOR ACTIVE POWER FLOW REGULATION

In order to enforce the active power flow control capability ofthe VSC in the OPF solution a new explicit Lagrangian functionmay be defined as follows:

(B1)

in which is the amount of VSC active power exchangewith the grid and is the specified target active power flowin the converter. The Hessian terms with respect to state variableis given in (B2):

(B2)

TABLE XIIICONVERTER PWM OPERATIONAL PARAMETERS AT OPTIMUM

It is noted that normally corresponds to the DC sidebus active power flow which is under normal circumstances setto zero unless an energy storage device is present in which caseit is set to a pre-specified target value. The contribution of ad-ditional energy storage device will be extensively discussed inAppendix C.The Hessian terms of (B1) with respect to Lagrange multi-

pliers are obviously zero, hence

(B3)

APPENDIX CEFFECTS OF ENERGY STORAGE

The simplest of possible energy storage representationswithin the OPF formulation is done by applying an activepower flow constraint in the converter and setting a target valuefor the DC power. The fictitious three-node system introducedin Fig. 2 is modified to include a small bank of batteries whichinject 0.05 p.u. of active power into the system. The effectsthat the additional storage has in reducing the final value ofthe objective function are evident from Table XII. The finalvalues of the converters operational parameters are presentedin Table XIII. In this case the converter has a wider phase shiftto allow for a larger active power flow from the converter tothe grid.

APPENDIX DIDEAL PHASE SHIFTER CIRCUIT

One salient characteristic of the new VSC model is that nospecial provisions within a conventional AC power flow solu-tion algorithm is required to represent the DC circuit, since thecomplex tap-changing transformer of the VSCmay be used withease to give rise to the customary AC circuit and a notional DCcircuit. However, some further explanation is required since themodeling development involves the conflation of AC and DCcircuit concepts at an equivalent node, brought about by the useof the ideal tap-changing transformer concept.In order to elaborate the explanation from the vantage of elec-

tronic circuits, we are going to assume that the conductance as-sociated with switching losses, , in Fig. 1(b), may be re-ferred to the primary side of the ideal transformer. The relevantpart of the circuit illustrating such a situation but with capacitorrepresentation, as opposed to its equivalent battery representa-tion, is shown in Fig. 5By invoking (A1)

(D1)



Fig. 5. Equivalent circuit showing the ideal phase-shifting transformer ofFig. 1(b) and neighboring elements, where .

(D2)

In steady-state, a charged DC capacitor draws zero currentand it is well-accepted that it may be represented as a chargedbattery [24] and, by extension, as a DC voltage source feedingno current. These facts are reflected by (D1) and (D2) and givethe opportunity to interpret the circuit in Fig. 5 in terms ofelectronic circuits concepts. Hence, it may be argued that insteady-state this circuit behaves as a nullor operating on a DCsource representing the DC capacitor. The nullor is made up of anullator and a norator [25], represented in this case by the idealphase-shifting transformer and the equivalent admittance, ,respectively. The circuit in Fig. 5 may be re-drawn as follows.The nullator and the norator are said to be linear, time-in-

variant one-port elements. The former is defined as having zerocurrent through it and zero voltage across it. The latter, on theother hand, can have an arbitrary current through it and an ar-bitrary voltage across its terminals. Nullators have propertiesof both short-circuit (zero voltage) and open-circuit (zero cur-rent) connections. They are current and voltage sources at thesame time. A norator is a voltage or current source with infi-nite gain. It takes whatever current and voltage is required bythe external circuit to meet Kirchhoffs circuit laws. A noratoris always paired with a nulator [25].Either, by careful examination of (D1) and (D2) or by anal-

ysis of the electronic equivalent circuit in Fig. 6, it can be seenthat the ideal, complex tap-changing transformer of the VSCgives raise to the customary AC circuit and a notional DC cir-cuit where the DC capacitor yields voltage but draws nocurrent.In a more general sense and from the viewpoint of the AC

power flow solution, if resistive elements or DC power loads areconnected to the notional DC bus then currents do pass throughthe ideal phase-shifting transformer but it would be a compo-nent of current that yields a nodal voltage with zero phaseangle and, as one would expect, yields power with no imagi-nary component, hence, no reactive power exists in this part ofthe notional DC circuit.

REFERENCES[1] E. Acha and B. Kazemtabrizi, A new STATCOM model for power

flows using the Newton-Raphson method, IEEE Trans. Power Syst.,vol. 28, no. 3, pp. 24552465, Aug. 2013.

[2] E. Acha, C. R. Fuerte-Esquivel, H. Ambriz-Perez, and C. Angeles-Ca-macho, FACTS Modeling and Simulation in Power Networks. NewYork, NY, USA: Wiley, 2005.

Fig. 6. Interpretation of the equivalent circuit of Fig. A.1 in terms of electroniccircuit elements.

[3] G. N. Hingorani and L. Gyugyi, Understanding FACTS: Concepts andTechnologies of Flexible AC Transmission Systems. Piscataway, NJ,USA: IEEE, 2000.

[4] E. Acha, V. Agelidis, O. Anya-Lara, and T. J. M. Miller, Power Elec-tronic Control in Electrical Systems. New York, NY, USA: Newnes,2002.

[5] Y. Zhang, G. P. Adam, T. C. Lim, S. J. Finney, and B. W. Williams,Voltage source converter in high voltage applications: multilevelversus two-level converters, in Proc. 9th Int. Conf. AC and DC PowerTransmission (Conf. Publ.), Oct. 1921, 2010, pp. 15.

[6] N. Mohan, T. M. Undeland, and W. P. Robins, Power Electronics:Converters, Applications and Design. New York, NY, USA: Wiley,2003.

[7] L. Gyugi, Dynamic compensation of AC transmission lines by solid-state synchronous voltage sources, IEEE Trans. Power Del., vol. 9,no. 2, pp. 904911, Apr. 1994.

[8] D. J. Gotham and G. T. Heydt, Power flow control and power flowstudies for systems with FACTS devices, IEEE Trans. Power Syst.,vol. 13, no. 1, pp. 6065, Feb. 1998.

[9] X. Zhang and E. J. Handshcin, Optimal power flow control by con-verter based FACTS controllers, in Proc. 7th Int. Conf. AC-DC PowerTransmission (Conf. Publ.), Nov. 2830, 2001, pp. 250255.

[10] C. Angeles-Camacho, O. L. Tortelli, E. Acha, and C. R. Fuerte-Es-quivel, Inclusion of a high voltage dc-voltage source converter modelin a Newton-Raphson power flow algorithm, IEE Proc. Gen., Transm.,Distrib., vol. 150, pp. 691696, Nov. 2003.

[11] A. Lesnicar and R. Marquardt, An innovative modular multilevel con-verter topology suitable for a wide power range, in Proc. IEEE PowerTech Conf., Jun. 2003, vol. 3, pp. 69.

[12] M. Hagiwara and H. Akagi, PWM control and experiment of modularmultilevel converters, in Proc. IEEE Power Electronics SpecialistsConf., PESC 2008, Jun. 2008, pp. 154161.

[13] W. D. Stevenson and J. Grainger, Power System Analysis. New York,NY, USA: McGraw-Hill, 1994, pp. 531587.

[14] R. Garcia-Valle and E. Acha, The incorporation of a discrete, dynamicLTC transformer model in a dynamic power flow algorithm, in Proc.IASTED/Acta Press, Palma de Mallorca, Spain, May 2007.

[15] D. P. Bertsekas, Constrained Optimization and Lagrangian MultiplierMethods. New York, NY, USA: Academic, 1982.

[16] G. Allaire, Numerical Analysis and Optimization: An Introduction toMathematical Modeling and Numerical Simulation. Oxford, U.K.:Oxford Univ. Press, 2007, pp. 306388.

[17] A. Ruszczynski, Nonlinear Optimization. Princeton, NJ, USA:Princeton Univ. Press, 2005, pp. 286331.

[18] D. I. Sun, B. Ashley, B. Brewer, A. Hughes, and W. F. Tinney, Op-timal power flow by Newtons approach, IEEE Trans. Power App.Syst., vol. PAS-103, pp. 28642875, 1984.

[19] H. W. Dommel and W. F. Tinney, Optimal power flow solutions,IEEE Trans. Power App. Syst., vol. PAS-87, pp. 18661876, Oct. 1968.

[20] A. M. Sasson, F. Viloria, and F. Aboytes, Optimal load flow solutionusing the Hessian matrix, IEEE Trans. Power App. Syst., vol. PAS-92,pp. 3141, Jan. 1973.

[21] A. Santos, Jr and G. R. M. d. Costa, Optimal power flow solutions byNewtons method applied to an augmented Lagrangian function, IEEProc. Gen., Transm., Distrib., vol. 142, pp. 3336, 1995.

[22] A. Santos, Jr, S. Deckmann, and S. Soares, A dual augmented La-grangian approach for optimal power flow, IEEE Trans. Power Syst.,vol. 3, no. 3, pp. 10201025, Aug. 1998.

[23] IEEE 30-node Test System. [Online]. Available: http://www.ee.wash-ington.edu/research/pstca.



[24] J. W. Nilsson and S. Riedel, Electric Circuits, 9th ed. EnglewoodCliffs, NJ, USA: Prentice Hall, 2010.

[25] C. J. M. Verhoeven, A. van Staveren, G. L. E. Monna, M. H. L.Kouwenhoven, and E. Yildiz, Structured Electronic Design: NegativeFeedback Amplifiers. Norwell, MA, USA: Kluwer, 2003.

Behzad Kazemtabrizi (M12) was born in Tehran,Iran. He received the B.Sc. degree in electricalpower engineering from Azad University, Tehran,Iran, in 2006 and the M.Sc. and Ph.D. degreesin electronic and electrical engineering from theUniversity of Glasgow, Glasgow, U.K., in 2007 and2011, respectively. He has been with the School ofEngineering and Computing Sciences in DurhamUniversity, U.K., as a Research Associate since Jan-uary 2012 and has recently been appointed Lecturerin Electrical Engineering since September 2013.

Enrique Acha (SM02) was born in Mexico. Hegraduated from the Universidad Michoacana deSan Nicolas de Hidalgo in 1979 and received thePh.D. degree from the University of Canterbury,Christchurch, New Zealand, in 1988.He was a Professor of electrical power systems

at the University of Glasgow, U.K., in the period20012011 and he is now a Professor of electricalpower systems at Tampere University of Technology(TUT), Finland.Prof. Acha is an IEEE PES distinguished lecturer.

Date post:	29-Sep-2015
Category:	Documents
Upload:	pouria-deniz
View:	17 times
Download:	4 times

An Advanced STATCOM Model for Optimal

Documents