4282 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4

4282 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4, NOVEMBER 2013

Second-Order Cone Programming for OptimalPower Flow in VSC-Type AC-DC Grids

Mohamadreza Baradar, Student Member, IEEE, Mohammad Reza Hesamzadeh, Member, IEEE, andMehrdad Ghandhari, Member, IEEE

Abstract—This paper presents a second order cone program-ming (SOCP) formulation of the optimal power flow problem forAC-DC systems with voltage source converter (VSC) technology.Approximation techniques have been used to derive the SOCP for-mulation of the AC-DC OPF problem. Later, the SOCP formula-tion can be solved using the interior point method (IPM) by consid-ering the limits on AC-DC grid. The accuracy of SOCP formula-tion of AC OPF has been proven with numerical examples usingIEEE 14-bus, IEEE 30-bus, and IEEE 57-bus example systems.The results of the SOCP formulation are compared with availablecommercial software. Then a DC system with VSC technology ismodeled in the IEEE 30-bus example system. The SOCP formu-lation of AC-DC OPF is applied to the modified IEEE 30-bus ex-ample system and the results are discussed. The limitations of de-rived SOCP formulation are also discussed.

Index Terms—AC-DC optimal power flow, multi-terminalHVDC systems, second-order cone programming.

NOMENCLATURE

Number of AC buses.

Number of DC buses and converters.

Number of generators.

Number of AC lines.

Number of DC lines.

Difference phase angle between sending andreceiving ends at AC line th.

Set of convex cones.

Loop matrix.

Number of independent loops in AC network.

Vector of coefficients.

Incidence matrix associated with AC line activepowers with size of .

Incidence matrix associated with AC line activepower losses with size of .

Manuscript received November 01, 2012; revised March 04, 2013 and May13, 2013; accepted June 22, 2013. Date of publication July 30, 2013; date ofcurrent version October 17, 2013. Paper no. TPWRS-01226-2012.The authors are with KTH Royal Institute of Technology, School of Electrical

Engineering, Department of Electric Power Systems, SE-100 44 Stockholm,Sweden (e-mail: [email protected]; [email protected];[email protected]).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.2013.2271871

Vector of active power generation at each ACbus with size of which has onlynon-zero elements.

Vector of reactive power generation at each ACbus with size of which has onlynon-zero elements.

Vector containing active power loads at each ACbus with size of .

Vector containing reactive power loads at eachAC bus with size of .

Vector containing AC line flow active powersat receiving end of AC lines with sizeof .

Vector containing AC line flow reactive powersat receiving end of AC lines ( ) with size of

.

Vector containing AC line flow active powerlosses at AC lines with size of

.

Vector containing AC line flow reactive powerlosses at AC lines with size of

.

Incidence matrix associated with DC line activepowers with size of .

Incidence matrix associated with DC line activepower losses with size of .

Vector containing DC line flow active powersat receiving end of DC lines with sizeof .

Vector containing DC line flow active powerlosses at DC lines with size of

.

Diagonal matrix whose elements are shuntsusceptance AC nodes with size of .

Diagonal matrix whose elements areresistance of AC lines with size of .

Diagonal matrix whose elements areresistance of DC lines with size of .

Diagonal matrix whose elements arereactance of AC lines with size of .

Injected/absorbed active power vector ofconverters at PCCs.

0885-8950 © 2013 IEEE

BARADAR et al.: SECOND-ORDER CONE PROGRAMMING FOR OPTIMAL POWER FLOW IN VSC-TYPE AC-DC GRIDS 4283

Injected/absorbed reactive power vector ofconverters at PCCs.

Injected/absorbed active power vector at DCbuses.

Capacity limit of generator at bus .

Capacity limit of converter at bus .

Converter block losses.

Scalar representing converter losses.

I. INTRODUCTION

I NTRODUCTION of high voltage direct current (HVDC)technology has opened opportunities to transfer the electric

power between two points which in particular are remotely lo-cated from each other.Conventionally two types of converter, voltage source con-

verters (VSCs) and current source converters (CSCs), are em-ployed for two terminal HVDC links. The advances in powerelectronic devices and converter technology has led the multiterminal HVDC (MTDC) systems, which are not actually a newidea, to be suggested again as one of the solutions for intercon-necting several generating and load points. In [1], the MTDCsystems have been proposed as a cost-efficient way to connectoffshore wind farms to the onshore AC systems. Due to practicalreasons VSC is a more suitable option for building such hybridAC-DC systems than its older generation, CSCs. The VSCs notonly have no reactive power demand but also can either inject orabsorb the reactive power to/from the AC grid in order to main-tain AC side voltage as a generator [2]. The VSC technology canrapidly control active power and at the same time reactive powerat each terminal independent of the DC power transmission [3].Most importantly, the VSC does not need to reverse voltage po-larity to change the direction of power flow in its DC link. Asa results of advances in cable technology suitable for VSCs thepower flow reversal is done by reversal of current direction. Thischaracteristic allows several VSCs to be connected in paralleland form a meshed DC grid [4], [5]. Such hybrid AC-DC gridsbuilt from parallel connection of VSC links have a good poten-tial to be used in the future bulk power systems [6]. Possibilityof such connections has led to the proposition of European andNorth Sea SuperGrids that could connect several renewable en-ergy sources to a common DC network [7], [8]. However, inorder to fully reveal steady state and dynamic behavior of suchsystems to be used in the real-size power systems, extensiveresearch has to be done. References [9] and [10] have inves-tigated the fault occurrence in VSCMTDC grids and protectionmethodologies proposed to survive them. Studying the dynamicand transient behavior, modeling and control methodologies ofVSC MTDC grids are the main focus of [11] and [12]. Two dif-ferent power flow approaches, unified and sequential methods,for power flow calculation of VSC MTDC grids have been pro-posed by [13]–[15].One of the key aspects of such a hybrid VSC AC-DC grid

which has not been fairly addressed and reported in the litera-ture is its impact on improving the economic efficiency of theelectricity industry. It allows lower-cost generation to be sub-stituted by the high-cost generation and it also improves the

allocative efficiency by reducing the deadweight loss. In orderto quantify these components of economic efficiency, theseVSC AC-DC systems and their associated DC grids need to bemodeled in the optimal power flow (OPF) formulation. Mostof the OPF modelings proposed in the literature has accountedfor CSC-HVDC links. In [16], [17], the sequential quadraticprogramming (SQP) and sequential gradient-restoration algo-rithm (SGRA) methods have been employed to solve AC-DCOPF of AC systems with multiterminal DC systems built fromCSCs. Reference [18] has proposed a combined heuristic andinterior point method (IPM) for solving the optimal reactivepower flow (ORPF) problem in AC-DC systems incorporatingCSC converters. In [19] a model for security-constrained unitcommitment with CSC-type AC-DC systems using Bendersdecomposition has proposed in which the problem is dividedinto two subproblems which then solved using the mixed in-teger programming (MIP) and Newton Raphson (NR) methods.However, since there are several major differences between

VSCs and CSCs in both operation principle and structure,they cannot be applied to the AC-DC system incorporatingVSCs. Most of the papers investigating OPF modeling of VSCconverters has focused on VSC-based FACTS controllers em-bedded in AC systems [20]–[22]. In these studies, well knownSQP and NR methods and predictor-corrector primal-dualinterior point linear programming have been employed to solvethe optimization problems. There are also little literature thathas studied the OPF models incorporating two terminal VSCHVDC links. Reference [23] has modeled VSC HVDC linkswhich are limited only to back-to-back VSC HVDC links inwhich the nonlinear set of AC-DC load flow equations aresolved through the NR based OPF method. In [23], there isno DC grid representation and the DC links has been onlyaccounted for by introducing an equation expressing that thetotal amount of active power injected into the DC systemis equal to DC power losses. DC line losses have been alsodefined based on the DC slack bus voltage assuming that itsvalue (i.e., ) is prespecified constant value beforerunning the OPF. Also the control modes (i.e., PV or PQ mode)for each converter have been considered as constraint in OPFformulation. Recently, [24] has reported a security-constrainedunit commitment (SCUC) which considers DC grids with VSClinks. The Benders decomposition method has been used in[24]. Given the nonconvex nature of OPF problem, in all thesestudies, authors have tried to develop a convex optimizationproblem through several methods, these methods suffer fromtwo main drawbacks: 1) loss of accuracy in results, and 2) com-plexity of algorithms. In our paper, a set of AC-DC constraintsare developed through the line flow based (LFB) equations ofpower flow. In the LFB model the independent state variablesincludes bus voltage magnitudes, active and reactive line powerflows [25]. These variables i.e. bus voltage magnitude andline power flows reflect more practical knowledge about thesystem than conventional power flow variables. Moreover, itis shown in this paper that this formulation make it possible toobtain a convex problem thanks to some small approximationswe use in this paper. The LFB equations have been used fordifferent studies in power system analysis. Reference [25] hasformulated the power flow equations for load flow calculation


in radial AC systems with embedded FACTS devises. In [26]a LFB based formulation has been proposed for finding theoptimal locations of TCSC using MIP. The authors in [26]suggest that some of the main equality constraints are replacedby inequality constraints in order for the problem to be solvedthrough quadratic mixed-integer programming. Moreover, [26]have ignored the equation representing the transmission lineslosses in order to remove the nonlinearity from the set of equa-tions. The line power losses are the main origin of nonlinearityof power flow equations which some literature has presentedlosses as a quadratic function of difference phase angle of twoconnected buses and then used a linearized approximation ofpower losses [27], [28].In our paper, we consider all nonlinear equations associated

with active and reactive losses in all AC and DC lines as a func-tions of line flows variables. Instead of using a linearized ap-proximation of power losses, we transform them to a conic con-straint. Also, AC and DC line capacities and generator and VSCcapacity limitations are all converted to the conic constraints.After formulating the problem through the line flow variables,

the problem is transformed to a second order cone programming(SOCP) format which is a convex problem and can be efficientlysolved [29], [30]. Then through solving this convex problem, weobtain the global optimum point of the proposed formulation. Itshould be noted that there might be a small error between thissolution and solution to the original problem due to the approx-imations used in this paper. Recently, several researchers havestudied the possibility of redefinition and recasting of the non-linear OPF problem as an SOCP programming or semidefiniteprogramming problem under some mild conditions. Throughtheir proposed formulation a globally optimal solution for theOPF problem can be obtained [31], [32].In the proposed SOCP formulation of the AC-DC-OPF, un-

like [23] in which the control modes are set before running theOPF and considered in the OPF formulation as constraint (ex-ogenous modeling of control modes), the control modes [i.e.,the point of common connection (PCC) as a PQ bus or PV busand the a DC bus as DC slack bus] can be set either beforeOPF or after OPF solution is obtained (endogenous modeling ofcontrol modes). Therefore, based on the results obtained fromour SOCP formulation of the AC-DC-OPF, one can determinewhich VSC set as PV or PQ mode and which DC bus is set asthe slack bus.In order to evaluate the accuracy of the proposed SOCP for-

mulation of the AC-DC OPF, the numerical results which areobtained in GAMS platform usingMOSEK solver [33] are com-pared to results obtained in MATPOWER software, [34]. Themodified IEEE 30-bus test system is used as a case study.

II. VSC STATION MODEL AND OPERATION MODES

In a VSC-MTDC system with N converters, one convertershould be considered as a DC slack converter to regulate its DCvoltage around a specified value. For the sake of simplicity, theth VSC is considered as slack converter. This is to ensure the

power balance in the MTDC network, i.e.,

(1)

Fig. 1. Equivalent circuit of an AC line.

where is the injected DC power at each DC terminal andis the total loss in the MTDC network. This implies that

the active power at the PCC bus connected to a slack converter isnot specified, and this unspecified power is defined by .However, the converter may control either the reactive power

or the voltage at the PCC bus . In [23], thesecontrol modes are set before running the OPF and considered inthe OPF formulation as constraint.However, in the AC-DC OPF proposed in this paper all

active/reactive converter powers and voltage magnitudes atPCCs and DC buses are considered as independent controlvariables. Therefore, there is opportunity to either set thesecontrol variables as a prespecified constant values before run-ning the AC-DC-OPF and solve the OPF with these constraintsor obtain the specified values for these variables for whichthe objective function of AC-DC-OPF has been optimized.The slack converter also can be chosen based the output ofAC-DC-OPF in such a way that the DC voltage value of a DCbus obtained from OPF solution can be set as the in loadflow analysis.

III. AC-DC LINE FLOW BASED EQUATIONS

In this section, the line flow equations [25] are extended tofully represent all AC and DC networks and converters.

A. AC Line Flow Equations

Fig. 1 shows the equivalent circuit of an AC line. Let considera conventional direction for each line starting from sending busand ending to receiving bus (see Fig. 1). The active and reactivepower balance equations for AC node other than PCCs can bewritten as follows:

(2)

(3)

where

(4)

(5)


Each element of and are as follows:

(6)

(7)

The equation associated with the voltage drop across each lineis written as follows:

(8)

If both sides of (8) are multiplied by , then the magni-tude of both sides are obtained and finally both sides are dividedby :

(9)

therefore (9) can be written for each AC line.To take into account the effect of phase angle in (8), we obtain

the imaginary part of both sides in (8) and one can obtain thefollowing equation by assuming and :

(10)

On the other hand, a meshed power network can be seen as agraph consisting of independent loops. According to graphtheory, if and are assumed to be the number ofbranches and nodes of the graph, can be obtained as follows:

(11)

The phase angle difference around each independent loop of agraph is zero. Therefore another set of equations associated withthe graph loops ( equations) can be obtained using (10) asfollows:

(12)

where is an incidence matrix with the size of . Thispaper proposes set of (12) to model phase angle constraints fora meshed network. Although this approximation affects the ac-curacy of the OPF results, the algorithm becomes much sim-pler and faster. There are other works linearizing the voltageangle constraints around an operating point. This might resultin a more accurate result however at higher computational time,[35]:

(13)

Fig. 2. Modeling of the AC-DC systems in the proposed OPF formulation.

B. Converters Equations

As in Fig. 2, converters can bemodeled as a dummy generatoror motor which either can absorb or inject active/reactive powerto the AC network. The phase reactor and coupling transformerbetween PCCs and converters are also represented as an ACline with equivalent inductance and resistance. The active andreactive power balance equations at each PCC can be obtainedas follows:

(14)

(15)

where

(16)

and if the converter losses are assumed to be propor-tional to passing active power through the converter (i.e.

) the DC power at the DC bus at eachstation can be obtained as follows:

(17)

C. DC Grid Equations

To derive the line flow equations for the DC grid, considerthe equivalent circuit of DC line in Fig. 3. Each DC line arenumbered from 1 to and specified by an optional directionfrom sending bus to receiving bus (see Fig. 3). Doing so, active


Fig. 3. Equivalent circuit of a DC line.

power balance equation at DC bus is obtained basedon the line flow variables shown in Fig. 3 as follows:

(18)

where

(19)

If we write the voltage drop equation for DC line ,we have

(20)

If both side of (20) are multiplied by and then getsquared and finally are divided by we reach

(21)

From (19), equality (21) can be rewritten as

(22)

IV. SECOND-ORDER CONE PROGRAMMING

An SOCP or conic optimization problem is a nonlinearconvex problem which can be introduced as a general formof linear programming accompanied by nonlinear constraintswhich are in form of convex cones. Many kinds of problemssuch as LP and QP can be formulated as SOCPs and be effi-ciently solved through polynomial time IPMs [29], [30]. Thisoptimization problem has the following form:

(23)

where is the optimization variable vector and andare constant vectors. If the variables are divided into

in such a way that each element ofonly belongs to one set of , then the additional condition,i.e., is fulfilled

if are convex cones and

(24)

Assuming , each cone in (24) canhave two following forms:1) Second Order Cone: The standard form of this kind of

cone is as follows:

(25)

2) Rotated Quadratic Cone: This kind of cone is obtained byrotating a with an angle of 45 degrees in planewhich has the following form:

(26)

Therefore if the constraints involves some linearly independentfunction and nonlinear inequality which can be transformed toform of and , the problem can be efficiently solvedusing IPMs.

V. PROPOSED SOCP FORMULATION FOR AC-DC-OPF

We assume the in (23) to be the vector of short-run mar-ginal cost for all exiting generating units in the system. We usethis linear objective function, since the standard form of theconic problem shown in (23) has a linear objective function.However, one can use the piecewise linearization methods orother more accurate linear models for a cost objective functions.Let us define vector as (27):

(27)

where is the vector of square of AC bus voltagesis the vector of square of DC bus voltages. From derived (2),(3), (4), (5), (9), (12), (14), (15), (17), (18), (19) and (22) theconstraint in the AC-DC OPF problem are as follows:(A) Linear constraints:

(28)

(29)

(30)

(31)

(32)


(33)

(34)

(35)

(36)

where (28) and (29) are associated with AC buses’ mis-match constraints, (30) is associated with voltage dropequation constraint for AC lines, (31) is associated withphase angle difference onAC lines, (32) is associated withalgebraic sum of phase angles in each independent loop,(33) represents the relation between active and reactivepower losses at AC lines, (34) represents the relation be-tween injected DC power at DC grid and converter powerat PCC buses, (35) is associated with DC buses’ mismatchconstraints and finally (36) represents the DC voltage dropconstraint on DC lines.

(B) Non-Linear constraints:For each AC line :

(37)

For each DC line :

(38)

where and are the square of DCand AC voltage at receiving end of line th,

and . Since, and ,, if with a good approximation, the

equality constraints (37) and (38) are replaced withinequality constraints,and form an ap-propriate set for each AC and DC line to be recastas a convex quadratic cone format and thereforeintroduced as a rotated quadratic cone in (26). Foran arbitrary positive and sufficiently small constant

, one can obtain

(39)

Inequalities in (39) can be rewritten as follows:

(40)

(41)

Therefore (37) and (38) are replaced with (40)and (41). Doing so, the constraints involvelinearly independent functions and nonlinearinequality with form of and then theproblem can be efficiently solved using IPMs.

It should be noted that the appropriate sets arereplace by and

. The other inequalityconstraints can be defined as follows:

(42)

The underline and overline signs are used for lowerand higher limits, respectively. The present paperconsiders the ideal PQ-capability curve (PQ circle)for both generators and converters. For generator,we only consider PQ curve as a circle which is onlybased on armature current heating limit [36]. Takinginto account all practical limitations will add moreconstraints to the optimization problem. Having usedthe ideal characteristic, their associated nonlinearinequality constraints can also be represented as arotated quadratic cone as followa:

(43)

where , ,. The proposed SOCP formulation

of the AC-DC-OPF is coded in GAMS platformand is solved through its MOSEK package [33].This package is particularly very efficient in solvingSOCP optimization problems via interior-point opti-mizer. The output numerical results obtained throughMOSEK solver are compared to results obtained inMATPOWER software [34].

VI. CASE STUDY AND SIMULATION RESULTS

Two cases are studied here; (A) The SOCP formulation of ACOPF problem, and (B) The SOCP formulation of AC-DC OPFproblem.

A. SOCP Formulation of AC OPF

In this section, three example systems are studied. The OPFsolutions of IEEE 14-bus, IEEE 30-bus, and IEEE 57-busexample systems are found using the SOCP formulation and


TABLE IRESULTS FROM STANDARD FORMULATION (STRD) AND THE SOCP

FORMULATION (SOCP) FOR THREE IEEE TEST SYSTEMS

TABLE IISOCP OPF AND STRD OPF RESULTS WHEN SOME LINES ARE CONGESTED

(TEST CASE IEEE 30-BUS), SHOWS THE CONGESTED CASES

standard formulation of AC OPF. The SOCP formulation iscoded in GAMS platform and solved using the MOSEK solver.The standard formulation of AC OPF is solved using the MAT-POWER software [34]. The total generation costs, when thereis no binding constraint, are given in Table I. This table shows0.001% error for the IEEE 14-bus example system, 0.041%error for the IEEE 30-bus example system, and 0.099% errorfor the IEEE 57-bus example system.The IEEE 30-bus example system is used to investigate the

tightness of the SOCP relaxation and the loss over-satisfactionissue (Table II). First, the standard formulation and SOCP for-mulation of OPF problem are solved for 80-MVA, 60-MVA, and30-MVA power flow limits. Then the results of SOCP formu-lation are fed into a load flow problem and the system loss iscalculated. For the 80-MVA limit, two congested lines were ob-served. In this case, the ohmic loss from SOCP formulation is12.239 MW and the ohmic loss from load flow calculation is12.241 MW which is equivalent to 0.016% difference. In thesecond case, when we reduced the power flow limits to 60MVA,three lines hit their limits. The SOCP formulation gives us betteroperating point as compared to the standard formulation whilea difference of 0.01% in ohmic loss is observed. At the 30 MVAlimit, five lines are congested. The ohmic losses calculated fromthe SOCP formulation and the load flow are 3.159 MW and3.156MW, respectively. This is equivalent to 0.095% differencein ohmic loss. Again in this case, the operation cost from SOCPformulation is less than the one from standard formulation. Thisissue of convex relaxation has been studied in a few literatures.Reference [37] studies some limitations of the semidefinite pro-gramming (SDP) relaxation of OPF problem especially thosewhich are based on the conventional injection model [31]. Ref-erence [38] shows that conic relaxation, i.e., replacing equalityto inequality constraints is exact only for radial networks pro-vided that there is no upper bound on active and reactive loads

TABLE IIIRESULTS FROM STRD FORMULATION AND SOCP

FORMULATION OF AC-DC OPF FOR IEEE 30-BUS TESTSYSTEM

(load over-satisfaction). For a meshed network, authors in [39]show that if the phase angle constraints are eliminated fromthe original OPF problem and then the conic relaxation is ap-plied, the obtained problem has an exact solution of the orig-inal problem provided some extra conditions are met. Theseconditions include no upper bound on active and reactive loads(load over-satisfaction) and successful angle recovery. They in-troduced some conditions under which the omitted phase an-gles can be recovered. It is shown that the phase angle recoveryconditions are always valid for the tree networks, whereas itfails for the meshed networks in some cases. To handle thisproblem, they proposed installing a number of phase shiftersin the meshed network to satisfy the phase angle recovery con-ditions. In our proposed SOCP formulation of OPF problem,we do not eliminate the phase angle constraints rather we in-corporate them using a set of approximated equations given in(12). Also, we use loss over-satisfaction using the relaxation in(40) and (41) which is strongly related to the load over-satisfac-tion assumption. However as shown in Table II, even for lowlimits on power flows, the total power losses obtained in bothstandard OPF and SOCP OPF are very close. Negative nodalprice is another issue discussed in the convex relaxation litera-ture of OPF problem. Power markets operate at conditions withnegative nodal prices with some regularity. Binding power flowconstraints might create negative nodal prices. The relaxationin (40) and (41) appear to allow for increasing losses and thusprevent the occurrence of negative nodal prices. The detaileddiscussion of this issue has been left to the future extensions ofthis work.

B. SOCP Formulation of AC-DC OPF

To evaluate the proposed SOCP formulation of AC-DC-OPF,the IEEE 30-bus test system is modified and used. Since DCgrid and their associated converters has not been implemented inMATPOWER software, only the results of proposed SOCP for-mulation for AC-OPF are used to be compared to results of ACOPF obtained in MATPOWER software. The detailed resultsobtained for IEEE 30-bus test system is given in Table III. As it


Fig. 4. Modified IEEE 30-bus test system with embedded DC system.

is seen the results are very close and this highlights the accuracyand correctness of the proposed SOCP formulation of AC-DC-OPF. It should be noted that active and reactive power and lossesthrough each line which are variables in SOCP are almost iden-tical. In order to implement the proposed SOCP formulation ofAC-DC-OPF, the IEEE 30-bus test system with two installedthree terminal DC system is used. This is shown in Fig. 4. ThisDC grid can have two different V andDelta configurations. Con-verters VSC1, VSC2, VSC3, VSC4, VSC5 and VSC6 are con-nected to AC buses 2, 6, 5, 1, 15 and 30, respectively. The DCvoltage levels according to the ABB High Voltage Cable (HVC)product range are 80 kV, 150 kV and 320 kV [2]. In this paper,M9 module which has the highest voltage (320 kV) level ischosen for both cases. The converter power rating are chosen100 (MVA). As it was mentioned, through the proposed OPFthe PCCs can either set to be in one of the PV or PQ modesbefore running the OPF or active/reactive powers. AC voltageat PCC are considered as state variables and their final valuesfrom OPF are considered as economic values. Similarly, eitheran arbitrary DC bus can be set as the slack bus before runningOPF or the DC slack is selected based on the OPF results. Inour paper, none of the mentioned control modes are used forconverters and only the boundary condition on converter vari-ables are considered. Doing so, the converter variables are freeto get a value for which the objective function is minimized.The OPF results are given in Tables IV and V. In Table IV, the

is only the losses associated with DC lines. As

TABLE IVRESULTS OF SOCP FORMULATION OF AC-DC-OPF FOR IEEE 30-BUS TESTSYSTEM WITHOUT (WO) AND WITH (W) TWO INSTALLED VSC-DC GRIDS

TABLE VRESULTS OF SOCP FORMULATION OF AC-DC-OPF FOR MTDC SYSTEM

TABLE VIRESULTS OF SOCP FORMULATION OF AC-DC-OPF FOR IEEE30-BUS TEST SYSTEM WITH INSTALLED VSC DC GRIDS WITH

CONVERTER RATING 50 MVA AND 100 MVA

can be observed from the results due to economic allocation ofconverter variables the total AC losses are reduced which resultsin lower generated power and correspondingly lower generationcost. The reactive power generation also has been decreased dueto local reactive power injected by converters.In another analysis, the converter’s rating powers are set to a

lower value 50 (MVA) to see its impact on converter variablesand total generation cost. The corresponding results are givenin Tables VI and VII.


TABLE VIIRESULTS OF SOCP FORMULATION OF AC-DC-OPF FORMTDC SYSTEM WITH INSTALLED VSC DC GRIDS WITH

CONVERTER RATING OF 50 MVA AND 100 MVA

VII. CONCLUSION

This paper has successfully introduced SOCP in trans-forming nonlinear nonconvex AC-(VSC-)DC OPF problemsinto nonlinear convex problems. Comparing the introducedSOCP results with those from available commercial software(MATPOWER) proves the presented technique as an alternatereliable method for solving nonlinear non-convex AC-DC OPFequations. Through the proposed SOCP formulation of AC-DCOPF problem, several benefits of the DC systems with VSCtechnology can be evaluated.

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Mohamadreza Baradar (S’11) received the M.S.degree in electrical engineering from the Universityof Tehran, Tehran, Iran, in 2010. He is pursuingthe Ph.D. degree at the Electric Power SystemDepartment of the Royal Institute of Technology(KTH), Stockholm, Sweden.He is currently a member of the Electric Power

System Department at KTH. His research interestsinclude power system operation, electricity market,integration of renewable energy sources and AC/DCgrids modeling.

Mohammad Reza Hesamzadeh (M’10) receivedthe Ph.D. degree at Swinburne University of Tech-nology, Australia.He is an Assistant Professor in the Department of

Electric Power Systems at KTH Royal Institute ofTechnology, Stockholm, Sweden. His special fieldsof interests include power systems planning and de-sign, economics of wholesale electricity markets, andmathematical modelling and computing. He did hispostdoctoral fellowship at KTH.Dr. Hesamzadeh is a member of the International

Association for Energy Economics (IAEE) and a member of the Cigre NationalCommittee in Sweden.

Mehrdad Ghandhari (M’00) received the M.Sc.and Ph.D. degrees in electrical engineering fromRoyal Institute of Technology (KTH), Stockholm,Sweden, in 1995,and 2000, respectively. He iscurrently a full Professor at KTH. His researchinterests include power system dynamics, stabilityand control and FACTS and HVDC systems.

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4282 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 28, NO. 4