1

Topology-cognizant Optimal Power Flow inMulti-terminal DC Grids

Tuncay Altun, Student Member, IEEE, Ramtin Madani, Member, IEEE, and Ali Davoudi, Senior Member, IEEE

Abstract—This paper is concerned with the optimization ofdroop control set-points, i.e., voltage and power levels at eachbus, and the switching status of transmission lines, in multi-terminal direct current grids. Additional constraints, that ensurea safe operation in response to power fluctuation while updatingdroop set-points, are integrated into the problem formulation.This problem is cast as a mixed-integer nonlinear program withthree sources of computational complexity: i) Non-convex powerflow equations, (ii) Non-convex converter loss equations, and iii)Binary variables accounting for the on/off status of transmissionlines. Second-order cone programming relaxation tackles thenon-convexity of power flows and converter loss equations,and branch-and-bound search determines the optimal switchingstatus of transmission lines. CIGRE B4 DC grid benchmarkis emulated in a real-time hardware-in-the-loop environment tocorroborate the proposed method.

Index Terms—HVDC transmission, multi-terminal dc grids,optimal power flow, optimal transmission switching.

I. INTRODUCTION

Multi-terminal direct current (MTDC) grids are becomingpopular as they allow efficient power exchange between syn-chronized or unsynchronized power grids, and are suitable can-didates for integrating offshore wind farms [1] or long-distancepower transmission (e.g., European supergrid [2]) as its DCgrid enjoys a simpler control mechanism and avoids challengesnative to AC grids. Voltage-source converters (VSCs) arebuilding blocks of MTDC grids and allow interconnectionwith weak AC grids [3], black start in the case of blackouts,[4] and power flow reversal without switching the voltagepolarity [5]. There are ongoing efforts to realize bulk powerexchange among independent grids using the VSC technology[6]. Control of the VSC’s DC voltage is the key measure toproper power dispatch and loss management in a MTDC grid,and is mainly done via master-slave [7], voltage margin [8],or voltage droop [9] mechanisms. Voltage margin control cancause an oscillatory behavior [10]. Droop control approach ismore reliable than the master-slave control if several convertersactively participate in the regulation process.

The two-tier control hierarchy of MTDC grids [11], [12]includes a faster, lower-level droop controller that locally reg-ulates the VSC voltage at the cost of power sharing objectives.Hence, the upper-level optimizer periodically tunes the set-points of the lower-level droop control to meet predefinedobjectives, i.e, minimizing generation cost, transmission loss,etc. The optimization involved in tuning droop set-points could

This work is supported by the National Science Foundation un-der award ECCS-1809454. The authors are with the University ofTexas at Arlington, TX, 76019, USA (e-mail: tuncay.altun@mavs.uta.edu;ramtin.madani@uta.edu; davoudi@uta.edu).

become computationally prohibitive for real-time applications[13]. This delay, or any interruption in the communicationbetween the two control layers, could cause a prescribed droopset-point to violate an operational safety limit, particularlyif the load demand or power generation fluctuate noticeablybefore the subsequent droop set-points update [14]–[16]. Pre-ventive measures, while still pursuing optimality in MTDCgrids, are rare in the literature, e.g., see [17]–[19].

The optimal power flow (OPF) in MTDC grids aims to min-imize transmission loss alone [11] or along with conversionloss [20]. Convex relaxation methods, including semi-definiteprogramming (SDP) and second-order cone programming(SOCP), can transform nonlinear power flow optimization intoconvex surrogates by reformulating it in a high-dimensionalspace while preserving equivalency with the original non-convex problem [21]. The SDP and SOCP relaxations, andtheir variations, have manifested notable success in solvingOPF problems in AC systems [22], [23]. These approacheshave been extended to the static OPF problem of MTDC gridsthat suffer from additional non-convexity due to the presenceof quadratic converter loss equations [24], [25].

Static OPF solutions, however, overlook the optimal switch-ing of transmission lines that can help alleviate line overloads,address voltage violations, minimize transmission losses, pro-tect the grid from abnormal operations, or schedule mainte-nance [26]. In fact, a transmission line built for a long-term re-quirement could exhibit dispatch inefficiencies [27]. Recently,mixed-integer cone programming techniques have been usedto solve AC grid topology problems [28], [29]. In this paper,we offer a mixed-integer second-order cone programming(MISOCP) formulation for the topology-cognizant OPF inMTDC grids to minimize both transmission and converterlosses. The proposed formulation includes safety constraintsthat prevent voltage violations caused by power fluctuation inbetween two droop set-point updates. In summary, the salientcontributions of this paper are listed as follows:

• The static OPF problem of MTDC grids, with the ob-jective of minimizing the total loss, is formulated asa nonlinear optimization while respecting pyhsical andoperational constraints for both AC and DC parts.

• Additional constraints that sustain a safe operation byfurther restricting voltage limits, in response to volatilegeneration/load profiles in between two droop set-pointupdates, are integrated into the problem formulation.

• Static OPF formulation is extended to a topology-cognizant OPF by incorporating binary variables thatrepresent the switching status of transmission lines.

2

• The proposed MISOCP surrogate is formulated fortopology-cognizant OPF problem.

• The proposed static OPF, topology-cognizant OPF with-out safety constraints, and topology-cognizant OPF withsafety constraints are experimentally validated throughreal-time hardware-in-the-loop (HIL) experiments.

The remainder of the paper is organized as follows. SectionII covers preliminary materials. Section III elaborates themodeling of MTDC grids. Section IV integrates the switchingstatuses of transmission lines into the OPF problem, andprovides its MISOCP-relaxed formulation. In Section V, theresulting SOCP-relaxed OPF and MISOCP-relaxed topology-cognizant OPF problems are verified for the CIGRE B4 DCgrid benchmark using a real-time HIL. Section VII concludesthe paper.

II. NOTATIONS AND GRID TERMINOLOGIES

A. Notations

Bold uppercase and lowercase letters (e.g., X, x) representmatrices and vectors, respectively. 1 and 0 refer to vectorswith all elements as 1 and 0, respectively. The R and Csymbolize the sets of real and complex numbers, respectively.Sn and Hn represent the n × n symmetric and hermitianmatrices, respectively. The real and imaginary parts of acomplex number or matrix are defined by real· and imag·,respectively. The matrix entries are denoted by indices (i, j).Superscripts (·)> and (·)∗ denote the transpose and conjugatetranspose operator, respectively. | · | represents the cardinalityof a set or the absolute/magnitude value of a vector/scalar. [·]creates a matrix whose diagonal elements are obtained froma given vector. tr· refers to the trace of a given matrix.‖ · ‖2 stands for the euclidean norm of a given vector. diag·composes a vertical vector from the diagonal elements of agiven matrix. X 0 means that X is a symmetric positivesemidefinite matrix.

B. Grid Terminologies

Figure 1 shows a portion of an MTDC grid. Grid buses areconnected via DC transmission lines. Terminologies for gridelements are elaborated here:• DC Grid: The DC transmission grid can be structured

as a directed graph H = (N ,L), with N and L as thesets of buses and lines, respectively. Define the pairs~L, ~L ∈ 0, 1|L|×|N| as the from and to line-incidencematrices for the DC grid, respectively. For every l ∈ Land k ∈ N , ~Llk = 1 if and only if the transmissionline l starts at bus k, and ~Llk = 1 if and only if theline l ends at bus k. The matrices Y ∈ R|N |×|N|,~Y , ~Y ∈ R|L|×|N| represent the bus-conductance, andthe from and to line-conductance matrices of the DCgrid, respectively. Define ~f and ~f ∈ R|L| as thefrom and to vectors of transmission line power flows,respectively. Let f~

~

max ∈ (R ∪ ∞)|L| be the vectorof power flow limits. Additionally, define x~

~

∈ 0, 1|L|as the binary vector representing the on/off switchingstatus of transmission lines. Let x~

~

min,x~

~

max ∈ 0, 1|L|encapsulate prior knowledge of the on/off switches, i.e.,

x~

~

minl =x~

~

maxl =0, if line l∈L is known to be disconnected,

Phase ReactorPhase Reactor

pg1

pi

pd1

ii

vi

pi

zi xl flfl

dc

dc

acacg

d

vfiac vci

ac

si

acqi

ac= + i p

jdc

vjdc vfj

acvcjac

pd2

d

ijac p

g2

g

(g1, g2) Є (d1, d2) Є (i, j) Є l Є

zj

Fig. 1. A portion of a meshed MTDC grid. The grid is endowed withswitching devices to enable line switching decisions, x~

~

. VSCs couple ACand DC parts by controlling their voltage and power levels on both sides.

x~

~

minl =x~

~

maxl =1, if line l∈L is known to be connected,x~

~

minl =0, x~

~

maxl =1, otherwise.

Finally, let vdc,pdc ∈ R|N | represent the vectors of nodalDC voltages and active power injections into the DC side.

• Buses/VSCs: Each DC bus k ∈ N is assumed to accom-modate a single voltage-source converter (VSC) which isconnected to a set of loads and generators through a phasereactor, modeled as a series impedance zk ∈ C. Definez, iac ∈ C|N | as the vectors of phase-reactor impedanceand current values, respectively. Let vac

c ,vacf ∈ C|N |

account for the vectors of VSC and load/generation-side AC voltages, respectively. Let sac ∈ C|N |, andpac, qac ∈ R|N |, respectively, represent the vectors ofapparent, active and reactive power injections from VSCsinto the AC sides.

• Generators/Loads: Define G as the set of generators andG ∈ 0, 1|G|×|N| as the generator incidence matrix,where Ggk = 1 if and only if the generator g ∈ G islocated at the AC side of the bus k ∈ N . sg ∈ C|G|and pg, qg ∈ R|G|, respectively, represent the vectors ofapparent, active and reactive power generations. DefineD as the set of loads and D ∈ 0, 1|D|×|N| as the loadincidence matrix where Ddk = 1 if and only if the loadd ∈ D is located at the AC side of the bus k ∈ N .Finally, pd ∈ R|D| represents the vectors of active powerdemand.

III. MTDC GRID MODEL

A. AC/DC Coupling

VSC losses are approximated by a quadratic polynomialwith respect to current magnitude as

pconvloss , −pac − pdc = a+ [b]|iac|+ [c]|iac|2, (1)

where a, b, c ∈ R|N | are the vectors of positive coefficients[24], and pac and pdc are the vectors of active power injec-tions by the VSCs into the AC and DC sides, respectively.Additionally, the AC and DC side voltages are related with amodulation factor, m,

|vacc | ≤

√3

2mvdc. (2)

B. VSC Limits

The AC side complex powers can be calculated as

sac = [vacc ]([z]−1(vac

c − vacf ))∗, (3)

3

with the VSCs active power is bounded as

pacmin ≤ pac ≤ pac

max. (4)

With no loss of generality, the VSC reactive power limits canbe formulated as

−mb|sac|≤qac≤ [|imagz|]−1[vac

cmax

](vac

cmax− |vac

f |), (5)

where mb is a positive constant, and |sac| is the vector ofnominal VSC apparent power values [20]. To simplify (5)while finding the maximum reactive power constraint, one cansubstitute |vac

f | with vacfmax

[20].Finally, according to Ohm’s law,

iac = [z]−1(vacc − vac

f ), (6)

and the current magnitude |iac|, should not exceed an upperlimit |iac

max|, to be compatible with the limits of phase reactorand controller.

C. Generator/Load Limits

Active power balance at the generator/load sides of phasereactors can be formulated as

G>pg −D>pd = real

[vacf ]([z]−1(vac

f − vacc ))∗

, (7)

with

pgmin ≤ pg ≤ pg

max, (8)vac

fmin≤ |vac

f | ≤ vacfmax

, (9)

enforcing generator/load power and voltage limits.

D. DC Grid Constraints

Nodal power balance equations of the DC grid can beformulated as

~L> ~f + ~L> ~f = pdc, (10)

where ~f and ~f are dictated by nodal DC voltages and thestatus of transmission lines:

~f = [x~

~

] diag~Lvdcvdc> ~Y> ≤ f~

~

max, (11a)

~f = [x~

~

] diag ~Lvdcvdc> ~Y> ≤ f~

~

max, (11b)

and constrained by thermal limits of the line. Additionally,nodal voltages and power injections of the DC grid should bebounded as follows:

vdcmin ≤ |vdc| ≤ vdc

max, (12)

pdcmin ≤ pdc ≤ pdc

max. (13)

Constraints (12)–(13) enforce steady-state safety requirements.However, due to transient effects, voltage limits in (12) need tobe further restricted based on variations in load and generation,as well as the computational time delays in between droopset-point updates. In the following subsection, we formulatecomplementary voltage constraints that can further improveoperational safety.

**

t1

t2

find vdc , pdc

s.t. (20b) - (20r)

Pow

er (

pu)

vdc ≤ vdc ≤ vdcmin max

(21a) - (21c)

(a)

(b)

t1 t2 t3 t4 t5 t6 t7

0.90

1.00

1.10

1.20

v

t1 t2 t3 t4 t5 t6 t7

0.94

1.00

1.06

t1+Δt1

Volta

ge (

pu)

maxdc

vmaxdc

vmindc

pmaxdc-

αk k

+ βk k + γk = 0pdcvdc

k

k

k pmax

dck

t3+Δt3 t5+Δt5

Fig. 2. Droop parameters optimized for the load/generation profile at timet1 violates the safe operating region until the subsequent update that happensat time t3 +∆t3. (a) Load/generation profile and DC side voltage variation.(b) Generalized voltage-droop characteristics.

E. MTDC Control Strategy

The generalized VSC voltage-droop characteristic can bewritten as

αkvdck + βkp

dck + γk = 0, ∀k ∈ N , (14)

where vdck and pdc

k are the DC voltage and power set-pointsof the VSC at bus k. αk, βk, and γk are the voltage-droopcoefficients of the corresponding converter. In this paper, weassume that αk = 1. Herein, the slope of voltage-droopcharacteristics is

κk , βk =vdc

maxk− vdc

mink

pdcmaxk

− pdcmink

, (15)

and additionally, γk = −vdck − κkpdc

k .Equation (14) guarantees the optimal operation as long as

updating droop set-points is fast enough compared to powerfluctuations [30]. However, due to the limits in computationalspeed, this assumption remains valid only if changes in load-/generation are negligible. The unwanted voltage deviation,

4

caused by rapid changes, can be formulated as

∆vk = κk∆pk, (16)

where ∆vk = |vdck (t1) − vdc

k (t2)| and ∆pk = |pdck (t1) −

pdck (t2)|. We seek to obtain conservative bounds on changes

in DC voltage magnitudes, with the aim of ensuring smoothtransition to the next operating point.

An illustrative example is provided in Figure 2 to highlightthe necessity of this constraint. In Figure 2 (a), the optimizercollects the load profile at time t1. Optimizing droop set-points requires a computational time, ∆t1. During this time, anoticeable variation in load/generation, as it happens at timet2, invalidates obtained droop set-points until the subsequentupdate (at time t3 + ∆t3). Thus, droop set-points found attime t1 +∆t1 become harmful for grid operation, particularlywithin the interval [t2, t3 +∆t3].

To prevent this issue, one can underpin droop controlwith additional constraints so that the DC voltage remainswithin the pre-described boundaries. We offer the followingadditional voltage constraints, instead of (12), to guarantee asafe operation under limited load/generation volatility:

vdckmin

+∆vk = vdckmin

+ κk∆pk ≤ vdck , (17)

vdckmax−∆vk = vdc

kmax− κk∆pk ≥ vdc

k , (18)

where ∆pk denotes the power variation in the DC side. Asafe operating region can be devised such that correspondingvoltage constraints allow power variation up to a specifiedlevel. This level can be decided based on the predefinedpercentage of an existing power injection, i.e., ∆pk ≤ µkp

dck .

IV. TOPOLOGY-COGNIZANT OPF

A. Formulation of the Optimal Grid Topology

In Section III, we have modeled the operational and phys-ical characteristics of a VSC-based MTDC grid. Herein, wedevise an objective function that minimizes the total activepower loss. The topology-cognizant OPF, with the objectiveof minimizing total loss, can be formulated as

minimize 1>|N|

(G>pg −D>pd

)(19a)

subject to realsac+ pdc + a+ [b]|iac|+ [c]|iac|2 = 0 (19b)

|vacc | ≤

√3

2vdc (19c)

sac = [vacc ]([z]−1(vac

c − vacf ))∗

(19d)pac

min ≤ realsac ≤ pacmax (19e)

qacmin ≤ imagsac ≤ qac

max − [qac]|vacf | (19f)

iac = [z]−1(vacc − vac

f ) (19g)|iac| ≤ iac

max (19h)

G>pg −D>pd =real

[vacf ]([z]−1(vac

f − vacc ))∗

(19i)pg

min ≤ pg ≤ pg

max (19j)vac

fmin≤ |vac

f | ≤ vacfmax

(19k)

pdc = ~L> ~f + ~L> ~f (19l)

pdcmin ≤ pdc ≤ pdc

max (19m)

| ~f − diag~L vdcvdc> ~Y>| ≤M(1− x~

~

) (19n)

| ~f − diag ~L vdcvdc> ~Y>| ≤M(1− x~

~

) (19o)

| ~f | ≤ [f~

~

max]x~

~

(19p)

| ~f | ≤ [f~

~

max]x~

~

(19q)x~

~

min ≤ x~

~

≤ x~

~

max (19r)

vdcmin + [κ][µ]pdc ≤ vdc ≤ vdc

max − [κ][µ]pdc (19s)

variables vdc,pdc ∈ R|N|; vacc ,vac

f , iac, sac ∈ C|N|

pg ∈ R|G| ; ~f , ~f ∈ R|L| ; x~

~

∈ 0, 1|L|

where vectors qacmin, qac

max, and qac are set such that (19f)concludes (5). The topology-cognizant OPF formulation (19)suffers from (i) non-convex power flow equations (19d), (19n)and (19o), (ii) non-convex converter loss equations (19b), and(iii) the presence of binary variables, (19r), accounting for thelines’ statuses. The non-convex power flow equations corre-sponding to binary variables in (11a) and (11b) are relaxedusing disjunctive inequalities, big-M reformulation, to (19n)and (19o), respectively. Thus, binary variables are defined foreach alternative scenario, as the binary variable is either zeroor one. Nonlinear components, namely, vdcvdc> , |vac

f |, |vacf |2,

|vacc |2, diagvac

c vac∗

f and |iac| can be convexified via conicinequalities.

B. Convexification of the Problem Formulation

First, we introduce auxiliary variables W dc ∈ S|N |,φac, tac,wac

cc ,wacff ∈ R|N |, and wac

cf ∈ C|N | for nonlinearterms vdcvdc> , |iac|, |iac|2, |vac

c |2, |vacf |2, and diagvac

c vac∗

f ,respectively. The lifted problem can be formulated as

minimize 1>|N|

(G>pg−D>pd

)(20a)

subject to realsac+ pdc + a+ [b]φac + [c]tac = 0 (20b)

0 ≤ waccc ≤

3

2diagW dc (20c)

sac =([z]−1)∗ (wac

cc −waccf ) (20d)

pacmin ≤ realsac ≤ pac

max (20e)qac

min ≤ imagsac ≤ qacmax − [qac]|vac

f | (20f)

tac =[|z|]−2(waccc +wac

ff −2realwaccf ) (20g)

tac ≤ (iacmax)2 (20h)

G>pg −D>pd = real

[z]−1(wacff −wac

cf )∗

(20i)pg

min ≤ pg ≤ pg

max (20j)

(vacfmin

)2 ≤ wacff ≤ (vac

fmax)2 (20k)

pdc = ~L> ~f + ~L> ~f (20l)

pdcmin ≤ pdc ≤ pdc

max (20m)

| ~f − diag~L W dc ~Y>| ≤M(1− x~

~

) (20n)

| ~f − diag ~L W dc ~Y>| ≤M(1− x~

~

) (20o)

5

| ~f | ≤ [f~

~

max]x~

~

(20p)

| ~f | ≤ [f~

~

max]x~

~

(20q)x~

~

min ≤ x~

~

≤ x~

~

max (20r)

vdcmin + [κ][µ]pdc ≤ vdc ≤ vdc

max − [κ][µ]pdc (20s)

√tac = φac = |iac| (20t)

√[~Ldiag

W dc

] ~Ldiag

W dc

= diag

~LW dc ~L>

diag

(~L− ~L)W dc(~L− ~L)>

=(~Lvdc−~Lvdc)2

diag

(~L+ ~L)W dc(~L+ ~L)>

=(~Lvdc+~Lvdc)2

diagW dc =

(vdc)2 (20u)

√[wac

ff ]waccc = |wac

cf |,

wacff +wac

cc − 2 realwaccf = |vac

c − vacf |2,

wacff +wac

cc + 2 realwaccf = |vac

c + vacf |2,

wacff +wac

cc − 2 imagwaccf = |vac

c + ivacf |2,

wacff +wac

cc + 2 imagwaccf = |vac

c − ivacf |2,

wacff = |vac

f |2, waccc = |vac

c |2, (20v)

variables vacc ,vac

f ,waccc ,w

acff ,wac

cf , sac, iac ∈ C|N|;

W dc ∈ S|N|; φac, tac,pdc ∈ R|N|; pg ∈ R|G|;~f , ~f ∈ R|L|; x~

~

∈ 0, 1|L|

The non-convexity of (19) is circumvented by lifting itsnonlinear terms into the form of (20) while preserving theequivalency between the two formulations, with the help ofadditional constraints (20t)-(20v). The main purpose behindthis formulation is that it can be immediately convexified viatransformation of equalities in (20t)-(20v) to inequalities.

C. SOCP Relaxation

Motivated by [31], a MISOCP formulation can be readilyobtained, by relaxing (20t)-(20v) into the following conic andparabolic inequalities:√tac ≥ φac ≥ |iac| (21a)

√[~Ldiag

W dc

] ~Ldiag

W dc

≥ diag

~LW dc ~L>

diag

(~L− ~L)W dc(~L− ~L)>

≥ (~Lvdc−~Lvdc)2

diag

(~L+ ~L)W dc(~L+ ~L)>≥ (~Lvdc+~Lvdc)2

diagW dc

≥ (vdc)2 (21b)

√[wac

ff ]waccc ≥ |wac

cf |,

wacff +wac

cc − 2 realwaccf ≥ |vac

c − vacf |2,

wacff +wac

cc + 2 realwaccf ≥ |vac

c + vacf |2,

wacff +wac

cc − 2 imagwaccf ≥ |vac

c + ivacf |2,

wacff +wac

cc + 2 imagwaccf ≥ |vac

c − ivacf |2,

wacff ≥ |vac

f |2, waccc ≥ |vac

c |2, (21c)

The resulting MISOCP-relaxed topology-cognizant OPF prob-lem (20) with relaxed constraints (21a)-(21c) is compatiblewith the state-of-the-art branch-and-bound solvers which en-ables the search for binary variables.

D. Penalization

The proposed convex relaxation (21a)-(21c) can be inexactand fail to provide feasible points for the original nonconvexformulation. In order to ensure that (20t)-(20v) are satisfied,we incorporate a penalty function of the form

ρvdc,vacc ,vac

f ,iac(Wdc,vdc,wac

cc ,vacc ,w

acff ,v

acf , t

ac, iac) =

ηdc(trW dc

− 2(vdc)>vdc + ‖vdc‖22

)+ (22a)

ηacc

(1>|N|w

accc − (vac

c )∗vacc − (vac

c )∗vacc + ‖vac

c ‖22)+ (22b)

ηacf

(1>|N|w

acff − (vac

f )∗vacf − (vac

f )∗vacf + ‖vac

f ‖22)+ (22c)

ηaci

(1>|N|t

ac − (iac)∗iac − (iac)∗iac + ‖iac‖22)

(22d)

into the objective of convex relaxation, where(vdc, vac

c , vacf , i

ac) ∈ R|N | × R|C| × R|C| × R|C| can beany arbitrary initial point. As shown in [22], the properselection of penalty coefficients ηdc, ηac

c , ηacf , η

aci ≥ 0

guarantees the recovery of near-optimal feasible points. Inthe following section, we show that the simple choice ofparameters

vdc = vacc = vac

f = iac = 0, (23a)

ηdc = 10−4, ηaci = 10−5, ηac

c = ηacf = 0, (23b)

can reliably solve the original non-convex problem in practice.

V. CASE STUDIES

A. System Setup

The modified CIGRE B4 DC grid benchmark [32],equipped with switches to open/close transmission lines, isshown in Figure 3. This grid is emulated in a HIL environment,with two dSPACE DS1202 MicroLabBoxes to implementdroop controllers for individual VSCs, and two TyphoonHIL604 units to emulate VSCs and transmission lines, asshown in Figure 4. The proposed optimization algorithm runson a 16-core Xeon PC with 256 GB RAM. The TCP/IPlink between Typhoon HIL/MATLAB/dSPACE MicroLab-Boxes shares the load, set-point information, and the status ofswitching devices at every five second. Resulting SOCP andMISOCP problems are solved in the CVX v2.1 environment[33] using the conic mixed-integer solver GUROBI v8.0.1[34]. In the following studies, four time intervals, [0s, 120s],[120s, 220s], [220s, 320s], and [320s, 420s], are considered.

The rated power of each VSC is 1200 MW. Variable loadsare attached to bus 1 and bus 4. The bounds on power con-straints for AC and DC sides are pdc

mink= pac

mink= −1200 MW

and pdcmink

= pacmaxk

= 1200 MW for every VSC at busk ∈ N . The loss coefficients in (1) are ak = 2.65 × 10−5,bk = 3.7 × 10−5, and ck = 3.6 × 10−5 for every VSCat bus k ∈ N . The converter constant and nominal apparentpower in (5) are mb = 0.6 and |sac| = 1.2 pu, respectively.Phase-reactor parameters in (5) are rk = 2.5 × 10−6 andxk = 4 × 10−4 for every k ∈ N . VSC parameters are

6

3

4

5

1

2

250 km

320 km

160

km

320

km50

0 km

Energy

380 kV

VSC

132 kVSource

400 km

Fig. 3. The modified CIGRE B4 DC grid equipped with line switches. DCcable resistance for +/-400 kv is 0.0095 Ω/km.

Controller Implementation

dSPACE MicroLabBoxTyphoon HIL

Real-Time Hardware Emulation

VSC-+∆

∆

subject to (20b) - (20s) & (21a) - (21c)

VoltageControl

vdc,pdc, xpd

vdc

vdcvdc

pdc

pdc

minimize 1T ( GTpg - DTpd ) + (22)| |

Fig. 4. Topology-cognizant OPF testbed on a HIL system consisting ofreal-time hardware emulation (Typhoon HIL), controller implementation(dSPACE), and TCP/IP communication link.

icacmax = 1.0526 and vc

acmax = 1.05. The maximum modulation

factor in (2) is m = 1. The voltage bounds are 0.94 pu(352.7 kV) and 1.06 pu (402.8 kV). The lines are rated atf~

~

lmax= 0.3 pu (300 MW) for every l ∈ L. Moreover, the

big-M constant in (19) and (20) is M = 500. Safety constraintcoefficients in (20s) are µk = 10% and κk = 5% for everyVSC at bus k ∈ N . Penalty coefficients in (22) are chosenfrom (23).

B. MTDC Grid Operation with Static OPF

Static OPF refers to the problem (20) with a connectedgrid such that x~

~

lmin=x~

~

lmax=1 for every l ∈ L. If OPF results

do not update the set-points of the local droop controller, theyarrive at a feasible operating condition shown in Figure 5.The main goal of a local droop controller is to ensure stableoperation of a VSC in voltage-power tracking and meetingthe load demand. Local controllers are not concerned withthe optimal operation of the MTDC grid. Hence, the totalloss obtained via local controllers can always be reduced byconsidering the optimality conditions. The total loss obtainedvia local controllers and the static OPF are given in Table I.Upon enforcing the outcome of the static OPF, roughly around

375

377.5

380

382.5

385

Bus

Vol

tage

(kV

)

-800

-400

0

400

800

1200

Pow

er (M

W)

50 100 150 200 250 300 350 400Time (s)

456

789

Pow

er L

osse

s (M

W)

10

v1dc v 2

dc v3dc v4

dc v5dc

p 1dc p 2

dc p 3dc p 4

dc p 5dc

(a)

(b)

(c)

Fig. 5. MTDC operation with droop control under varying load: (a) DC sidevoltage variation, (b) DC side power variation, and (c) Total power losses.

TABLE ITOTAL LOSSES WITH DIFFERENT APPROACHES (MW)

Method

Time interval (s)0-120 120-220 220-320 320-420

Local droop controller 4.40 8.00 6.50 9.00

Static OPF 3.97 7.16 5.86 8.08

Topology-cognizant OPFwithout the safety constraints 3.78 6.19 5.11 7.11(20) without (20s)Topology-cognizant OPFwith the safety constraints 3.80 6.22 5.14 7.13(20) with (20s)

10% reduction in loss is reported compared to results obtainedvia local droop controllers alone. The average computationtime to solve the static OPF and update droop set-points is1.4 s.

C. MTDC Grid Operation with Topology-cognizant OPF

The problem (20), with and without the constraints in(20s), is solved to determine the optimal grid topology withthe aim of reducing the total loss. The outcome of topology-cognizant OPF problem without voltage safety limits in (20s)further reduces the total loss by 4.79%, 13.54%, 12.80%, and12.00% for the four time intervals as compared to the staticOPF scenarios. Based on the outcome of (20) (except (20s)),

7

Bus

Vol

tage

(kV

)Po

wer

(MW

)

50 100 150 200 250 300 350 400Time (s)

456789

Pow

er L

osse

s (M

W)

395

397.5

400

402.5

405

-800

-400

0

400

800

1200

3

v1dc v 2

dc v3dc v4

dc v5dc

p 1dc p 2

dc p 3dc p 4

dc p 5dc

(a)

(b)

(c)

Fig. 6. MTDC operation with static OPF under varying load: (a) DC sidevoltage variation, (b) DC side power variation, and (c) Total power losses.

x~

~

2−3 is always disconnected, while x~

~

1−4 is disconnected onlyduring [0s, 120s] time interval. Eventhough the total loss isfurther reduced, the voltage safety limits are violated at bus 5due to load fluctuations and the computation time involved inupdating droop set-points, see Figure 7 (a). Additional safetyconstraints in (20s) mitigate any voltage violation in responseto power fluctuations in between two droop set-points updatesas shown in Figure 8 (a). This safer operation comes witha slightly higher total loss compared to the case ignoring(20s); Nevertheless, it still offers remarkable reduction intotal loss as compared to the static OPF. The total lossobtained by the topology-cognizant OPF with and withoutthe constraints in (20s) are given in Table I. Total losses fordifferent loading profiles are about 0.42%, 0.57%, 0.52%, and0.65% of the total load demand for the four time intervals. Theconverter losses obtained from SOCP relaxation approach areabout 7.31%, 4.71%, 5.65%, and 4.18% of the total loss incorresponding intervals. Updating droop set-points, that aresent to VSCs every five second, takes around 2.5 s.

VI. CONCLUSION

This paper offers a convex optimization framework to solvethe grid topology-cognizant OPF problem for MTDC grids.It provides local voltage and power set-points for droop con-trollers of VSCs as well as the operational status of transmis-sion lines. Additional constraints, that sustain safe operationin response to the power fluctuation in between two droopupdates, are integrated into the proposed formulation. The

Bus

Vol

tage

(kV

)Po

wer

(MW

)

395

400

405

-800

-400

0

400

800

1200

v1dc v 2

dc v3dc v4

dc v5dc

p 1dc p 2

dc p 3dc p 4

dc p 5dc

50 100 150 200 250 300 350 400Time (s)

Pow

er L

osse

s (M

W)

3

4

5

6

7

80

0.5

1

Line

Sta

tuse

s

x1-2 x1-3 x1-4 x2-3 x3-4 x4-5

(a)

(b)

(c)

(d)

>402.8 kVv5dc397.5

402.5

Fig. 7. MTDC operation with topology-cognizant OPF disregarding voltagesafety constraints in (20s). The variations in load/generation have DC voltageat bus 5 violate the safety limit: (a) DC side voltage variation (dotted lineshows the safety limit, vdc

max = 402.8 kV), (b) DC side power variation, (c)Line statuses, and (d) Total power losses.

resulting MINLP embodies three types of non-convexity due tothe quadratic power flow equations, incorporation of switchingdecisions, and the nonlinear converter loss equations. We relaxthis problem into a tractable MISOCP that can be solvedusing standard branch-and-bound solvers. Experimental resultsverify the efficacy of the proposed method.

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