optimal power flow in multi-terminal hvdc networks for dc-system operator: constant current...

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Dr Francisco M. Gonzalez-Longatt PhD | http://fglongatt.org | Copyright © 2008-2014 1/1550th

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Dr Francisco M. Gonzalez-Longatt

@fglongatt

http://fglongatt.org/

http://www.fglongatt.org/

http://scholar.google.co.uk/citations?user=4dkpMnEAAAAJ&hl=en


http://orcid.org/0000-0002-7157-9844

http://orcid.org/0000-0002-7157-9844

http://www.scopus.com/authid/detail.url?authorId=24922792900


http://www.linkedin.com/pub/francisco-gonzalez-longatt/12/660/43b


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• The power injections (Pi)in a DC grid are controlledby the converters.

• On a MTDC grid asSupergrid, the power flowinto, or out of, eachconverter can bedynamically changedwithout anyreconfiguration of theHVDC grid.

• The purpose of this paperis to present an optimalpower flow methodconsidering DC-ISOoperation objectives.



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• The introduction of HVDCgrids brings with it majorchallenges, andopportunities.

• In this paper, DC-ISO isdefined as a private orpublic entity, and it tocoordinates, controls andmonitors the operation ofthe DC transmission systeminvolving one or severalpower park modules andone or several TSOs. DC-ISO is expected to performthe same functions as ISOs,but cover only the MTDCsystem.

DC-connected Power

Parks

Pdc,k

MTDC System

Meshed

DC

Network

Bulk generationBulk Transmission

TSO1

TSOk

TSOn

......

......

Power

Park 1

Power

Park m

......

Synchronous

Areas

Customers

DC

Independent

System

Operator

Single

Market

Service

Providers

.........

...

Grid Side

Converters

Transmission

System Operators

DC-Wide-Area

Supervision and Control Power Park

Converter

......



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• DC-ISO will uses the OPF in order to dispatch the MTDCaccording to signals provided by the pool market.

• The steady-state behaviour of a MTDC system can be described bya set of nonlinear set of the algebraic equations:

• G is the set of algebraic equations define the power balance at network

• X is state vector

• Y is the vector of independent variable.

• The state vector contains the state variables describing the state ofthe MTDC system, it contain dependent variables.

, G X Y 0



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• OPF is formulated mathematically as a general constrainedoptimization problem where set of constraints are taking inaccount.

• The most basic and general OPF formulation is based on a problemof minimization without inequality constraints as:

• Subject to:

• where f(X,Y) is the function to be optimized.

min ,f X Y

, G X Y 0



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• In this paper, system loses are located on the DC transmissionsystem and it is assumed to be the Joule heating or ohmic heating inthe cables.

• Under the previous assumption, the total losses in a MTDC systemcan be written as:

• where Pdc,i are the elements in Pdc calculated in terms of the nodalvoltages using:

• where the DC current Udc =[Udc,1, Udc,2, ...,Udc,ndc]T is the DC

voltage vector and YDC is the DC nodal admittance matrix.

,

1

dcn

losses dc i

i

f P P

X,Y

convK dc dc dc dcP = U Y U



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• Bound constraints: Lower (Xmin) and upper (Xmax) bounds limitthe components of the solution X. Bound constraints are written inthe form of:

• DC-voltage at station converters (Udc,i) are written as boundconstraints based on operational limits:

• Nonlinear equality constraints: Nonlinear inequality constraintshave the form G(X,Y) = 0, where G is a vector of constraints, onecomponent for each constraint.

• The mathematical formulation of the OPF includes a set ofnonlinear equality constraints as:

min maxX < X < X

min , maxdc iU U U

, G X Y 0



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• Linear inequality constraints: have a form as:

AieqX < Bieq

• where Aieq is an n-by-m matrix, which represents m constraints foran n-dimensional vector X. Bieq is m-dimensional.

• Linear equality constraints: have a form as:

AeqX = Beq

• where Aeq is an n-by-m matrix, which represents m’ constraints foran n-dimensional vector X. Beq is m-dimensional.

max

conv convI < I max

conv dc dc convI = Y U I

MTDC

network

1

i

ndc

Udc,1

Udc,k

j

k

Udc,ndc

Udc,i Udc,j

Ii,j

, , ,

esp

ij i i dc i dc j ijI Y U U I =

The current flowing through the cable connected between node i

and node j, Iij, is written using nodal analysis as:

where Yi,j is the correspondent element of the YDC is the DC nodal

admittance matrix, and Iijesp represents the operational current

defined by the DC-ISO for that specific branch.



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• Test systems:

GSC1

GSC2 WFC1

PWF1 = 0.95p.u

R13 = 0.045

WF1

AC1

AC2R23 =

0.052

R12 =

0.0

.07

3

①

②

③ I12 = 0.50 p.u

P-mode

V-mode

V-mode

①

②

③

④

⑤

GSC1

GSC2 WFC1

PWF1 = 0.75p.u

R14 = 0.052

WF1

AC1

AC2R24 = 0.052

GSC3AC3R 34 =

0.073

WFC2

R35 = 0.073

PWF2 = 0.85p.u

WF2

I14 = 0.3 p.u

GSC1

WFC3

AC1

R46 = 0.0500

PWF2 = 0.30p.u

R45 = 0.0250 R57 = 0.0325

PWF3 = 0.50p.u

GSC2 AC2

WFC1

PWF1 = 0.40 p.u

R14 =

0.0

10

0R

24 =

0.0

12

5 R35 =

0.0

07

5

①

② ③

WFC2

⑦ ④ ⑤ ⑥

WF1

WF3

I15 = 0.10 p.u

Test system I: 3Node+1WF+2ACTest system II: 5-Node+2WF+3AC Test System

Test system III:

7-Node+3WF+2AC Test

System



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• The proposed OPF methodology is tested considering power lossesas objective function.

• Bound constraints are considered in all simulations in order toensure a secure system operation (0.90 < Udc < 1.10 p.u).

• (i) Case I: No Linear equality constraints Case I, no currentconstraints in any under-sea cable scenario.

• (ii) Case II: Linear equality constraints is considered to ensureconstant current operation in one cable.

GSC1

GSC2 WFC1

PWF1 = 0.95p.u

R13 = 0.045

WF1

AC1

AC2R23 =

0.052

R1

2 =

0.0

.073

①

②

③ I12 = 0.50 p.u

P-mode

V-mode

V-mode

①

②

③

④

⑤

GSC1

GSC2 WFC1

PWF1 = 0.75p.u

R14 = 0.052

WF1

AC1

AC2R24 = 0.052

GSC3AC3R 34 =

0.073

WFC2

R35 = 0.073

PWF2 = 0.85p.u

WF2

I14 = 0.3 p.u

GSC1

WFC3

AC1

R46 = 0.0500

PWF2 = 0.30p.u

R45 = 0.0250 R57 = 0.0325

PWF3 = 0.50p.u

GSC2 AC2

WFC1

PWF1 = 0.40 p.u

R14 =

0.0

10

0R

24 =

0.0

12

5 R35 =

0.0

07

5

①

② ③

WFC2

⑦ ④ ⑤ ⑥

WF1

WF3

I15 = 0.10 p.u

Test system I: 3Node+1WF+2AC Test system II: 5-Node+2WF+3AC Test System Test system III: 7-Node+3WF+2AC Test System



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SIMULATION RESULTS OF OPF: TEST SYSTEM I

SIMULATION RESULTS OF OPF: TEST SYSTEM II

SIMULATION RESULTS OF OPF: TEST SYSTEM III

Node

Case I Case II

Udc

(pu)

Idc

(p.u)

Pdc

(p.u)

Udc

(pu)

Idc

(p.u)

Pdc

(p.u)

① 1.0896 -0.2315 -0.5045 1.1000 0.6434 1.4155

② 1.0896 -0.2003 -0.4365 1.0635 -1.0778 -2.2925

③ 1.1000 0.4318 0.9500 1.0935 0.4344 0.9500

Node

Case I Case II

Udc

(pu)

Idc

(p.u)

Pdc

(p.u)

Udc

(pu)

Idc

(p.u)

Pdc

(p.u)

① 1.1000 0.3000 0.6600 1.0720 -0.1274 -0.2731

② 1.0645 -0.3829 -0.8152 1.0720 -0.1274 -0.2731

③ 1.0652 -0.6516 -1.3881 1.0718 -0.4792 -1.0273

④ 1.0844 0.3458 0.7500 1.0786 0.3477 0.7500

⑤ 1.0936 0.3886 0.8500 1.1000 0.3864 0.8500

Node

Case I Case II

Udc

(pu)

Idc

(p.u)

Pdc

(p.u)

Udc

(pu)

Idc

(p.u)

Pdc

(p.u)

① 1.1000 0.1818 0.4000 1.1000 0.1818 0.4000

② 1.0999 0.1364 0.3000 1.0999 0.1364 0.3000

③ 1.0979 0.2277 0.5000 1.0974 0.2278 0.5000

④ 1.0982 0.0000 0.0000 1.0982 0.0000 0.0000

⑤ 1.0962 0.0000 0.0000 1.0957 0.0000 0.0000

⑥ 1.0862 -0.2396 -0.5205 1.0873 -0.2182 -0.4745

⑦ 1.0863 -0.3063 -0.6654 1.0850 -0.3278 -0.7114

Optimal solution are

found on all simulation

scenarios if no branch

constraints are

considered (Case I).

The participation of

grid side converters

(GSC) on DC-voltage

regulation allows the

optimal operation of

the MVSCDC

fulfilling all the

considered constraints

(Case II).



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• As expected, Case I exhibit lower power losses compared with Case II.

• The additional linear equality constraint added by constant current operation produces apredefined power flow in one branch inside the MVSCDC increasing the losses

• The constant current operation, linear equality constraint, adds stress on searching the

OPF solution which increases the total simulation time



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• This paper roughly introduces the new concept of DCIndependent System Operator, DC-ISO.

• A methodology for an optimal steady-state operation of aMTDC system based on DC-ISO objectives has been presented inthis paper.

• DC-ISO might use a path inside the MTDC as interconnectors forinternational electricity trade allowing inter TSO operation; underthis condition the current magnitude and direction in one or severalundersea cable inside the MTDC must be loaded at very specificvalue under variables conditions.

• This paper proposes the use of a type linear equality constraintsbased on nodal analysis to include this specific operational modeto the OPF.



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@fglongatt

Dr Francisco M. Gonzalez-Longatt

@fglongatt





http://orcid.org/0000-0002-7157-9844

http://orcid.org/0000-0002-7157-9844





optimal power flow in multi-terminal hvdc networks for dc-system operator: constant current...

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