optimal power flow (dc-opf and ac-opf)

Optimal Power Flow (DC-OPF and AC-OPF)

DTU Summer School 2018

Spyros Chatzivasileiadis

What is optimal power flow?

2 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 25, 2018

Optimal Power Flow (OPF)

• In its most realistic form, the OPF is a non-linear, non-convex problem,which includes both binary and continuous variables.

costs, losses, . . .

supply=demand

generation limits

voltage, line limits, etc.

• Disclaimer:

• Realistic OPF implementations include thousands of variables andconstraints• Here we focus on the most “fundamental” formulations of OPF

supply=demand

generation limits

• Disclaimer:

supply=demand

generation limits

• Disclaimer:

Outline

• Economic Dispatch

• DC-OPF

• AC-OPF

Outline

• used in Power Pools and Power Exchanges• Supply must meet demand• Generator limits

• DC-OPF

• used in Nodal and Zonal Pricing markets (e.g. US and Europe)• considers line limits and the power flows! (linearized)• only active power; no losses

• AC-OPF

•“Security-Constrained AC-OPF: ultimate goal for market software”1

• not only markets: minimize losses, optimize voltage profile, and others• full AC power flow equations• active and reactive power flow, current, voltage, losses

1M.B.Cain, R. P. O’Neill, Anya Castillo, “History of Optimal Power Flow and Formulations – Optimal Power Flow Paper 1”

Outline

• DC-OPF

• AC-OPF

Outline

• DC-OPF

• AC-OPF

Economic Dispatch

Find the cheapest generators that can cover the total demand! How?

min∑i

subject to:

PminGi≤ PGi

≤ PmaxGi∑

PGi= PD

• The Economic Dispatch does not consider any network flows or networkconstraints!

•We assume a copperplate network, i.e. a lossless and unrestricted flow ofelectricity from A to B.

Economic Dispatch

min∑i

subject to:

PminGi≤ PGi

≤ PmaxGi∑

PGi= PD

Economic Dispatch

min∑i

subject to:

PminGi≤ PGi

≤ PmaxGi∑

PGi= PD

Can we solve the economic dispatch problem without using anoptimization solver?

Yes! With the help of the merit order curve.

Can we solve the economic dispatch problem without using anoptimization solver?

Yes! With the help of the merit order curve.

The Merit-Order Curve

0 A B C D

A = PmaxG1

B = A+ PmaxG2

C = B + PmaxG3

D = C + PmaxG4

0 A B C D

powerPD

A = PmaxG1

B = A+ PmaxG2

C = B + PmaxG3

D = C + PmaxG4

0 A B C D

powerPD

• cG3 is the systemmarginal price

• G1 and G2 fullydispatched

• G4 not dispatched

• G3 partially dispatched:“marginal generator”

The Merit-Order Curve: An Example

Merit-Order of the German conventional generation in 2008

Source: Forschungsstelle fur Energiewirtschaft e. V.

Line Congestion and Marginal Generators

0 A B C D

powerPD

• Although G3 hasenough capacity,it cannot produceenough to coverthe demand dueto line congestion

• Instead G4, amore expensivegen, must producethe missing power

• In a DC-OPF context, there is no longer a single system marginal price(we will observe different nodal prices in different nodes)

0 A B C D

powerPD

0 A B C D

powerPD

DC-OPF vs Economic Dispatch

What is the difference?

DC-OPF includes the line flow constraints!

So how do I do that?

Linearized power flow equations

Vi∠θi Vj∠θj

simplified model of the line

Pij =ViVjxij

sin(θi − θj) →linearize

Pij =1

xij(θi − θj)

1 What are my assumptions for linearizing the power flow?

Vi = Vj = 1p.u and sin(θi − θj) ≈ θi − θj

2 How do I include the line flow constraint in the DC-OPF?

Vi∠θi Vj∠θj

Pij =ViVjxij

Pij =1

xij(θi − θj)

1 What are my assumptions for linearizing the power flow?

Vi = Vj = 1p.u and sin(θi − θj) ≈ θi − θj

Vi∠θi Vj∠θj

Pij =ViVjxij

Pij =1

xij(θi − θj)

1 What are my assumptions for linearizing the power flow?Vi = Vj = 1p.u and sin(θi − θj) ≈ θi − θj

DC-OPF

min∑i

subject to:PminGi≤ PGi

≤ PmaxGi∣∣∣∣ 1xij (θi − θj)

∣∣∣∣ ≤ Pij,max

B · θ = PG −PD

• The line flow constraints must include both directions!

• The DC-OPF with the standard power flow equations contains both thepower generation PG and the voltage angles θ in the vector of theoptimization variables.

DC-OPF

min∑i

subject to:PminGi≤ PGi

≤ PmaxGi∣∣∣∣ 1xij (θi − θj)

∣∣∣∣ ≤ Pij,max

B · θ = PG −PD

• The line flow constraints must include both directions!

• The DC-OPF with the standard power flow equations contains both thepower generation PG and the voltage angles θ in the vector of theoptimization variables.

Exercise

cG1 = 60 $/MWh, cG2 = 120 $/MWh

Pload = 150 MW

PmaxG1 = 100 MW, Pmax

G2 = 200 MW

X12 = 0.1 pu, X13 = 0.3 pu, X23 = 0.1 pu,BaseMVA = 100 MVA

Pmax13 = 40 MW (line limit)

1 What are the optimization variables? Form theoptimization vector

2 Formulate the objective function

3 Formulate the constraints

DC-OPF in Matlab

How would you transfer yourproblem formulation to Matlab?

How do you calculate the nodalprices?

Discussion Points

• sin δ ≈ δ• δ is in rad!

• B · θ = P

• B is in p.u.• θ is in rad, ⇒ dimensionless• P must be in p.u.

• Bus Admittance Matrix B in DC-OPF

• bij = 1xij⇒ positive

• all off-diagonal elements are non-positive (zero or negative)• all diagonal elements are positive• AC-OPF: This differs from the case where zij = rij + jxij . In that

case, it is yij = gij + jbij with bij is negative.

• If the DC-OPF does not converge, check that the admittance matrix B iscorrect!

Discussion Points

• B · θ = P

Discussion Points

• B · θ = P

Discussion Points

• B · θ = P

Some additional points...

• Nodal prices

In a market context, the nodal prices are:

• the lagrangian multipliers of the equality constraints Bθ = P• of a DC-OPF (at the moment)• with objective function the minimization of costs

• Power Transfer Distribution Factors (PTDFs)

• PTDFs are linear sensitivies that relate the line flows to the powerinjections• the DC-OPF can be formulated with respect to PTDFs• PTDFs eliminate the need of θ as optimization variable• In the zonal pricing in Europe PTDFs are used to model the flows

between the zones

Some additional points...

• Nodal prices

In a market context, the nodal prices are:

• the lagrangian multipliers of the equality constraints Bθ = P• of a DC-OPF (at the moment)• with objective function the minimization of costs

• Power Transfer Distribution Factors (PTDFs)

• PTDFs are linear sensitivies that relate the line flows to the powerinjections• the DC-OPF can be formulated with respect to PTDFs• PTDFs eliminate the need of θ as optimization variable• In the zonal pricing in Europe PTDFs are used to model the flows

between the zones

4-slide “break”

DC-OPF: linear program = convex

AC-OPF: non-linear non-convex problem in its original form⇒ recent efforts to convexify the problem

Convex vs. Non-convex Problem

Convex Problem Non-convex problem

Cost f(x)

Costf(x)

One global minimum Several local minima

Several local minima: So what?

Example: Optimal Power Flow Problem• Assume that the difference in the

cost function of a local minimumversus a global minimum is 1%

• The total electric energy cost inthe US is ≈ 400 Billion$/year

• 1% amounts to 4 billion US$ ineconomic losses per year

• Convexifying AC-OPF

1 guarantees that we find aglobal minimum or

2 at least determines how far weare from the global minimum

Costf(x)

Convexifying the Optimal Power Flow problem(OPF)

• Convex relaxations transform theOPF to a convex Semi-DefiniteProgram (SDP)

• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem2

Costf(x)

Convex Relaxation

2Javad Lavaei and Steven H Low. “Zero duality gap in optimal power flow problem”. In: IEEETransactions on Power Systems 27.1 (2012), pp. 92–10722 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 25, 2018

• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem2

Costf(x)f(x)

Convex Relaxation

• Under certain conditions, theobtained solution is the globaloptimum to the original OPFproblem2 x

Costf(x)f(x)

Convex Relaxation

Break is over...More in 1 hour and in Pascal’s talk tomorrow!

Be patient :)

AC-OPF

• Minimize

Costs, Line Losses, other?

• subject to:

AC Power Flow equations

Line Flow Constraints

Generator Active Power Limits

Generator Reactive Power Limits

Voltage Magnitude Limits

(Voltage Angle limits to improve solvability)

(maybe other equipment constraints)

Line Current Limits

Apparent Power Flow limits

Active Power Flow limits

• Optimization vector: [P Q V θ]T

AC-OPF

• Minimize

• subject to:

Line Current Limits

AC-OPF

• Minimize

• subject to:

Line Current Limits

AC-OPF

• Minimize

• subject to:

Line Current Limits

AC-OPF3

obj.function min cTPG

AC flow SG − SL = diag(V )Y∗busV

Line Current |Y line,i→jV | ≤ Iline,max

|Y line,j→iV | ≤ Iline,max

or Apparent Flow |V iY∗line,i→j,i-rowV

∗| ≤ Si→j,max

|V jY∗line,j→i,j-rowV

∗| ≤ Sj→i,max

Gen. Active Power 0 ≤ PG ≤ PG,max

Gen. Reactive Power −QG,max ≤ QG ≤ QG,max

Voltage Magnitude Vmin ≤ V ≤ Vmax

Voltage Angle θmin ≤ θ ≤ θmax

3All shown variables are vectors or matrices. The bar above a variable denotes complexnumbers. (·)∗ denotes the complex conjugate. To simplify notation, the bar denoting a complexnumber is dropped in the following slides. Attention! The current flow constraints are defined asvectors, i.e. for all lines. The apparent power line constraints are defined per line.25 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 25, 2018

AC-OPF3

∗| ≤ Si→j,max

∗| ≤ Sj→i,max

AC-OPF3

∗| ≤ Si→j,max

∗| ≤ Sj→i,max

Current flow along a line

rij + jxij

π-model of the line

It is:

yij =1

rij + jxij

ysh,i = jbij2+ other shunt

elements connected to that bus

i→ j : Ii→j = ysh,iVi + yij(Vi − Vj) ⇒ Ii→j =[ysh,i + yij −yij

] [ViVj

j → i : Ij→i = ysh,jVj + yij(Vj − Vi) ⇒ Ij→i =[−yij ysh,j + yij

] [ViVj

rij + jxij

It is:

yij =1

rij + jxij

] [ViVj

rij + jxij

It is:

yij =1

rij + jxij

] [ViVj

]26 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 25, 2018

Line Admittance Matrix Yline

• Yline is an L×N matrix, where L is the number of lines and N is thenumber of nodes

• if row k corresponds to line i− j:

• Yline,ki = ysh,i + yij• Yline,kj = −yij

• yij =1

rij + jxijis the admittance of line ij

• ysh,i is the shunt capacitance jbij/2 of the π-model of the line

•We must create two Yline matrices. One for i→ j and one for j → i

Bus Admittance Matrix Ybus

Si = ViI∗i

Ii =∑k

Iik,where k are all the buses connected to bus i

Example: Assume there is a line between nodes i−m, and i− n. It is:

Ii = Iim + Iin

= (yi→msh,i + yim)Vi − yimVm + (yi→n

sh,i + yin)Vi − yinVn= (yi→m

sh,i + yim + yi→nsh,i + yin)Vi − yimVm − yinVn

Ii = [ysh,im + yim + ysh,in + yin︸︷︷︸Ybus,ii

−yim︸︷︷︸Ybus,im

−yin︸︷︷︸Ybus,in

][Vi Vm Vn]T

Bus Admittance Matrix Ybus

• Ybus is an N ×N matrix, where N is the number of nodes

• diagonal elements: Ybus,ii = ysh,i +∑

k yik, where k are all the busesconnected to bus i

• off-diagonal elements:

• Ybus,ij = −yij if nodes i and j are connected by a line4

• Ybus,ij = 0 if nodes i and j are not connected

• yij =1

rij + jxijis the admittance of line ij

• ysh,i are all shunt elements connected to bus i, including the shuntcapacitance of the π-model of the line

4If there are more than one lines connecting the same nodes, then they must all be added toYbus,ij , Ybus,ii, Ybus,jj .29 DTU Electrical Engineering Optimal Power Flow (DC-OPF and AC-OPF) Jun 25, 2018

AC Power Flow Equations

Si = ViI∗i

= ViY∗

busV∗

For all buses S = [S1 . . . SN ]T :

Sgen − Sload = diag(V )Y ∗busV∗

optimal power flow (dc-opf and ac-opf)

Documents