9-11September2018- 50th NorthAmericanPowerSymposium

Distribution Locational Marginal Pricing (DLMP) forMultiphase Systems

Presentation at 50th North American Power Symposium

Ibrahim AlsalehElectrical Engineering Department

University of South Florida

Dr. Lingling FanElectrical Engineering Department

University of South Florida

Septemper 10, 2018

9-11September2018- 50th NorthAmericanPowerSymposium

OutlineIntroduction

MotivationObjectives

Formulation of 3-φ Branch Flow ModelOhm’s LawPower BalancePSD and Rank-1 MatrixDG/SVC capacity and system limitsDSO Optimization Problem and Convexification

Distribution Locational Marginal Pricing (DLMP)Case Studies

Passive Network: Only substation and SVC (utility)Active Network: Inclusion of DGs

Tightness of SDP RelaxationReferences

9-11September2018- 50th NorthAmericanPowerSymposium

OutlineIntroduction

MotivationObjectives

Formulation of 3-φ Branch Flow ModelOhm’s LawPower BalancePSD and Rank-1 MatrixDG/SVC capacity and system limitsDSO Optimization Problem and Convexification

Distribution Locational Marginal Pricing (DLMP)Case Studies

Passive Network: Only substation and SVC (utility)Active Network: Inclusion of DGs

Tightness of SDP RelaxationReferences

9-11September2018- 50th NorthAmericanPowerSymposium

MotivationI The unprecedented reformation that the modern distribution network is undergoing

(will undergo) because of the increased deployment of distributed energy resources(DERs).

I Although DERs are immensely beneficial to the system, higher penetrations with ill-managed operation could fail to deliver the desired outcome, causing sharp voltagefluctuations, power flow congestion and supply-demand imbalance.

I The lack of adopting a convex multiphase power-flow distribution model that ex-hibits the practical system structure in the literature [4]-[10].

I The fact that DERs are currently seen as if they were connected to the T-D interface,disregarding the distribution.

I The need to incentivize the various resources to respond according to: i) loadingat the phase of installation ii) their location (loss compensation), and iii) theircapacity.

9-11September2018- 50th NorthAmericanPowerSymposium

Objectives

I Design an electricity market for distribution that adopts locational marginal pricingto quantify and analyze signals sent to future-market for each phase.

I The optimization problem is centrally performed by a non-profit entity -distributionsystem operator (DSO), that to collect bids from participants including ISO atthe wholesale level, control volt-var and DER assets, and settle the market, whileabiding by the system physical and security constraints. .

I Clear the market with a day-ahead point forecast of loading and substation/DERsupply.

I Leverage a convex multiphase distribution model to guarantee a global optimumsolution.

9-11September2018- 50th NorthAmericanPowerSymposium

Distribution Electricity Market

DG Owner

Prosumer ConsumerEV StationStorage

Microgrid

Distribution MarketDay ahead

Real-time Balancing

Wholesale Market

Generation ISODSO

CommunicationPower Line

Figure: Distribution electricity market

9-11September2018- 50th NorthAmericanPowerSymposium

OutlineIntroduction

MotivationObjectives

Formulation of 3-φ Branch Flow ModelOhm’s LawPower BalancePSD and Rank-1 MatrixDG/SVC capacity and system limitsDSO Optimization Problem and Convexification

Distribution Locational Marginal Pricing (DLMP)Case Studies

Passive Network: Only substation and SVC (utility)Active Network: Inclusion of DGs

Tightness of SDP RelaxationReferences

9-11September2018- 50th NorthAmericanPowerSymposium

Variables 3-φ Branch Flow Model

DG Owner

Prosumer ConsumerEV StationStorage

Microgrid

Distribution MarketDay ahead

Real-time Balancing

Wholesale Market

Generation ISODSO

CommunicationPower Line

𝑎 𝑎

𝑏 𝑏

𝑐 𝑐

𝜌(𝑖) 𝑖

𝑠𝜌(𝑖) 𝑠𝑖

𝑣𝜌(𝑖)𝑎𝑎 𝑣𝜌(𝑖)

𝑎𝑏 𝑣𝜌(𝑖)𝑎𝑐

𝑣𝜌(𝑖)𝑏𝑎 𝑣𝜌(𝑖)

𝑏𝑏 𝑣𝜌(𝑖)𝑏𝑐

𝑣𝜌(𝑖)𝑐𝑎 𝑣𝜌(𝑖)

𝑐𝑏 𝑣𝜌(𝑖)𝑐𝑐

𝑣𝑖𝑎𝑎 𝑣𝑖

𝑎𝑏 𝑣𝑖𝑎𝑐

𝑣𝑖𝑏𝑎 𝑣𝑖

𝑏𝑏 𝑣𝑖𝑏𝑐

𝑣𝑖𝑐𝑎 𝑣𝑖

𝑐𝑏 𝑣𝑖𝑐𝑐

𝑧𝑖𝑎𝑎 𝑧𝑖

𝑎𝑏 𝑧𝑖𝑎𝑐

𝑧𝑖𝑏𝑎 𝑧𝑖

𝑏𝑏 𝑧𝑖𝑏𝑐

𝑧𝑖𝑐𝑎 𝑧𝑖

𝑐𝑏 𝑧𝑖𝑐𝑐

ℓ𝑖 , 𝑆𝑖

Figure: Variables in SDP relaxation of three-phase OPF.

9-11September2018- 50th NorthAmericanPowerSymposium

Ohm’s Law

The following equations are defined for branch ρ(i)→ i, assuming Φi = Φρ(i) [1]

Vi =Vρ(i) − ziIi (1)

By multiplying both sides by their Hermitian transposes, and definingvi = ViV

Hi , vρ(i) = Vρ(i)V

Hρ(i), Si = Vρ(i)I

Hi and `i = IiI

Hi , then (1)

can be re-written as

vi = vρ(i) − (SizHi + ziSHi ) + zi`iz

Hi ∀ i ∈ N+ (2)

Vi & Ii ∈ C|Φi|.

zi ∈ C|Φi|×|Φi |.

Diagonal entreeof vi ∈ C|Φi|×|Φi |and`i ∈ C|Φi|×|Φi |are squaredmagnitudes andoff-diagonal arecomplex mutualelements.

9-11September2018- 50th NorthAmericanPowerSymposium

Power Balance

For each ρ(i)→ i→ k where k ∈ δ(i), to interpret the power balance at i, (1) ismultiplied by IHi

ViIHi = Vρ(i)I

Hi − ziIiIHi (3)

Vi( ∑k∈δ(i)

IHk − IHi inj)

= Si − zi`i (4)

As a result, the power balance at bus i is the diagonal of (4)∑k∈δ(i)

diag(Sk)− si = diag(Si − zi`i) ∀ i ∈ N+ (5)

where

si = sgi + ssvci − sLi (6)

Ii inj: Currentinjection.

si ∈ C|Φi|: Netpower injection .

L: Load.

g: PV/wind DG.

svc: Static-Varcompensator.

9-11September2018- 50th NorthAmericanPowerSymposium

PSD and Rank-1 Matrix

To set the relationship among the variables, the following positive and rank one matrixis defined

Xi =[Vρ(i)Ii

] [Vρ(i)Ii

]H=[vρ(i) SiSHi `i

]

Xi � 0 ∀ i ∈ N+ (7)rank(Xi) = 1 ∀ i ∈ N+ (8)

9-11September2018- 50th NorthAmericanPowerSymposium

Distributed GeneratorsIt is essential that inverters of both PVs and wind be equipped with curtailmentcapability so as to dispatch an appropriate amount of generated active power formarket clearing. The set of inequality limits constraints, DG = PV +WT , is asfollows

DG ={sgi ∈ C|Φi| | 0 ≤ real(sgi ) ≤ Pgi

− a real(sgi ) ≤ imag(sgi ) ≤ a real(sgi )} ∀ i ∈ Ng

(9)

a =√

1− PF2/PF (10)

where

P gφi = ωφ( ∑i∈N+

∑φ∈|Φi|

real(sLφi )/|PV| × |Φi|)∀ i ∈ DG (11)

P gi : Themaximum DGpowergeneration.

a: Fixesinverter’s powerfactor

ω ∈ R|Φi|:Penetration level.

9-11September2018- 50th NorthAmericanPowerSymposium

Static-Var Compensators and Voltage Limits

Voltage profile can be improved by dispatching SVCs to either generate or absorbreactive powers.

SVC ={ssvci ∈ C|Φi| | real(ssvc

i ) = 0− q ≤ imag(ssvc

i ) ≤ q} ∀ i ∈ Nsvc(12)

Except for the substation node (v0 = V0VH

0 ), ±5% of the nominal voltage areenforced as bounds on each element of the diagonal voltage squares.

V 2 ≤ diag(vi) ≤ V2 ∀ i ∈ N+ (13)

9-11September2018- 50th NorthAmericanPowerSymposium

DSO Optimization Problem

The objective is to minimize market participants’ generation costs ahead of a day(|T | = 24)

minvi,`i,Si,s

gi ,s

svci

f =∑t∈T

∑φ∈Φi

(σPs real(S

φφt,1 ) + σQs imag(Sφφt,1 )

+∑i∈Ng

σPg real(sgφt,i ) + σQg imag(sgφt,i )

+∑i∈Nsvc

σsvcimag(ssvcφt,i )

)s. t. v0 = V0V

H0

(2), (5), (7), (8), (9), (12), (13)

(14)

where σ denotes the generation bidding price with superscripts P and Q for active andreactive power, while St,1 is the power flow from the substation.

9-11September2018- 50th NorthAmericanPowerSymposium

Convexification

I The non-convex optimization problem in (14) is convexified by removing(relaxing) the rank constraint (8). Thus, an SDP-relaxed problem is obtained.

I In [1], it has been shown that a tight relaxation holds for most IEEE distributionfeeders. For validation, a tightness check will be conducted for the case studies.

9-11September2018- 50th NorthAmericanPowerSymposium

OutlineIntroduction

MotivationObjectives

Formulation of 3-φ Branch Flow ModelOhm’s LawPower BalancePSD and Rank-1 MatrixDG/SVC capacity and system limitsDSO Optimization Problem and Convexification

Distribution Locational Marginal Pricing (DLMP)Case Studies

Passive Network: Only substation and SVC (utility)Active Network: Inclusion of DGs

Tightness of SDP RelaxationReferences

9-11September2018- 50th NorthAmericanPowerSymposium

Distribution Locational Marginal Pricing (DLMP)The Lagrangian function of the general form [2] for the overall single-period problem,with emphasis on the power balance equation, is

L = f(x) +∑i∈F

λfi fi(x) +∑i∈H

µhi hi(x)

+∑i∈N+

∑φ∈|Φi|

λpφi real

( ∑k∈δ(i)

Sφφk − sφi − S

φφi + (zi`i)φφ

)+∑i∈N+

∑φ∈|Φi|

λqφi imag

( ∑k∈δ(i)

Sφφk − sφi − S

φφi + (zi`i)φφ

) (15)

The partial derivative of (15) w.r.t real(sLφi ) and imag(sLφi ) result in

A-DLMP : ∂L∂ real(sLφi )

= λpφi , R-DLMP : ∂L

∂ imag(sLφi )= λqφ

i (16)

Each A-DLMP and R-DLMP accumulates an energy price, a loss price, and a congestion price.

9-11September2018- 50th NorthAmericanPowerSymposium

OutlineIntroduction

MotivationObjectives

Formulation of 3-φ Branch Flow ModelOhm’s LawPower BalancePSD and Rank-1 MatrixDG/SVC capacity and system limitsDSO Optimization Problem and Convexification

Distribution Locational Marginal Pricing (DLMP)Case Studies

Passive Network: Only substation and SVC (utility)Active Network: Inclusion of DGs

Tightness of SDP RelaxationReferences

9-11September2018- 50th NorthAmericanPowerSymposium

IEEE 37-bus Feeder and Profiles of Load,PV and wind from CAISO [3]

1

2 3 4

5

6

7

8

9

10

11

12

13

14

15

16171819

20 21

22 2325

2629

27

30 31 32 33

34

35

36

0

WT24

WT8

PV4

PV18

PV33

SVC28

2 4 6 8 10 12 14 16 18 20 22 24

Time (h)

0

20

40

60

80

100

Nor

mal

ized

Pro

file

(%)

DemandPVWTNet Demand

9-11September2018- 50th NorthAmericanPowerSymposium

Passive Network

(a) (b) (c)

(d) (e) (f)

Figure: (a)-(c) Active-power DLMPs, and (d)-(f) reactive-power DLMPs for passive network.

9-11September2018- 50th NorthAmericanPowerSymposium

Passive Network

0 5 10 15 20 25 30 35

15

15.5

16

16.5

3

3.1

3.2

3.3

R-D

LMP

($/k

Varh)

A-D

LMP

($/k

Wh)

Bus

BC

ABC

R-DLMPA

A-DLMP

Figure: A-DLMPs and R-DLMPs at peak hour 13:00.I The system runs as a passive distribution network, and only includes the SVC with

bidding prices σPs =15$/kWh and σQs = σsvc =3$/kVarh.I A common trend of ADLMPs and RDLMPs is that they gradually increase as buses locate

farther from the substation.I A-DLMPs among the three phases differ notably because of the unbalanced loads and line

impedances (losses).I R-DLMPs increase in a similar trend except at buses near the SVC at bus 28.

9-11September2018- 50th NorthAmericanPowerSymposium

Active Network

(a) (b) (c)

(d) (e) (f)

Figure: (a)-(c) Active-power DLMPs, and (d)-(f) reactive-power DLMPs for active network.

9-11September2018- 50th NorthAmericanPowerSymposium

Active Network

0 5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

0

0.05

0.1

0.15

0.2

0.25

0.3

q ($/kVarh)

P ($

/kWh)

Bus

BC

ABC

qAp

Figure: Nodal change in marginal prices at 13:00 between passive and active networks.I PV- and wind-based DGs participate in the market with the same bidding prices as the

substation and 40% penetration.I DLMPs change with the participation of DGs, mostly at times when DGs produce

excessive power (peak).I A lower net demand is viewed as light loading, and thereby it cuts down on DLMPs.I manifests the change in marginal prices at 13:00, where ∆p and ∆q are the difference of

DLMPs between the cases and this case. The change is the DG contribution to reducinglosses, especially those of remote lines, and alleviating the SVC binding cost.

9-11September2018- 50th NorthAmericanPowerSymposium

Active NetworkTable: DG Active-power Output and A-DLMPs at 13:00

A B C

kW $/kWh kW $/kWh kW $/kWh

Sub. 144.9 15 71.3 15 371 15PV4 109.2 15.0154 103 14.9972 109.2 15.0567PV18 109.2 15.0494 76.7 14.9983 109.2 15.0925PV33 109.2 15.1776 54.7 15.0000 109.2 15.2062WT8 8.8 15.0000 141.3 15.0119 94.9 15.0000WT24 141.3 15.0868 100 15.0000 141.3 15.1399

I DGs are burdened at phase A and C because of the phase heavy loading, increasinglcongestion costs predominantly at distant buses.

I DGs at phase B (lightly loaded) mostly curtails its output so as to balance supply withdemand, except at WT8 due to the large load at bus 10.

9-11September2018- 50th NorthAmericanPowerSymposium

Lower DG Bidding Prices

0 5 10 15 20 25 30 35

Bus

14.6

14.8

15

15.2

A-D

LMP

($/

kWh)

2.8

2.9

3

3.1

R-D

LMP

($/

kVarh)

BC

ABC

R-DLMPA

A-DLMP

Figure: DLMPs at peak PV, 13:00.

I Setting σPg =$13/kWh and σQg =2$/kVarh results in reduced DLMPs in general, and theyare minimum at buses of DG installation.

I This incentivizes DG owners to increase their generation capacity to participate inimbalance and loss reduction for the next increment of load.

9-11September2018- 50th NorthAmericanPowerSymposium

Total Daily Consumer Payment w.r.t. Penetration Level

0 10 20 30 40 50 60 70 80 90 100760

765

770

775

780

785

74.5

75

75.5

76

76.5

77

77.5

q

p

p ($)

q ($)

Level of Penetration (%)

103 103

Figure: Change in total daily payment with DG penetrations.The figure shows the decay of the total consumer?s payment for the entire day w.r.t.incremental DG penetration.

Λp =∑t∈T

∑i∈N +

∑φ∈Φ

λpφt,ireal(s

Lφt,i )

Λq =∑t∈T

∑i∈N +

∑φ∈Φ

λqφt,i imag(sLφt,i )

9-11September2018- 50th NorthAmericanPowerSymposium

OutlineIntroduction

MotivationObjectives

Formulation of 3-φ Branch Flow ModelOhm’s LawPower BalancePSD and Rank-1 MatrixDG/SVC capacity and system limitsDSO Optimization Problem and Convexification

Distribution Locational Marginal Pricing (DLMP)Case Studies

Passive Network: Only substation and SVC (utility)Active Network: Inclusion of DGs

Tightness of SDP RelaxationReferences

9-11September2018- 50th NorthAmericanPowerSymposium

Tightness of SDP RelaxationThe rank is examined by computing the devision of the second largest eigenvalue bythe largest eigenvalue, |eig2/eig1|, where eig1 > eig2 > 0. Smaller ratios indicate thesolution proximity to being rank one. The maximum ratio is computed.

Tightness = max(∑t∈T

∑eig∈X∗

t,i

|eig2/eig1|)

Table: Tightness of Numerical Solutions

Case Tightness

1 5.1615e-072 6.1587e-073 4.9779e-07

Since the overall solution ratios satisfy sufficiently small values,|eig2/eig1| ≤ 6.1587× 10−7, the SDP relaxation is tight.

9-11September2018- 50th NorthAmericanPowerSymposium

OutlineIntroduction

MotivationObjectives

Formulation of 3-φ Branch Flow ModelOhm’s LawPower BalancePSD and Rank-1 MatrixDG/SVC capacity and system limitsDSO Optimization Problem and Convexification

Distribution Locational Marginal Pricing (DLMP)Case Studies

Passive Network: Only substation and SVC (utility)Active Network: Inclusion of DGs

Tightness of SDP RelaxationReferences

9-11September2018- 50th NorthAmericanPowerSymposium

References I

[1] L. Gan and S. H. Low, “Convex relaxations and linear approximation for optimalpower flow in multiphase radial networks,” in Power Systems ComputationConference (PSCC), 2014. IEEE, 2014, pp. 1-9.

[2] S. Boyd and L. Vandenberghe, Convex optimization. Cambridge university press,2004.

[3] “California iso,”(last accessed on April 17th, 2018). [Online]. Available:http://www.caiso.com

[4] P. M. Sotkiewicz and J. M. Vignolo, “Nodal pricing for distribution networks:efficient pricing for efficiency enhancing dg,” IEEE Transactions on PowerSystems, vol. 21, no. 2, pp. 1013-1014, 2006.

[5] R. Li, Q. Wu, and S. S. Oren, “Distribution locational marginal pricing foroptimal electric vehicle charging management,” IEEE Transactions on PowerSystems, vol. 29, no. 1, pp. 203-211, 2014.

9-11September2018- 50th NorthAmericanPowerSymposium

References II

[6] S. Huang, Q. Wu, S. S. Oren, R. Li, and Z. Liu, “Distribution locational marginalpricing through quadratic programming for congestion management in distributionnetworks,” IEEE Transactions on Power Systems, vol. 30, no. 4, pp. 2170-2178,2015.

[7] N. Li, “A market mechanism for electric distribution networks,” in Decision andControl (CDC), 2015 IEEE 54th Annual Conference on. IEEE, 2015, pp.2276-2282.

[8] L. Bai, J. Wang, C. Wang, C. Chen, and F. F. Li, “Distribution locationalmarginal pricing (dlmp) for congestion management and voltage support,” IEEETransactions on Power Systems, 2017.

[9] R. Yang and Y. Zhang, “Three-phase ac optimal power flow based distributionlocational marginal price, ” in Power & Energy Society Innovative Smart GridTechnologies Conference (ISGT), 2017 IEEE. IEEE, 2017, pp. 1-5.

9-11September2018- 50th NorthAmericanPowerSymposium

References III

[10] E. DallAnese, H. Zhu, and G. B. Giannakis, “Distributed optimal power flow forsmart microgrids, ” IEEE Transactions on Smart Grid, vol. 4, no. 3, pp.1464-1475, 2013.