Distribution Locational Marginal Pricing (DLMP) for ...
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.