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Power Divider
Sairaj Dhople
Department of Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA
Funding Acknowledgments NSF and MiNnesota’s Discover, Research and InnoVation Economy
January 28, 2016
Sairaj Dhople (UMN) Power Divider January 28, 2016 1 / 33
Acknowledgment
Yu (Christine) Chen Dept. of ECE, The University of British Columbia, Vancouver, Canada
Sairaj Dhople (UMN) Power Divider January 28, 2016 2 / 33
A Note
Paper to appear in the IEEE Transactions on Power Systems
Sairaj Dhople (UMN) Power Divider January 28, 2016 3 / 33
Outline
Background
Current Divider
Power Divider
Special Cases
Applications
Sairaj Dhople (UMN) Power Divider January 28, 2016 4 / 33
Key Question
How do real & reactive power injections map to real & reactive flows?
Why is this a hard problem?
Nonlinear constant-power constraints
Operating point dependence
Why would it be useful?
Transmission-network cost allocation
Power tracing
Sairaj Dhople (UMN) Power Divider January 28, 2016 5 / 33
Contributions over methods in the literature
Analytical, closed-form expressions
Valid over entire operating regime
Agnostic to location or specification of slack bus
Previous Work
Previous approaches are numerical or rely on linear approximations
Numerical methods [Kirschen ‘97, Bialek ‘98, Wollenberg ‘01]
Utilizing current flows as proxies for power flows [Conejo ‘07]
Generation shift distribution factors [Rudnick ‘95, Silva ‘13]
Sairaj Dhople (UMN) Power Divider January 28, 2016 6 / 33
Previous Work
Previous approaches are numerical or rely on linear approximations
Numerical methods [Kirschen ‘97, Bialek ‘98, Wollenberg ‘01]
Utilizing current flows as proxies for power flows [Conejo ‘07]
Generation shift distribution factors [Rudnick ‘95, Silva ‘13]
Contributions over methods in the literature
Analytical, closed-form expressions
Valid over entire operating regime
Agnostic to location or specification of slack bus
Sairaj Dhople (UMN) Power Divider January 28, 2016 7 / 33
Power System Model & Notation
Connected system with N buses, Admittance matrix Y
Complex-power bus injections: S = [S1, . . . , SN ]T , SP = PP + jQP
Voltage phasor, V : |V | = [|V |1, . . . , |V |N ]T; θ = [θ1, . . . , θN ]
T
Flow on line (m, n): I(m,n), P(m,n), Q(m,n)
Useful notation: θm = θm1N − θ
Useful notation: ⎡ ⎤1 |V |1
0 0
0
cos θm ⎢⎣ ⎥⎦
cos(θ1−θ2) 0diag , m = 1, N = 3 = |V |2
0|V | cos(θ1−θ3)0 |V |3
Sairaj Dhople (UMN) Power Divider January 28, 2016 8 / 33
Current Divider
Contributions of current injections, I = [I1, I2, . . . , IN ]T, to the current
flow on line (m, n), I(m,n)
I(m,n) = ymn(Vm − Vn) + ymVm = T + yme T Vymnemn m
T T Y −1I= ymnemn + ymem
= (α(m,n) + jβ(m,n))TI.
Current divider only depends on network and not operating point
α(m,n), β(m,n) ∈ RN : topology and impedances of network
Sairaj Dhople (UMN) Power Divider January 28, 2016 9 / 33
Power Divider
How do real & reactive power injections map to real & reactive flows?
Contributions of real-power injections, P = [P1, P2, . . . , PN ]T, and
reactive-power injections Q = [Q1, Q2, . . . , QN ]T to
The real-power flow on line (m, n), P(m,n)
The reactive-power flow on line (m, n), Q(m,n)
Sairaj Dhople (UMN) Power Divider January 28, 2016 10 / 33
Power Divider
Contributions of nodal injections to real- and reactive-power flow on (m, n)
P(m,n) = |V |m
( u T (m,n)P − v T
(m,n)Q ,( (1)
Q(m,n) = |V |m u T (m,n)Q + v T
(m,n)P . (2)
In (1)–(2), u(m,n), v(m,n) ∈ RN are given by
cos θm sin θm
u(m,n) = diag |V |
α(m,n) + diag |V |
β(m,n), (3)
sin θm cos θm
v(m,n) = diag |V |
α(m,n) − diag |V |
β(m,n). (4)
Sairaj Dhople (UMN) Power Divider January 28, 2016 11 / 33
Power Divider
Contributions of nodal injections to real- and reactive-power flow on (m, n)
P(m,n)
Q(m,n)
= |V |m
(u T (m,n)P − v T
(m,n)Q
= |V |m
(u T (m,n)Q + v T
(m,n)P
,
.
(1)
(2)
In (1)–(2), u(m,n), v(m,n) ∈ RN are given by
cos θm sin θm
u(m,n) = diag |V |
α(m,n) + diag |V |
β(m,n), (3)
sin θm cos θm
v(m,n) = diag |V |
α(m,n) − diag |V |
β(m,n). (4)
Sairaj Dhople (UMN) Power Divider January 28, 2016 12 / 33
Power Divider
Contributions of nodal injections to real- and reactive-power flow on (m, n)
P(m,n)
Q(m,n)
= |V |m
(u T (m,n)P − v T
(m,n)Q
= |V |m
(u T (m,n)Q + v T
(m,n)P
,
.
(1)
(2)
In (1)–(2), u(m,n), v(m,n) ∈ RN are given by
cos θm sin θm
u(m,n) = diag |V |
α(m,n) + diag |V |
β(m,n), (3)
sin θm cos θm
v(m,n) = diag |V |
α(m,n) − diag |V |
β(m,n). (4)
Sairaj Dhople (UMN) Power Divider January 28, 2016 13 / 33
Power Divider
Contributions of nodal injections to real- and reactive-power flow on (m, n) (T TP(m,n) = |V |m u(m,n)P − v(m,n)Q , (1) (T TQ(m,n) = |V |m u(m,n)Q + v(m,n)P . (2)
From the expressions, we see that the flows are
Linear functions of real- and reactive-power injections: P , Q
Nonlinear functions of the voltage profile: θ, |V | Embed network-topology information: α(m,n), β(m,n)
Sairaj Dhople (UMN) Power Divider January 28, 2016 14 / 33
Minimal Realization
Contributions of nodal injections to real- and reactive-power flow on (m, n)
P(m,n)
Q(m,n)
= |V |m
(u T (m,n)P − v T
(m,n)Q
= |V |m
(u T (m,n)Q + v T
(m,n)P
,
.
(1)
(2)
In (1)–(2), u(m,n), v(m,n) ∈ RN are given by
cos θm sin θm
u(m,n) = diag |V |
sin θm
α(m,n) + diag |V |
cos θm
β(m,n),
v(m,n) = diag |V |
α(m,n) − diag |V |
β(m,n).
This suggests that voltage information from all buses is required
Sairaj Dhople (UMN) Power Divider January 28, 2016 15 / 33
Minimal Realization
Kron reduction: Only generator- and load-bus information is needed
Sairaj Dhople (UMN) Power Divider January 28, 2016 16 / 33
Some Special Cases
Lossless networks: Y = jB, β(m,n) = 0N
cos θm sin θm
P(m,n) = |V |mα(T m,n) diag P − diag Q .
|V | |V |
Small angle differences: θi − θk ≈ 0, ∀i, k
θm1NP(m,n) = |V |mα(
T m,n) diag P − diag Q .
|V | |V |
Uniform (unity) voltage profile: |V |k = 1, ∀k
= αT (P − diag (θm) Q) .P(m,n) (m,n)
Decoupling: P >> diag (θm) Q
= αT P. P(m,n) (m,n)
Sairaj Dhople (UMN) Power Divider January 28, 2016 17 / 33
Some Special Cases
Lossless networks: Y = jB, β(m,n) = 0N
cos θm sin θm
P(m,n) = |V |mα(T m,n) diag P − diag Q .
|V | |V |
Small angle differences: θi − θk ≈ 0, ∀i, k
θm1NP(m,n) = |V |mα(
T m,n) diag P − diag Q .
|V | |V |
Uniform (unity) voltage profile: |V |k = 1, ∀k
= αT (P − diag (θm) Q) .P(m,n) (m,n)
Decoupling: P >> diag (θm) Q
= αTP(m,n) (m,n)P.
Sairaj Dhople (UMN) Power Divider January 28, 2016 18 / 33
Confirms intuition that current flows can serve as proxies for power flows
Decoupled Power Flow: Connection to Current Divider
Real-power flow on (m, n) as a function of real-power injections
P(m,n) = αT P (1)(m,n)
for the case when we consider: i) lossless networks, ii) vanishingly small angle differences, and iii) uniform (unity) voltage profile
Current flow on (m, n) as a function of current injections
I(m,n) (m,n)I (2)= αT
for the case when we consider lossless networks
Sairaj Dhople (UMN) Power Divider January 28, 2016 19 / 33
Decoupled Power Flow: Connection to Current Divider
Real-power flow on (m, n) as a function of real-power injections
P(m,n) = αT P (1)(m,n)
for the case when we consider: i) lossless networks, ii) vanishingly small angle differences, and iii) uniform (unity) voltage profile
Current flow on (m, n) as a function of current injections
= αT I (2)I(m,n) (m,n)
for the case when we consider lossless networks
Confirms intuition that current flows can serve as proxies for power flows
Sairaj Dhople (UMN) Power Divider January 28, 2016 20 / 33
Decoupled Power Flow: Connection to DC Power Flow
Flow on line (m, n) according to power divider
P(m,n) = αT (m,n)P (1)
Flow on line (m, n) according to DC power flow
P(m,n) = −bmn(θm − θn) (2)
(1) boils down to (2) when
Shunt susceptance terms are neglected
(An admittedly artefactual) slack bus is assigned
Sairaj Dhople (UMN) Power Divider January 28, 2016 21 / 33
Case Studies
Circulating power flow
Transmission-network allocation
Transmission-loss allocation
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Case Study: Circulating Power Flow
CPS: Constant Power Source, CPL: Constant Power Load, F: Forward, R: Reverse
Sairaj Dhople (UMN) Power Divider January 28, 2016 23 / 33
Case Study: Circulating Power Flow
CPS: Constant Power Source, CPL: Constant Power Load, F: Forward, R: Reverse Sairaj Dhople (UMN) Power Divider January 28, 2016 24 / 33
Case Study: Transmission-network Allocation
87
4
3
9
14
1011
13
6
5
12
2
1
Active-pow
erInjection[%
]
0
10
20
30
40
Reactive-pow
erInjection[%
]
-1.2
-0.7
-0.2
0.3
0.8
1.3
87
4
3
9
14
1011
13
6
5
12
2
1
Active-pow
erInjection[%
]
-8.7
-3.7
1.3
6.3
11.3
16.3
Reactive-pow
erInjection[%
]
-10
-0
10
20
30
P(6,12) Q(6,12)
Impact of bus active-power injections on line active-power flows is more dominant Buses 12 and 13 active-power injection contributions to Q(6,12) are 20.65% and −9.41%, respectively Both the active- and reactive-power injections in the network contribute to the reactive-power flow Sairaj Dhople (UMN) Power Divider January 28, 2016 25 / 33
Case Study: Transmission-loss Allocation
“[...] system transmission losses are a nonseparable, nonlinear function of the bus power injections which makes it impossible to divide the system losses into the sum of terms, each one uniquely attributable to a generation or load.”
Conejo, Galiana, and Kockar, “Z-bus loss allocation,” TPWRS, 2001.
Sairaj Dhople (UMN) Power Divider January 28, 2016 26 / 33
Sketch of derivation
L(m,n) = Re�y−1mn(I(m,n) − ymVm)∗(I(m,n) − ymVm)
�(1)
= Re {(Vm − Vn)y∗mn(Vm − Vn)
∗ + Vmy∗mV ∗m + Vny
∗nV
∗n } (2)
= Re{VmI∗(m,n) + VnI∗(n,m)} (3)
= P(m,n) + P(n,m) (4)
Case Study: Transmission-loss Allocation
Losses on line (m, n) decomposed into nodal contributions ( (T T T TL(m,n) = |Vm|u(m,n) + |Vn|u(n,m) P + |Vm|v(m,n) + |Vn|v(n,m) Q.
Sairaj Dhople (UMN) Power Divider January 28, 2016 27 / 33
Sketch of derivation
L(m,n) = Re�y−1mn(I(m,n) − ymVm)∗(I(m,n) − ymVm)
�(1)
= Re {(Vm − Vn)y∗mn(Vm − Vn)
∗ + Vmy∗mV ∗m + Vny
∗nV
∗n } (2)
= Re{VmI∗(m,n) + VnI∗(n,m)} (3)
= P(m,n) + P(n,m) (4)
Case Study: Transmission-loss Allocation
Losses on line (m, n) decomposed into nodal contributions ( (T T T TL(m,n) = |Vm|u(m,n) + |Vn|u(n,m) P + |Vm|v(m,n) + |Vn|v(n,m) Q.
Sairaj Dhople (UMN) Power Divider January 28, 2016 28 / 33
Case Study: Transmission-loss Allocation
Losses on line (m, n) decomposed into nodal contributions ( (T T T TL(m,n) = |Vm|u(m,n) + |Vn|u(n,m) P + |Vm|v(m,n) + |Vn|v(n,m) Q.
Sketch of derivation
L(m,n) = P(m,n) + P(n,m).
Sairaj Dhople (UMN) Power Divider January 28, 2016 29 / 33
Case Study: Transmission-loss Allocation
87
4
3
9
14
1011
13
6
5
12
2
1
Active-pow
erInjection[%
]
0
10
20
30
40
50
Reactive-pow
erInjection[%
]
-18.8
-13.8
-8.8
-3.8
1.2
Non-trivial impact of reactive-power injections on active-power losses
Line (6, 12): 27.4% of the loss from bus 14 active-power injection, and 16.8% from bus 13 reactive-power load
Sairaj Dhople (UMN) Power Divider January 28, 2016 30 / 33
Summary
Uncovered nonlinear mapping between injections and flows
Connections to: i) Current divider, ii) DC power flow
Applications to analysis of power networks
Sairaj Dhople (UMN) Power Divider January 28, 2016 31 / 33
Future Work
Obvious applications: flow control, pricing, visualization
Power tracing: Uncover the fraction of active- and reactive-power generator outputs consumed by loads in the power network
Operations and control: Realizing controllable aggregates of Distributed Energy Resources (DERs)
Power Tracing Aggregating DERs
Sairaj Dhople (UMN) Power Divider January 28, 2016 32 / 33
Questions?
Sairaj Dhople (UMN) Power Divider January 28, 2016 33 / 33