Power Divider - NREL · Power Divider Sairaj Dhople Department of Electrical and Computer...

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

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A Note

Paper to appear in the IEEE Transactions on Power Systems

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Outline

Background

Current Divider

Power Divider

Special Cases

Applications

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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

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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]

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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

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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

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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

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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)

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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)

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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)

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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)

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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)

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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

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Minimal Realization

Kron reduction: Only generator- and load-bus information is needed

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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)

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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.

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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

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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

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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

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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 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.

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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.

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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.

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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).

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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

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Summary

Uncovered nonlinear mapping between injections and flows

Connections to: i) Current divider, ii) DC power flow

Applications to analysis of power networks

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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

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Questions?

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