Dr. Hsiao-Dong Chiang

(i) Prof. of School of Electrical and Computer Engineering, Cornell University, Ithaca, NY

(ii) President of BSI, Ithaca, NY

Avaiable Delivery Capability of Unbalanced Distribution Networks to

Support High DG Penetrations

Contents

• Transmission networks are highly-utilized while distribution networks are usualy under-utilized.

• New tools are required to assess the

capabilities of distribution networks to meet the continual growth of renewable energy (RE) and to withstand the stress caused by these intermittent, and often uncoordinated sources of power.

Contents

• To achieve the goal of 33% renewable energy, the delievry capability of distribution networks needs to be assessed and enhanced,

• a CDFLOW (continuation distribution power flow) tool will be presented along with some analytical results.

• Its comprehensive modeling capability is

useful to simulate real-life situations.

Test Systems

IEEE 8,500-node networks

A practical 1101-node 3-phase unbalanced network

Features of Distribution Network

Compared to the transmission power system: 1. Radial or weakly meshed networks, 2. High R/X ratios, 3. Multi-phases ,unbalanced three phases, two

phases and single phase, 4. Unbalanced loads and 5. A wide variety of Distributed generators

Special Features of Distribution System

Computation engine : continuation methods

Transmission networks

Continuation Power Flow Engine (CPFLOW, CPF)

Chiang et al. (1990)

Ajjarapu et al. (1992)

Cañizares et al. (1992)

Chiang, Flueck, Balu (1995)

Iba, Yorino

others

Unbalanced distribution networks (?)

CDFLOW - Applications

to analyze voltage problems due to load and/or generation variations.

to evaluate/maximize available delivery capability.

to simulate distribution system static behaviors due to load and/or generation variations with/without control devices.

to conduct coordination studies of control devices for steady-state security assessment.

CDFLOW can be used in a variety of application such as

2

1

3

4

CDFLOW - Functional Specification

Functional Specification of CDFLOW Deriv. Imple.

Network Configuration

Radial distribution network

Meshed distribution network

Analytical Capabilities

3-phase power flow calculation

Voltage profiles

Power flows on each phase

3-phase continuation power flow

P-V, Q-V, P-Q-V curves

Exact SNB point and eigen analysis

Units and Typical Values Used in the Program

“km” unit for length

Typical voltages for Japanese distribution network High voltages: 6.6kv, 22kv and 33kv Low voltages: 100v, 200v and 415v

No Default Value for Voltage

CDFLOW - Modeling Capabilities

Functional Specification of CDFLOW Deriv. Imple.

Modeling Capabilities

Power Source Model for Transmission network Modeling

Dispersed Generational Models

PV Bus

PQ Bus

Capacitor Models Single-phase

3-phase

Load Models ZIP model for single-phase 2-phase and 3-phase

Loop Controller Model: Phase Shifting Transformers DC Link, Electro-mechanical Link

Not likely used in Distribution Systems

FACTS: Series/Parallel Compensated Model

CDFLOW - Modeling Capabilities

Functional Specification of CDFLOW Deriv. Imple.

Modeling Capabilities

Transformer

Single-phase

2-wire

3-wire

3-Phase

Wye-Delta

Delta-Delta Ungrounded

Grounded

Wye-Wye

Open Delta

V Connection 3-wire

4-wire

Delta-Grounded Wye

Autotransformer

Regulator Single-phase / 3-pahse LTC LDC

Uniqueness of power flow solution

Miu and Chiang (1998)

For radial unbalanced networks, the number of feasible power flow solution is one.

Local Bifurcations calculated in CDFLOW

SIB

SNB

Saddle-Node Bifurcation The stable equilibrium point and another equilibrium point coalesce and disappear in a saddle-node bifurcation as parameter varies. The physical meaning of SNB is available delivery capability of a network configuration.

Structure-Induced Bifurcation The mathematical mechanism is the switching of system equations. It usually induced by certain resources reach its Q-limit, the terminal bus type switches from PV to PQ, then the associated power flow equations changed.

Two types of local bifurcations will occur in the quasi-static power system model.

Conditions of Saddle-node Bifurcation

(A1) Bifurcation point is also an equilibrium point of the dynamic system, (A2) The corresponding system Jacobian matrix has a simple zero eigenvalue with right eigenvector v and left eigenvector w, such that (A3) (A4)

Condition A3 and A4 are transversal conditions，which are usually satisfied for general nonlinear system

Here we will focus on the condition A1 and A2

0 0, 0f x

0 0det , 0xf x

0 0, 0fx

2

0 02 , 0fx

x

Conditions of Saddle-node Bifurcation

Condition A1 requires that SNB point is also an

equilibrium point

Condition A2 requires that there is only one simple zero eigenvalue when a

SNB encounters.

The following equations also ensure condition A1 and A2

, 0 , 00 0

1 1x x

T T

f x f x

wf or f v

ww vv

Structure Induced Bifurcation: SIB: (Dobson 1992, Li and Chiang 2005) A peculiar bifurcation in power networks at which the real part of eigenvalues is non-zero

V

V

Post QPost Q

Pre Q Pre Q

Structure Induced Bifurcation No stable equilibrium point after branch switch (corresponding to bus type switch from PV to PQ)

Structure Induced Exchange Process Reach a stable equilibrium point after branch switch (corresponding to bus type switch from PV to PQ)

CDFLOW - Formulation

01 1

01 1

cos sin 0

sin cos 0

N M

ji ij ij ij ij Lii Gij

N M

ji ij ij ij ij Li

k k k kk

ij

k

k k k kk

k

P V V G B P P

Q V V G B Q

01 1

0

01 1

, ,

cos sin 0

sin cos 0

N M

ji ij ij ij ij Lii Gk

ij

ii

N M

ji ij ij ij ij Liij

min i m

k k kk

k

k

ax iGi

k k kk

k

P V V G B P P

V V

Q V V G B Q

Q Q Q

For a PV node , the continuation power flow equations is

For a PQ node, the continuation power flow equations is

CDFLOW - Formulation

20 0 0

0

1.0 1,2,...,

ii i iiLi Li Li Li Li

i

i i i

VP jQ S V S S

V

i N

For a P-V node with a Q limit, say Qmin and Qmax, the 3-phases continuation power flow equation can be expressed as:

, 0

, 0

, 0

, 0

pre Q G max i ii

post Q G max i ii

f x Q Q V V

f x Q Q V V

The load model can be constant impedance, constant current, constant power or their combination:

In summary, a three-phase continuation power flow can be expressed in the following form:

0 ,f x f x b

CDFLOW - Architecture

The implementation of the four basic elements of the continuation method in CDFLOW

Predictor

Corrector

Step-size Control

Tangent predictor (phase-one)

Secant predictor (phase-two)

Nonlinear predictor (phase-three)

Parameterization

Arc-length

Pseudo-arc-length

Local parameter

Modified Newton method

Implicit Z-bus Gauss method

Adaptive Step-size Control

CDFLOW

CDFLOW - Parameterization

2 2 2

10

n

i ii

x x s s s

1. Arc-length Parameterization 2. Pseudo Arc-length Parameterization 3. Local Parameterization

1

0n

i i ii

x x s x s s s s

ˆ 0k kx x

kx

x

x

P-V Curve of the Weak Bus M1026328

0 0.5 1 1.5 20

0.2

0.4

0.6

0.8

1

1.2

Lambda

Voltage M

agnitude (

p.u

)

3-phase P-V Curves (abc)

BusM1026328

Vmaga

Vmagb

Vmagc

Dominant Eigenvalue of the Jacobian

0 0.5 1 1.5 2 2.5-4

-2

0

2

4

6

8x 10

-4

Lambda

Real P

art

P-Eigenvalue Curve

SNB

Stable Half Portion

Unstable Half Portion

Voltage Magnitudes of Weak Bus 1216

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

Lambda

Vol

tage

Mag

nitu

de (

p.u)

3-phase P-V Curves (abc)

Vmaga

Vmagb

Vmagc

Fig. 3. The voltage magnitudes at Bus1216 as parameter varies

Bus with DG

Mathematical description λ

value

Physical description MW

value

Voltage or thermal violation

(Vmin = 0.9 p.u)

M1209790 0 0 L3177894 M3016088

…

M1108378 2.302819 6.908457 L3082993 M3016088

…

M1026927 3.866104 11.598312 Bus L3216339 Bus L3225319

…

L2691967 8.070759 24.212277 * L_6102322_6100788

DG Penetration at Different Locations TABLE I

DG Penetration Capability with Load Variation = 0 +j 0

Bus with DG

Mathematical description λ

value

Physical description MW value

Voltage or thermal violation

(Vmin = 0.9 p.u)

M1209790 0 0 L3177894 M3016088

…

M1108378 0.141103 0.423309 L3195327 M1027114

…

M1026927 0.236323 0.708969 L3225319 L3048201

…

L2691967 0.255488 0.766464 L3216339 L3225319

…

DG Penetration at Different Locations TABLE II

DG Penetration Capability with Load Variation = Pd0 +j Qd0

Test Case 4: ZIP Load Model

TABLE III Voltage Stability Limit with different load models

Case No. SCENARIO Limit ( )

1 Half number of loads are constant impedance, other loads are constant PQ 1.47632457

2 Half number of loads are constant current, other loads are constant PQ 1.41906738

3 All loads are constant PQ 1.11565160

4 Half number of loads are 30% constant current, 30% constant current and 40% constant PQ, other loads are constant PQ

1.36719488

*

load margin of 1101-node practical distribution network with different load models

Dominant Eigenvalue of the Jacobian

0 0.2 0.4 0.6 0.8 1 1.2-4

-3

-2

-1

0

1

2

3

4

5

6

7x 10

-3

SNB

Lambda

Rea

l Par

tP-Eigenvalue Curve

Stable Half Portion

Unstable Half Portion

Fig. 5. The largest real part of eigenvalues of Jacobian matrix

012-Sequence Voltage of Weak Bus 1216

0 0.2 0.4 0.6 0.8 1 1.20

0.2

0.4

0.6

0.8

1

Lambda

Vol

tage

Mag

nitu

de (

p.u)

3-phase P-V Curves (012 sequence)

Vmag1

Vmag2

Vmag0

Fig. 6. The sequence voltage magnitudes at Bus1216 as parameter varies.

Voltage Stability

NYSEG 394 Bus System

The NYSEG 394 bus, 1103 node system is an unbalanced distribution network serving balanced and unbalanced loads. The substation bus delivers power to the full feeder and there is no DG in this network. In this test case, the load variation is same as the initial loading condition, and the following aspects are studied: • The properties of 3-phase P-V curves; • The weak node in the network • The trajectory of dominant eigenvalue along solution

curve, and • The effect of composite load model

Test Case 4: ZIP Load Model

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60

0.2

0.4

0.6

0.8

1

Lambda

Vol

tage

Mag

nitu

de (

p.u)

3-phase P-V Curves (abc)

Pm2

Pm4

Pm3

Pm1Bus1102

Vmaga

Vmagb

Vmagc

Preventive and Enhancement Control

The key objective for development of enhancement control is to enlarge the available delivery capability and voltage stability load margin. The control actions includes: • Shunt capacitor; • ULTC tap changers; • ULTC phase-shifter; • Distributed generator terminal voltage control; • Load shedding; • Placement of new shunt capacitors;

There are two key steps in control design: • Determine a set of most effective controls; • Determine the amount of these controls;

Control #1: Load Shedding

Elements in left eigenvector corresponding to bus 671 are (0.042069, 0.031748), shedding load 1155+ j 660 kw at bus 671.

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

log10 (Real)

log10 (

Imag)

Eigenvalues of Jacobian before control Maximum load is 5.047896

Eigenvalues of Jacobian after control Maximum load is 5.980143

-5 -4.5 -4 -3.5 -3 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

log10 (Real)

log10 (

Imag)

Thanks！