Optimal Power Flow and Frequency Control · power flow model e.g. DC/AC power flow Advantages of load-side control Distributed loads can supplement generator-side control faster (no/low

Optimal Power Flow and Frequency Control

Steven Low

Computing + Math Sciences

Electrical Engineering

NSF Workshop

November 2013

Acknowledgment

Caltech S. Bose, M. Chandy, J. Doyle, M. Farivar, L. Gan, B.

Hassibi, E. Mallada, T. Teeraratkul, C. Zhao

Former L. Chen (Colorado), D. Gayme (JHU), J. Lavaei

(Columbia), L. Li (Harvard), U. Topcu (Upenn)

SCE A. Auld, J. Castaneda, C. Clark, J. Gooding, M.

Montoya, R. Sherick

network of

billions of active

distributed energy

resources (DERs)

DER: PV, wind tb, EV, storage, smart appliances

Solar power over land:

> 20x world energy demand

Risk: active DERs introduce rapid random

fluctuations in supply, demand, power quality

increasing risk of blackouts

Opportunity: active DERs enables realtime

dynamic network-wide feedback control,

improving robustness, security, efficiency

Caltech research: distributed control of networked DERs

• Foundational theory, practical algorithms, concrete

applications

• Integrate engineering and economics

• Active collaboration with industry

ligence

everywhere

Our approach

Endpoint based control Self-manage through local sensing, communication,

control

Real-time, scalable, closed-loop, distributed, robust

Local algorithms with global perspective Holistic framework with global objectives

Decompose global objectives into local algorithms

Control and optimization framework Theoretical foundation for a holistic framework that

integrates engineering + economics

Systematic algorithm design

dynamic model

e.g. swing eqtn

sec min 5 min 60 min day year

primary freq control

secondary freq control

power flow model

e.g. DC/AC power flow

Example: frequency control

economic dispatch

unit commitment

Frequency control is traditionally done on generation side

dynamic model

e.g. swing eqtn




power flow model


Advantages of load-side control

Distributed loads can supplement generator-side control

faster (no/low inertia!)

no waste or emission

more reliable (large #)

localize disturbance

seconds 10’s of secs minutes - hours



economic

dispatch

(OPF)

bulk generators

inverter-based volt/var control

distribution circuits



demand response

distributed loads

voltage control

Can we make this work?

DER control & optimization:

ubiquitous, continuous, fast, non-disruptive, incentive compatible

Outline

Convex relaxation of optimal power flow

Guarantee on solution quality, structural understanding, systematic alg design

Load-side frequency control

Ubiquitous, distributed, fast, efficient

Optimal power flow (OPF)

OPF underlies many applications

Unit commitment, economic dispatch

State estimation

Contingency analysis

Feeder reconfiguration, topology control

Placement and sizing of capacitors, storage

Volt/var control in distribution systems

Demand response, load control

Electric vehicle charging

Market power analysis

…

Optimal power flow (OPF)

Problem formulation Carpentier 1962

Computational techniques Dommel & Tinney 1968 Surveys: Huneault et al 1991, Momoh et al 2001,

Pandya et al 2008, FERC 2012-13

Bus injection model: SDP relaxation Bai 2008, 2009, Lavaei 2012 Bose 2011, Zhang 2011, Sojoudi 2012, Bose 2012 Lesieutre 2011

Branch flow model: SOCP relaxation Baran & Wu 1989, Chiang & Baran 1990, Taylor

2011, Farivar SGC2011 Farivar TPS2013, Li SGC2012, Gan CDC2012, Bose

2012

Outline


Problem formulation

Feasible sets, relaxations, equivalence



Power network

i j k

s j

gs j

c

zij

admittance matrix:

Yij :=

yik

k~i

å if i = j

-yij if i ~ j

0 else

ì

í

ïï

î

ïï

Bus injection model

I = YV

s j = VjI j

* for all j

admittance matrix:

Yij :=

yik

k~i

å if i = j

-yij if i ~ j

0 else

ì

í

ïï

î

ïï

I j : nodal current

Vj : voltage

s j = s j

g - s j

c

power balance

Kirchhoff law

Bus injection model

s j = tr YjVV *( ) for all j

Given find Y, s( ) V V

In terms of : V

Yj = Y *e je j

T

BIM is self-contained (e.g. no branch vars)

quadratic cost

power flow equation

Bus injection model: OPF

min V *CV

over V, s( )

subject to s j £ s j £ s j V j £ |Vj | £ V j

s j = tr Yj

HVV H( )

quadratic cost

power flow equation


min V *CV

over V, s( )


s j = tr Yj

HVV H( )

quadratic cost

power flow equation


min V *CV

over V, s( )


s j = tr Yj

HVV H( )

min tr CVV *

subject to s j £ tr YjVV *( ) £ s j v j £ |Vj |2 £ v j

OPF we study

quadratically constrained QP (QCQP)

nonconvex, NP-hard

Current approach

always converge, fast Nonlinear algorithms

may not converge

Yes

global optimal No guarantee on

solution quality

Algorithms based on

convex relaxation

Traditional

algorithms

feasible ?

heuristics

w/ guarantee

DC OPF solution

may be infeasible

or conservative

No

Other features

Security constrained OPF Solve for operating points after each single

contingency (N-1 security)

Unit commitment Discrete variables

Stochastic OPF Chance constraints Pr(bad event) <

Time correlation Load (buildings, PEV), generation (ramp, wind)

Other constraints Line flow, line loss, stability limit, …

e

… OPF in practice is a lot harder

Outline


Problem formulation

Feasible sets, relaxations, equivalence



Basic idea

min tr CVV *


V

• All complexity due to nonconvexity of V • Relaxations:

• design convex supersets of V

• minimize cost over convex supersets

• Exact relaxation: optimal solution of relaxation

happens to lie in V (when?)

Basic idea

min tr CVV *


V

Approach

1. Three equivalent characterizations of V

2. Each suggests a convex superset and relaxation

min tr CVV *


min tr CW

subject to s j £ tr YjW( ) £ s j vi £ Wii £ vi

W ³ 0, rank W =1

Equivalent problem:

Feasible sets

convex in W

except this constraint

quadratic in V

linear in W !!

min tr CW

subject to s j £ tr YjW( ) £ s j vi £ Wii £ vi

W ³ 0, rank W =1

Equivalent problem:

Feasible sets

SDP relaxation

min tr CVV *


Feasible sets

idea: W = VV *

Feasible set

y jk

*

k:k~ j

å Vj

2

-VjVk

*( ) : only Vj

2

and VjVk

*

corresponding to edges ( j, k) in G!

Wjj Wjklinear in ! Wjj,Wjk( )

min tr CVV *


V

Feasible sets

idea: W = VV *

idea: Wc(G ) = VV * on c(G)( ) matrix completion [Grone et al 1984]

idea: WG = VV * only on G( )

Feasible sets

Theorem

V º WG º Wc(G) º W

Given there is

unique completion and unique

WG Î WG or Wc(G) Î Wc(G)

W Î W V Î V

Can minimize cost over any of these sets, but …

Bose, Low, Chandy Allerton 2012

Bose, Low, Teeraratkul, Hassibi 2013

Feasible sets

W+ Wc(G )

+

Feasible sets

WG

+

V W Wc(G )WG

Theorem

Radial G :

Mesh G :

VÍW+ ºWc(G)

+ ºWG

+

VÍW+ ºWc(G)

+ ÍWG

+

Bose, Low, Chandy Allerton 2012

Bose, Low, Teeraratkul, Hassibi 2013

Examples 25

(a) (b)

Fig. 4: Projections of feasible regions on space for 3-bus system in (3).

P1

P2

0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49

0.48

0.49

0.5

0.51

0.52

0.53

0.54

F

F

F F

Fig. 5: Zoomed in Pareto fronts of the 3-bus case in space.

B. IEEE benchmark systems

For IEEE benchmark systems [35], [42], we solve , and in MATLAB using CVX

[43] with the solver SeDuMi [44]. The objective values and running times are presented in

Table II. As in Theorem 1, the problems and have the same objective function value,

i.e., . However, the optimal objective value of is lower, i.e., . For IEEE

benchmark systems, note that and are exact [14]–[16], while is not. As evidenced

by the running times in Table II, is much faster than . The chordal extension of the

May 31, 2013 DRAFT

power flow

solution X

SDP Y

SOCP Y

Real Power Reactive Power

• Relaxation is exact if X and Y have same Pareto front

• SOCP is faster but coarser than SDP

[Bose, et al 2013]

Convex relaxations

OPF

minV

C(V ) subject to V Î V

G

Recap: convex relaxations

V W Wc(G )WG

WG

+

SOCP relaxation

• coarsest superset

• min # variables

• fastest

W+

SDP relaxation

• tightest superset

• max # variables

• slowest

Wc(G )

+

Chordal relaxation

• equivalent superset

• much faster for

sparse networks

simple

construction completion

spec

decomp

Recap: convex relaxations

V W Wc(G )WG

WG

+

SOCP relaxation

• coarsest superset

• min # variables

• fastest

W+

SDP relaxation

• tightest superset

• max # variables

• slowest

Wc(G )

+

Chordal relaxation

• equivalent set

• much faster for

sparse networks

simple

construction completion

spec

decomp

radial

For radial network: always solve SOCP !

OPF-socp

OPF solution

Recover V* cycle condition

Y

rank-1

OPF-ch OPF-sdp

Y

WG

* Wc(G )

*W *

Y, mesh

2x2 rank-1

Y radial

OPF-socp

cycle condition Y

x*

equality

Y radial

Y, mesh

OPF-socp

OPF solution


Y

rank-1

OPF-ch OPF-sdp

Y

WG

* Wc(G )

*W *

Y, mesh

2x2 rank-1

Y radial

OPF-socp

cycle condition Y

x*

equality

Y radial

Y, mesh

OPF-socp

OPF solution


Y

rank-1

OPF-ch OPF-sdp

Y

WG

* Wc(G )

*W *

Y, mesh

2x2 rank-1

Y radial

OPF-socp

cycle condition Y

x*

equality

Y radial

Y, mesh

Without PS: SDP vs SOCP

SDP not

scalable

SOCP

inexact

SDP SOCP SOCP SDP

Examples

SDP not

scalable

SOCP

inexact

SDP/ch SOCP SOCP SDP chordal

Advantages of relaxations

always converge, fast Nonlinear algorithms

may not converge

Yes

global optimal No guarantee on

solution quality

Algorithms based on

convex relaxation

Traditional

algorithms

feasible ?

heuristics

w/ guarantee

DC OPF solution

may be infeasible

or conservative

No

Advantages of relaxations

always converge, fast

Yes

global optimal

Sufficient conditions

guaranteeing exact

relaxation

Algorithms based on

convex relaxation

feasible ?

heuristics

w/ guarantee

No

Outline




Motivation

Network model

Alg design, optimality, stability

Zhao, Topcu, Li, Low, TAC 2014

Mallada, Low, 2013

Motivation

Synchronous network All buses synchronized to same nominal

frequency (US: 60 Hz)

Supply-demand imbalance frequency

fluctuation

Frequency regulation Generator based

Frequency sensitive (motor-type) loads

Controllable loads Do not react to frequency deviation

… but intelligent

Need active control – how?

Idea dates back to 1970s

Homeostatic utility control :

• freq adaptive loads

• spot prices

• IT infrastructure Schweppe et al (1979, 1980)

Potential benefit

1

Abstract— This paper addresses design considerations for

frequency responsive Grid FriendlyTM appliances (FR-GFAs),

which can turn on/off based on frequency signals and make

selective low-frequency load shedding possible at appliance level.

FR-GFAs can also be treated as spinning reserve to maintain a

load-to-generation balance under power system normal operation

states. The paper first presents a statistical analysis on the

frequency data collected in 2003 in Western Electricity

Coordinating Council (WECC) systems. Using these frequency

data as an input, the triggering frequency and duration of an FR-

GFA device with different frequency setting schemes are

simulated. Design considerations of the FR-GFA are then

discussed based on simulation results.

Index Terms—Grid FriendlyTM appliances, load frequency

control, load shedding, frequency regulation, frequency response,

load control, demand-side management, automated load control.

I. INTRODUCTION

RADITIONALLY, services such as frequency regulation,

load following, and spinning reserves were provided by

generators. Under a contingency where the system frequency

falls below a certain threshold, under-frequency relays are

triggered to shed load to restore the load-to-generator balance.

In restructured power systems, the services provided may be

market based. Because load control can play a role very

similar to generator real power control in maintaining the

power system equilibrium, it can not only participate in under-

frequency load shedding programs as a fast remedial action

under emergency conditions, but also be curtailed or reduced

in normal operation states and supply energy-balancing

services [1][2][3].

Grid FriendlyTM appliances (GFAs) are appliances that can

have a sensor and a controller installed to detect frequency

signals and turn on or off according to certain control logic,

thereby helping the electrical power grid with its frequency

control objectives. Refrigerators, air conditioners, space

heating units, water heaters, freezers, dish washers, clothes

washers, dryers, and some cooking units are all potential

GFAs. Survey [4] shows that nearly one-third of U.S. peak

This work is supported by the Pacific Northwest National Laboratory,

operated for the U.S. Department of Energy by Battelle under Contract DE-

AC05-76RL01830.

N. Lu and D. J. Hammerstrom are with the Energy Science and Technology

Division, Pacific Northwest National Laboratory, P.O. Box 999, MSIN: K5-20,

Richland, WA - 99352, USA (e-mail: [email protected],

[email protected])

load capacity is residential (Fig. 1a). The residential load can

be categorized into GFA and non-GFA loads. Based on a

residential energy consumption survey (Fig. 1b) conducted in

1997, 61% of residential loads are GFA compatible. If all

GFA resources were used, the regulation ability of load would

exceed the operating reserve (13% of peak load capacity)

provided by generators.

(a)

(b)

Fig. 1. (a) Load and reserves on a typical U.S. peak day, (b) Residential load

components. [4]

Compared with the spinning reserve provided by

generators, GFA resources have the advantage of faster

response time and greater capacity when aggregated at feeder

level. However, the GFA resources also have disadvantages,

such as low individual power load, poor coordination between

units, and uncertain availabilities caused by consumer comfort

choices and usages. Another critical issue is the coordination

between regulation services provided by FR-GFAs and

generators. Therefore, whether FR-GFAs can achieve similar

regulation capabilities as generators is a key issue to be

addressed before one can deploy FR-GFAs widely.

As a first step to evaluate the FR-GFA performance, a

research team at Pacific Northwest National Laboratory

(PNNL) carried out a series of simulations which focused on

studying the individual FR-GFA performance to obtain basic

operational statistics under different frequency setting

Design Considerations for Frequency

Responsive Grid FriendlyTM

Appliances

Ning Lu, Member, IEEE and Donald J. Hammerstrom, Member, IEEE

T

1

Abstract— This paper addresses design considerations for

frequency responsive Grid FriendlyTM appliances (FR-GFAs),

which can turn on/off based on frequency signals and make

selective low-frequency load shedding possible at appliance level.

FR-GFAs can also be treated as spinning reserve to maintain a

load-to-generation balance under power system normal operation

states. The paper first presents a statistical analysis on the

frequency data collected in 2003 in Western Electricity

Coordinating Council (WECC) systems. Using these frequency

data as an input, the triggering frequency and duration of an FR-

GFA device with different frequency setting schemes are

simulated. Design considerations of the FR-GFA are then

discussed based on simulation results.

Index Terms—Grid FriendlyTM appliances, load frequency

control, load shedding, frequency regulation, frequency response,

load control, demand-side management, automated load control.

I. INTRODUCTION

RADITIONALLY, services such as frequency regulation,

load following, and spinning reserves were provided by

generators. Under a contingency where the system frequency

falls below a certain threshold, under-frequency relays are

triggered to shed load to restore the load-to-generator balance.

In restructured power systems, the services provided may be

market based. Because load control can play a role very

similar to generator real power control in maintaining the

power system equilibrium, it can not only participate in under-

frequency load shedding programs as a fast remedial action

under emergency conditions, but also be curtailed or reduced

in normal operation states and supply energy-balancing

services [1][2][3].

Grid FriendlyTM appliances (GFAs) are appliances that can

have a sensor and a controller installed to detect frequency

signals and turn on or off according to certain control logic,

thereby helping the electrical power grid with its frequency

control objectives. Refrigerators, air conditioners, space

heating units, water heaters, freezers, dish washers, clothes

washers, dryers, and some cooking units are all potential

GFAs. Survey [4] shows that nearly one-third of U.S. peak

This work is supported by the Pacific Northwest National Laboratory,

operated for the U.S. Department of Energy by Battelle under Contract DE-

AC05-76RL01830.

N. Lu and D. J. Hammerstrom are with the Energy Science and Technology

Division, Pacific Northwest National Laboratory, P.O. Box 999, MSIN: K5-20,

Richland, WA - 99352, USA (e-mail: [email protected],

[email protected])

load capacity is residential (Fig. 1a). The residential load can

be categorized into GFA and non-GFA loads. Based on a

residential energy consumption survey (Fig. 1b) conducted in

1997, 61% of residential loads are GFA compatible. If all

GFA resources were used, the regulation ability of load would

exceed the operating reserve (13% of peak load capacity)

provided by generators.

(a)

(b)

Fig. 1. (a) Load and reserves on a typical U.S. peak day, (b) Residential load

components. [4]

Compared with the spinning reserve provided by

generators, GFA resources have the advantage of faster

response time and greater capacity when aggregated at feeder

level. However, the GFA resources also have disadvantages,

such as low individual power load, poor coordination between

units, and uncertain availabilities caused by consumer comfort

choices and usages. Another critical issue is the coordination

between regulation services provided by FR-GFAs and

generators. Therefore, whether FR-GFAs can achieve similar

regulation capabilities as generators is a key issue to be

addressed before one can deploy FR-GFAs widely.

As a first step to evaluate the FR-GFA performance, a

research team at Pacific Northwest National Laboratory

(PNNL) carried out a series of simulations which focused on

studying the individual FR-GFA performance to obtain basic

operational statistics under different frequency setting

Design Considerations for Frequency

Responsive Grid FriendlyTM

Appliances

Ning Lu, Member, IEEE and Donald J. Hammerstrom, Member, IEEE

T US:

operating reserve: 13% of peak

total GFA capacity: 18%

Lu & Hammerstrom (2006), PNNL

• Residential load accounts

for ~1/3 of peak demand

• 61% residential appliances

are Grid Friendly

Small demo: PNNL

PNNL Grid Friendly Appliance Demo Project (early 2006 – March 2007)

• 150 clothes dryers, 50 water heaters

• Under-frequency threshold: 59.95 Hz (0.08% dev)

• 358 under-freq events during project, lasting secs – 10

mins

• Despite wide geographical distribution, all GFA detected

events correctly and loads shedded as designed

• Survey reported no customer inconvenience

Hammerstrom et al (2007), PNNL

Heffner et al, LBNL (2007)

• Now: more and more load participation in managing imbalance

• Future: continuous, non-disruptive, fast-acting, ubiquitous

Load participation in practice

Callaway, Hiskens (2011)

Callaway (2009)

Can household Grid Friendly

appliances follow its own PV

production?

Dynamically adjust

thermostat setpoint

• 60,000 AC

• avg demand ~ 140 MW

• wind var: +- 40MW

• temp var: 0.15 degC

dynamic model

e.g. swing eqtn




power flow model


Advantages of load-side control

Distributed loads can supplement generator-side control

faster (no/low inertia!)

no waste or emission

more reliable (large #)

localize disturbance

seconds 10’s of secs minutes - hours



economic

dispatch

(OPF)

bulk generators

inverter-based volt/var control

distribution circuits



demand response

distributed loads

voltage control

Can we make this work?

DER control & optimization:

ubiquitous, continuous, fast, non-disruptive, incentive compatible

Outline




Motivation

Network model



Mallada, Low, 2013

Network model

i

Pi

m

generation

di + d̂i

loads: controllable + freq-sensitive

j

xij

reactance

i : bus/control area/balancing authority

Network model

DC approximation

Lossless network (r=0)

Fixed voltage magnitudes

Reactive power ignored

Do not assume small angle difference

Pi

m

i

j

Pij

di + d̂i

Dynamic model

Swing equation on bus i

frequency

Miwi = Pi

m - Pi

e

mechanical

power electrical

power

Pi

m

Piji

j

Newton’s 2nd law

Variables: deviations from nominal values

di + d̂i

Dynamic model


Miwi = Pi

m - Pi

e

Pi

m

i

j

Pi

e := di + Diwi + Pij

i~ j

å

controllable

loads

branch

power flow

freq-sens

loads

Pij

di + d̂i

Dynamic model


Miwi = Pi

m - Pi

ePi

m

i

j

Pi

e := di + Diwi + Pij

i~ j

å

Pij

Pij = bij wi -w j( )

bij = 3Vi Vj

xij

cos qi0

- q j0( ) linearization around nominal

di + d̂i

swing dynamics

Network model

Generator bus (may contain load):

wi = -1

Mi

di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

åæ

èçç

ö

ø÷÷

Pij = bij wi -w j( ) " i® j

0 = di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

å

Load bus (no generator):

Real branch power flow:

swing dynamics

Network model

System dynamics recap:

wi = -1

M i

di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

åæ

èçç

ö

ø÷÷

0 = di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

å


Suppose the system is in steady state, and suddenly …

wi = 0 Pij = 0

Given: disturbance in gens/loads

Current: adapt remaining generators

to re-balance power

(and restore nominal freq, zero ACE)

Our goal: adapt controllable loads

to re-balance power

while minimizing disutility of load control

Frequency control

Pi

m

di

Controller in general may

require non-local info

require coordination among loads

Load-side controller design

wi = -1

M i

di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

åæ

èçç

ö

ø÷÷

0 = di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

å


d w(t), P(t)( )

Forward engineering

design an optimization problem (OLC)

derive local control as distributed solution

Load-side controller design

wi = -1

M i

di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

åæ

èçç

ö

ø÷÷

0 = di + Diwi - Pi

m + P ij - Pji

j®i

åi® j

å


Outline




Motivation

Network model



Mallada, Low, IFAC 2014

min ci di( ) + 1

2Di

d̂i

2æ

èç

ö

ø÷

i

å

over loads dl Î d l, dléë ùû, d̂i

s. t. di + d̂i( )i

å = Pi

m

i

å

Optimal load control (OLC)

uncontrollable

load

demand = supply

across network

controllable

load

min ci di( ) + 1

2Di

d̂i

2æ

èç

ö

ø÷

i

å


s. t. di + d̂i( )i

å = Pi

m

i

å


demand = supply

across network

uncontrollable

load

controllable

load


min ci di( ) + 1

2Di

d̂i

2æ

èç

ö

ø÷

i

å


s. t. di + d̂i( )i

å = Pi

m

i

å demand = supply

across network

uncontrollable

load

controllable

load

disturbances

Theorem

swing dynamics

+ frequency-based load control

= primal-dual algorithm that solves OLC

Completely decentralized

Not need for explicit communication

Not need for detailed network data

Exploit free global control reference

Punchline

… reverse engineering swing dynamics

swing dynamics (recap)

Punchline

wi = -1

M i

di (t)+ Diwi (t)- Pi

m + Pij (t)- Pji (t)j®i

åi® j

åæ

èçç

ö

ø÷÷

Pij = bij wi (t)-w j (t)( )

load control

di(t) := ci

'-1 wi (t)( )éë

ùûd i

di

active control

implicit

Theorem

system trajectory

converges to

is unique optimal load control

is unique optimal for DOLC

is optimal for dual of DOLC

Punchline

d(t), d̂(t), w(t), P(t)( )d*, d̂*, w*, P*( ) as t®¥

d*, d̂*( )w*

P*


Theorem

system trajectory

converges to




Punchline


d*, d̂*( )w*

P*


Theorem

system trajectory

converges to




Punchline


d*, d̂*( )w*

P*


Implications

Freq deviations contains right info on global power imbalance for local decision

Decentralized load participation in primary freq control is stable

: Lagrange multiplier of OLC

info on power imbalance

: Lagrange multiplier of DOLC

info on freq asynchronism

w*

P*

Implications



: Lagrange multiplier of OLC

info on power imbalance

: Lagrange multiplier of DOLC

info on freq asynchronism

w*

P*

Implications



Load-side secondary freq control requires

communication with neighbors

Mallada, Low, 2014

Simulations

Dynamic simulation of IEEE 68-bus system

• Power System Toolbox (RPI)

• Detailed generation model

• Exciter model, power system

stabilizer model

• Nonzero resistance lines

Simulations

Simulations

Documents

Optimal Power Flow and Frequency Control · power flow model e.g. DC/AC power flow Advantages of load-side control Distributed loads can supplement generator-side control faster (no/low