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Optimal	Power	Flow	Paper	9	

Staff	paper	by	

Clay	Campaigne	

Paula	A.	Lipka	

Mehrdad	Pirnia	

Richard	P.	O’Neill	

Shmuel	Oren	

June2013

The views presented are the personal views of the authors

and not the Federal Energy Regulatory Commission or any

of its Commissioners.

	 	
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Testing	Step‐size	Limits	for	Solving	the	

Linearized	Current	Voltage	AC	Optimal	Power	Flow	

Optimal	Power	Flow	Paper	9	

Clay	Campaigne,	Paula	A.	Lipka,	Mehrdad	Pirnia,		

Richard	P.	O’Neill	and	Shmuel	Oren	

claycampaigne@berkeley.edu;	plipka@berkeley.edu;	mpirnia@uwaterloo.ca,	

richard.oneill@ferc.gov;	oren@ieor.berkeley.edu

June	2013			

Abstract	and	Executive	Summary	

In	this	paper,	we	seek	to	improve	the	performance	of	the	iterative	linear	program	

approximation	to	the	current	voltage	AC	optimal	power	flow	ሺILIV‐ACOPFሻ.		By	

adding	a	set	of	constraints	that	limit	the	differences	between	the	real	and	

imaginary	voltages	of	successive	major	iteration	solutions,	we	limit	the	error	in	the	

linear	approximation,	and	we	seek	to	decrease	the	time	to	solve	and	increase	the	

robustness	of	the	procedure.		The	primary	motivation	is	that	the	iterative	

linearization	procedure	sometimes	exhibits	periodic	behavior	ሺ“bouncing”	between	

two	solutionsሻ.		This	behavior	may	add	to	the	solution	time	or	result	in	a	failure	to	

converge.		Generally,	the	step‐size	constraints	improve	performance	of	the	iterative	

linear	approximation	procedure,	but	the	best	parameters	of	the	step‐size	constraint	

are	problem	dependent.	Although	the	convergence	tests	are	different,	the	linear	

procedure	is	considerably	faster	than	the	nonlinear	solver		The	tradeoff	between	

the	iterative	linearization	and	the	nonlinear	solver	was	speed	compared	to	greater	

accuracy.	Increasing	the	preprocessed	cuts	from	16	to	32	increases	the	solution	

time.	As	the	problem	size	gets	bigger,	we	see	diminishing	returns	to	the	number	of	

preprocessed	constraints.		The	tighter	tolerance	for	convergence	takes	longer,	but	

does	not	seem	to	have	a	major	impact	on	the	optimal	solution	value,	except	when	

there	is	no	step‐size	constraint,	where	a	more	restrictive	tolerance	sometimes	

results	in	the	linear	program	not	converging.		We	find	that	step‐size	constraints	

decrease	the	time	to	solve	and	increase	the	robustness	of	the	procedure.		Solution	

times	were	up	to	six	times	faster	using	step‐size	limits.	

Disclaimer:	The	views	presented	are	the	personal	views	of	the	authors	and	not	the	

Federal	Energy	Regulatory	Commission	or	any	of	its	Commissioners.		
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Table	of	Contents

1.	Introduction	....................................................................................................	 	 	 	 4	

2.	Methods	............................................................................................................	 	 	 	 4	

3.	ACOPF	Formulations....................................................................................	 	 	 	 5	

4.	Step‐size	Limits	on	the	Real	and	Reactive	Power	Approximations	 	 		 	 7	

5.	Linear	Iteration	Approach		 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 		 	 10	

	

6.	Computational	Testing			 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 11	

	

7.	Conclusions		.....................................................................................................	 	 	 16	

	

References	............................................................................................................	 	 	 17	
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1. Introduction	

Since	the	AC	optimal	power	flow	ሺACOPFሻ	problem	was	introduced	in	1962	

ሺCarpentierሻ,	it	has	received	considerable	attention.		The	ACOPF	optimizes	the	

steady	state	performance	of	an	AC	power	system	by	minimizing	an	objective	

function	such	as	generation	cost,	or	maximizing	market	surplus,	while	satisfying	

system	constraints	including	nodal	real	power	balance,	nodal	reactive	power	

balance,	bounds	on	bus	voltages,	flows	on	transmission	lines,	real	and	reactive	

power	injections,	and	contingencies.	Since	the	ACOPF’s	introduction,	different	

objective	functions	and	formulations	have	been	tried	ሺsee	Cain	et	alሻ.		The	

canonical	ACOPF	problem	uses	the	polar	representation	of	voltage,	real	and	

reactive	power.		ACOPF	problems	have	nonconvex	continuous	functions	and	can	be	

large.	More	difficult	variations	include	binary	variables	for	topology	control	and	

unit	commitment	ሺsee,	for	example,	Potluri	and	Hedmanሻ.	

Solving	an	ACOPF	problem	that	meets	a	power	system’s	realistic	physical	

criteria	has	continued	to	be	a	challenge	in	power	system	operations.	While	most	

NLP	solvers	find	local	optimal	solutions	most	of	the	time,	their	lengthy	solution	

times	and	poor	convergence	ሺespecially	with	the	introduction	of	binary	variablesሻ	

have	focused	attention	on	linear	approximations.		Linear	and	mixed	integer	

programming	methods	play	a	significant	role	in	solving	these	problems	with	more	

robust	solutions	and	better	execution	times.	Currently,	one	of	the	most	common	

approaches	is	the	DC‐OPF,	in	which	the	real	part	of	the	admittance	matrix	is	

considered	negligible,	reactive	power	and	voltage	magnitude	variables	are	fixed	in	

the	formulation,	and	bus	voltage	angle	differences	are	assumed	to	be	near‐zero	

ሺStott,	Alsac,	and	Monticelli,	2009ሻ.	

A	vast	body	of	literature	proposes	different	optimization	methods	to	solve	

the	ACOPF	including	Lagrangian	approaches,	sequential	quadratic	programming,	

sequential	linear	programming,	interior	point	methods,	and	heuristics.	Literature	

reviews	appear	periodically	ሺsee,	for	example,	Dommel	and	Tinney,	Huneault	and	

Galiana,	Momoh,	et	al,	Frank	and	Steponavice	and	Castillo	and	O’Neill,	2013a).	The	

literature	reviews	present	an	evolution	of	approaches	to	solve	the	ACOPF.		

Capitanescu	et	al	(2011)	reviews	the	state	of	the	art	and	challenges	to	the	optimal	

power	flow	computations	including	corrective	post‐contingency	actions,	voltage	

and	transient	stability	constraints,	problem	size	reduction,	discrete	variables	and	

uncertainty.	

O’Neill	el	al	(2012a)	formulate	the	ACOPF	in	several	ways,	compare	each	

formulation’s	properties,	and	argue	that	the	rectangular	current‐voltage	or	“IV”	

formulation	and	its	linear	approximations	may	be	easier	to	solve	than	the	

traditional	quadratic	power	flow	“PQV”	formulation.		The	IV‐ACOPF	isolates	

nonconvexities	to	each	bus	and	transmission	element	rather	than	allowing	

nonconvexities	in	the	flow	equations	as	in	the	canonical	polar	PQV	formulation.			
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O’Neill	et	al	(2012b)	compare	solving	the	IV	linear	approximation	of	the	

ACOPF	to	solving	the	ACOPF	with	several	nonlinear	solvers.	In	general,	the	linear	

approximation	approach	is	more	robust	and	faster	than	several	of	the	commercial	

nonlinear	solvers.	On	several	starting	points,	the	nonlinear	solvers	failed	to	

converge	or	contained	positive	relaxation	(penalty)	variables	above	the	threshold.	

The	iterative	linear	program	approach	finds	a	near‐feasible	near‐optimal	solution	

in	almost	all	problems	and	starting	points.			

Lipka	et	al	ሺ2013ሻ	added	current	constraints	to	the	problems	and	examined	

the	resulting	effects	on	the	solution	times.	Pirnia	et	al	ሺ2013ሻ	tested	pre‐processed	

circumscribing	polygon	approximations	in	combination	with	and	without	iterative	

cuts	to	the	procedure	finding	the	‘sweet	spot’	for	the	number	of	preprocessed	cuts	

in	the	ILIV‐ACOPF.		In	testing	to	date,	the	results	have	favored	the	ILIV‐ACOPF.		

Castillo	and	O’Neill	(2013b)	present	an	experimental	framework,	statistical	

methods	and	numerical	results	from	testing	commercial	nonlinear	solvers	with	

several	ACOPF	formulations	and	initializations.	The	experiments	indicate	a	clear	

advantage	to	employing	a	rectangular	formulation	over	a	polar	formulation.	

In	spite	of	all	the	work	that	has	been	done,	the	ACOPF	remains	‘very	much	a	

work	in	progress’	ሺStott	and	Alsaç,	2012ሻ.	Further,	they	state	that	solutions	to	the	

problems	encountered	in	‘real‐life’	are	‘not	easy	to	obtain’	and	still	require	

significant	individual	intervention	and	tuning.	

In	this	paper,	we	seek	to	improve	the	performance	of	the	ILIV‐ACOPF	in	

previous	studies	by	adding	a	set	of	constraints	that	limit	the	differences	between	

the	real	and	imaginary	voltages	of	successive	solutions.	The	primary	motivation	for	

this	is	that	the	iterative	linearization	procedure	sometimes	converges	slowly	or	

fails	to	converge	exhibiting	periodic	behavior	ሺ“bouncing”	between	two	solutionsሻ.	

We	find	that	step‐size	constraints	decrease	the	time	to	solve	and	increase	the	

robustness	of	the	procedure.			

2. Notation.	

Variables	and	parameters	are	indexed	over	buses	using	subscripts	n	and	m.

Here	we	refer	to	transmission	assets	as	lines.	Transmission	lines	are	indexed	by	

terminal	buses	n	and	m	and	line	number	k.		For	a	complex	variable	or	parameter,	

the	superscript	r	denotes	the	real	portion	and	the	superscript	j	denotes	the	

imaginary	portion.		For	example,	if	x	ൌ	a൅jb,	xr ൌ	a,	xj	ൌ	b	where	j	ൌ	ሺ‐1ሻ1/2.		The	

index	of	a	major	iteration	is	h.	

Decision	Variables

pn 	 real	power	injected	at	bus	n

qn reactive	power	injected	at	bus	n

vr n		 real	part	of	the	voltage	at	bus	n
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vj

n		 imaginary	part	of	the	voltage	at	bus	n

vn		 voltage	magnitude	at	bus	n;	vn ൌ	ሾሺvr nሻ2 ൅ ሺvj

nሻ2ሿ1/2

i

r n		 real	part	of	the	current	at	bus	n

i

j

n 	 imaginary	part	of	the	current	at	bus	n

in 	 current	magnitude	at	bus	n;	in ൌ	ሾሺi

r nሻ2 ൅ ሺij

nሻ2ሿ1/2

i

r nmk		 real	part	of	the	current	on	line	k at	bus	n connecting	to	bus	m

i

j

nmk imaginary	part	of	the	current	on	line	k at	bus	n connecting	to	bus	m

inmk current	magnitude	on	k at	n connecting	to	m ; inmk ൌ	ሾሺi

r nmkሻ2 ൅ ሺij

nmkሻ2ሿ1/2

Parameters	

cp nሺpnሻ quadratic	cost	of	real	power	at	bus	n

cq nሺqnሻ quadratic	cost	of	reactive	power	at	bus	n

cpl nሺpnሻ stepwise	linear	approximation	of	cp nሺpnሻ

cql n ሺqnሻ stepwise	linear	approximation	of	cq nሺqnሻ

bnmk susceptance	of	line	k between	bus	n	and	m

gnmk conductance	of	line	k between	bus	n	and	m

ynmk ൌ gnmk൅jbnmkadmittance	of	line	k between	bus	n and	m

yn0 admittance	from	bus	n to	ground

pmin n minimum	required	real	power	at	bus	n

pmax n		 maximum	allowed	real	power	at	bus	n

qmin n minimum	required	reactive	power	at	bus	n

qmax n maximum	allowed	reactive	power	at	bus	n

vmin n minimum	required	voltage	magnitude	at	bus	n

vmax n maximum	allowed	voltage	magnitude	at	bus	n

vr n real	voltage	value	at	bus	n from	the	previous	linear	program	solution

vj

n imaginary	voltage	value	at	bus	n from	the	previous	linear	program	

solution

i

r n real	current	value	at	bus	n from	the	previous	linear	program	solution

i

j

n imaginary	current	value	at	bus	n from	the	previous	linear	program	

solution	

i

max nmk maximum	current	magnitude	on	line	k connecting	bus	n	to	bus	m

i

r nmk the	real	current	value	on	line	k at	bus	n to	m from	the previous	linear		

program	solution	

i

j

nmk imaginary	current	value	on	line	k at	bus	n to	m	from	the	previous	linear	

program	solution	

3. ACOPF	Formulations		
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The	nonlinear	IV‐ACOPF	is	used	as	a	benchmark	for	linear	approximation	

ILIV‐ACOPF.		The	IV‐ACOPF	formulation	is	

Minimize	∑n cp n ሺpnሻ൅cq n ሺqnሻ	 ሺ1ሻ

Subj.		 i

r nmk	ൌ	gnmkሺvr n	‐ vr mሻ ‐ bnmkሺvj

n	‐ vjmሻ for all	n,	m,	k	 ሺ2ሻ

To	 i

j

nmk	ൌ	bnmkሺvr n	‐ vr mሻ ൅ gnmkሺvjn	‐ vjmሻ for all	n,	m,	k	 ሺ3ሻ

i

r n	ൌ	∑mk i	r nmk	 for all	n	 ሺ4ሻ

i

j

n	ൌ	∑mk i	jnmk for all	n	 ሺ5ሻ

pn	ൌ	vr ni

r n ൅	vj

ni

j

n			 for all	n 	 ሺ6ሻ

pmin n ൑ pn ൑ pmax n	 for all	n	 ሺ7ሻ

qn	ൌ	vj

ni

r n ‐	vr nij

n	 for all	n	 ሺ8ሻ

qmin n ൑ qn ൑ qmax n for all	n	 ሺ9ሻ

	 ሺvr nሻ2 ൅ሺvj

nሻ2 ൑ ሺvmax nሻ2 for all	n	 ሺ10ሻ

	 ሺvmin nሻ2	൑ ሺvr nሻ2 ൅ ሺvj

nሻ2 for all	n	 ሺ11ሻ

	 ሺi

r nmkሻ2 ൅ ሺij

nmkሻ2 ൑ ሺi

max nmkሻ2 for all	n,	m,	k	 ሺ12ሻ

The	line	flow	and	network	flow	equations,	ሺ2ሻ‐ሺ5ሻ	are	linear.		The	feasible	sets	for	

line	current	magnitudes	ሺ12ሻ	and	voltage	magnitudes	at	buses	ሺ10ሻ	are	discs,	and	

thus	can	be	approximated	to	any	degree	of	accuracy	with	circumscribing	polygons.		

The	lower	bound	on	voltage	magnitude,	while	non‐convex,	is	seldom	binding	

because	the	optimization	pushes	voltages	higher	to	reduce	losses.			

In	rectangular	form,	the	equations	for	real	ሺ6ሻ	and	reactive	power	ሺ8ሻ	injections	and	

withdrawals	in	terms	of	current	and	voltage	are	second‐order	non‐convex	

polynomials.		We	approximate	the	quadratic	constraint	equations	ሺwhich	express	pn

and	qn	in	terms	of	currents	and	voltagesሻ	with	hyperplanes	that	are	tangent	to	the	

constraint	hypersurfaces	using	first	order	Taylor	approximations.	With	the	resulting	

linear	approximations,	the	ILIV‐ACOPFሺhሻ	at	each	major	iteration	h	is:	

Minimize	∑n cpl nሺpnሻ൅cql nሺqnሻ	 ሺ21ሻ

Subj.	 i

r nmk	ൌ	gnmkሺvr n ‐ vr mሻ ‐ bnmkሺvj

n ‐ vjmሻ for all	n,	m,	k ሺ22ሻ

to	 i

j

nmk	ൌ	bnmkሺvr n ‐ vr mሻ ൅ gnmkሺvjn ‐ vjmሻ for all	n,	m,	k ሺ23ሻ

i

r n	ൌ	∑mk ir nmk for all	n ሺ24ሻ

i

j

n	ൌ	∑mk i

j

nmk for all	n ሺ25ሻ

pnൌ	vr ni

r n ൅	vj

ni

j

n ൅	vr ni

r n ൅ vj

ni

j

n ‐ ሺvr ni

r n ൅ vj

ni

j

nሻ for all	n ሺ26ሻ

pmin n ൑ pn ൑ pmax n for all	n ሺ27ሻ

qn ൌ	vj

ni

r n ‐ vr nij

n ‐	vr ni

j

n ൅ vjni

r n ‐ ሺvj

ni

r n ‐ vr nij

nሻ for all	n ሺ28ሻ

qmin n ൑ qn ൑ qmin n for all	n ሺ29ሻ

cosሺθsሻvr n ൅	sinሺθsሻvj

n	൑ vmax n for	sൌ	0,	1,	...,	smax ;

for all	n ሺ30ሻ

	 ሺvr nd/vndሻvr n ൅ ሺvj

nd/vndሻvjn ൑ vmax n for	d ൌ	0,	...,	h‐1;

for	all	n ሺ31ሻ

cosሺθsሻi

r nmk ൅	sinሺθsሻij

nmk ൑ i

max nmk for	s ൌ	1,	...,	smax ; ሺ32ሻ
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for	all	n,	m,	k

	 ሺi

r nmkd/inmkdሻir nmk ൅ ሺij

nmkd/inmkdሻi

j

nmk ൑ i

max nmk for	d ൌ	0,	...,	h‐1;	

for	all	n,	m,	k ሺ33ሻ

where	smax is	the	number	of	sides	of	the	preprocessed	circumscribing	polygons,	and	

d	indexes	the	iterative	tight	cuts.			

	 	 	 	 The	line	flow	and	network	flow	equations,	ሺ22ሻ‐ሺ25ሻ	are	linear	and	

unchanged	from	the	IV_ACOPF.		For	the	voltage	magnitudes,	the	preprocessed	

outer	bounds	are	in	ሺ30ሻ	and	the	iterative	tight	cuts	are	in	ሺ31ሻ.		For	the	line	

current	magnitudes,	the	preprocessed	outer	bounds	are	in	ሺ32ሻ	and	the	iterative	

tight	cuts	are	in	ሺ33ሻ.		The	lower	bound	on	voltage	magnitude,	while	non‐convex,	is	

seldom	binding	because	the	optimization	pushes	voltages	higher	to	reduce	losses.		

	 	 	 	 The	constraints	ሺ30ሻ‐ሺ33ሻ	are	all	constraints	on	magnitude	of	the	component	

of	the	complex	voltage	or	current	vector	in	the	direction	of	some	unit‐norm	

reference	vector.		Accordingly,	the	left‐hand‐side	is	the	dot	product	of	voltage	or	

current	with	the	unit‐norm	vector.		In	ሺ30ሻ	and	ሺ32ሻ,	the	pre‐processed	cut	

constraints	that	specify	a	regular	polygon	are	evenly	spaced	angles	θs	ൌ	2πs/smax

where	s	ൌ	1,	...,	smax .			are	the	tight	cuts	from	the	previous	candidate	solutions,	ሺ31ሻ	

and	ሺ33ሻ,	that	violated	the	nonlinear	constraints,	correspond	to	step	ሺ4ሻ	of	the	ILIV‐

ACOPF	procedure	described	in	Section	5 below.

In	rectangular	form,	real	ሺ26ሻ	and	reactive	power	ሺ28ሻ	injections	and	

withdrawals	in	terms	of	current	and	voltage	are	approximated	by	the	first	order	

Taylor	series.		We	treat	ሺ11ሻ	as	active	constraints,	that	is,	we	only	use	a	Taylor’s	

series	approximation	when	the	constraint	is	violated.			

4. Step‐size	Limits	on	the	Real	and	Reactive	Power	Approximations	

An	unconstrained	local	linearization	can	result	in	relatively	large	errors.	We	

introduce	the	step‐size	constraints,	which	constrain	the	voltages	to	lie	within	a	box	

centered	on	the	base	point,	vr n,	vj

n,	ir n,	ijn,

		for	the	current	major	iteration	to	correct	

this	problem.	We	control	the	errors	in	pn and	qn by	controlling	the	step‐size	for	the	

voltage	as	follows.	The	equation	for	real	power	is	

					 pnൌvr ni

r n൅vn

jinj,	

and	its	first‐order	Taylor	approximation	around	vr n,	vj

n,	ir n,	ijn

		is		

					 papprox n	ൌ vr ni

r n ൅	vj

ni

j

n ൅	vr ni

r n ൅	vj

ni

j

n ‐	ሺvr ni

r n ൅	vn

jinjሻ.	

The	approximation	error	is	

					 pn‐papprox nൌሺvr n‐	vr nሻሺi

r n	‐	ir nሻ൅ ሺvj

n	‐	vjnሻሺi

j

n	‐	ijnሻ.	

Similarly,	for	reactive	power,	

					 qnൌ	vj

ni

r n‐vr nij

n,	and	

					 qapprox n	ൌ	vj

ni

r n‐vr nij

n‐	vr ni

j

n൅vr ni

j

n‐ሺvjni

r n‐vr nij

nሻ.		
Page 9

The	approximation	error	is	

					 qn‐qapprox nൌ	ሺvj

n	‐	vjnሻሺi

r n	‐	ir nሻ൅ ሺvr n	‐	vr nሻሺij

n	‐	ijnሻ.	

Since	the	bus	currents,	ሺ2ሻ‐ሺ5ሻ,	are	a	linear	function	of	voltage	differences,	the	

difference	in	bus	currents	can	be	expressed	as	

					 i

r n‐ir n	ൌ	∑m	gnmሾሺvr n	‐	vr nሻ	–	ሺvr m	‐	vr mሻሿ	‐	∑m	bnmሾሺvj

n	‐	vjnሻ	–	ሺvjm	–	vjmሻሿ		

and		

					ij

n‐ijn	ൌ	∑m	gnmሾሺvn

j	‐	vj

nሻ	–	ሺvjm	‐	vjmሻሿ ൅ ∑m	bnmሾሺvr n	‐	vr nሻ	–	ሺvr m	–	vr mሻሿ.	

If	|vn

r 	‐	vr n|,	|vnj	‐	vj

n|,|vr m	‐	vr m|,	|vjm	‐	vjm|	are	all	less	than	δ,	then	

					 |i

r n‐ir n| ൑ ∑m	2δሺ|gnm| ൅ |bnm|ሻ	ൌ	2δሾሺ∑m|gnm|ሻ ൅ ሺ∑m|bnm|ሻሿ.	

Similarly,		

|i

j

n‐ijn| ൑	2δሾሺ∑m|gnm|ሻ ൅ ሺ∑m|bnm|ሻሿ.	

Using	definition	of	the	admittance	matrix,	∑mሺ്nሻ gnm	ൌ	‐gnn	and	∑mሺ്nሻ	bnm	ൌ	‐bnn,,

					 |i

r n‐ir n| ൑	2δሺ|gnn| ൅ |bnn|ሻ.	

and		

					 |i

j

n‐ijn| ൑	2δሺ|gnn| ൅ |bnn|ሻ.	

Our	original	worst	case	bounds	are	then:	

					|pn‐papprox n|	൑ |ሺvr n	‐	vr nሻሺi

r n	‐	ir nሻ൅ ሺvj

n	‐	vjnሻሺi

j

n	‐	ijnሻ| ൑	4δ2ሺ|gnn| ൅ |bnn|ሻ		

and

|qn‐qapprox n| ൑	4δ2ሺ|gnn| ൅ |bnn|ሻ.			

	 In	order	to	control	the	errors	that	arise	from	linearly	approximating	pn	and	qn,

and	also	to	prevent	periodic	behavior	in	the	iterative	procedure	that	slows	

convergence	and	may	not	terminate	with	an	optimal	solution,	we	introduce	new	

constraints	on	|vn

r	‐	vnr |	and	|vnj	‐	vnj|,	so	that	they	are	more	and	more	tightly	

restricted	at	each	major	iteration.		In	the	first	major	iteration,	vr and	vj	are	allowed	

to	vary	as	long	as	they	satisfy	the	preprocessed	voltage	cuts.		In	subsequent	major	

iterations,	each	must	stay	within	a	square	centered	on	the	value	from	the	previous	

solution.		The	size	of	these	squares	decreases	over	successive	major	iterations.		We	

added	the	following	constraints	to	the	ILIV‐ACOPFሺhሻ:	

‐avmax/hb ൑	vr n‐vr n ൑	avmax/hb		 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 ሺ34ሻ	

			‐avmax/hb ൑	vj

n‐vjn ൑	avmax/hb 		 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 	 ሺ35ሻ	

In	ሺ34ሻ	and	ሺ35ሻ,	we	tried	b	ൌ	1	ሺlinear	stepsize	decayሻ	and	b	ൌ	2	ሺquadratic	

step‐size	decayሻ.		The	results	of	the	experiments	are	displayed	below.		In	the	

systems	we	studied,	vmax n is	the	same	for	all	n,	so	we	drop	the	n	subscript.			
Page

10

	

5. Linear	Iteration	Approach	

The	ILIV‐ACOPF	method	is:	

1ሻ	Set	h	ൌ	0.	Choose	a	starting	point	vr n0	and	vi

n0.		Add	a	circumscribing	

polygon	for	each	maximum	voltage	magnitude	and	maximum	current	magnitude	

constraint.		Approximate	pn	and	qn	with	hyperplanes	that	are	tangent	to	the	

constraint	hypersurfaces.	

2ሻ	Set	h	ൌ	h൅1.		Solve	the	resulting	ILIV‐ACOPFሺhሻ	to	obtain	optimal	values	for	

the	current	and	voltage,	vr nh,	vj

nh,	ir nh,	ijnh

3ሻ	Check	the	result	for	convergence	of	the	optimal	values	using	the	actual	

nonlinear	pn	and	qn	equations	ሺ6ሻ	through	ሺ12ሻ.	If	within	convergence	tolerance,	

stop;	otherwise	continue.	

4ሻ	Add	another	set	of	tight	constraints	to	the	iterative	convex	approximation	

cutting	off	the	optimal	solution	to	the	linear	approximation,	but	an	infeasible	

solution	to	the	nonlinear	formulation.	Relinearize	the	pn	and	qn	approximations	and	

adjust	the	stepsize.	Go	to	step	2.	

We	presently	describe	the	step	ሺ3ሻ	convergence	criteria	in	detail.		At	each	major	

iteration	h,	the	optimal	solution	to	the	linear	approximation	produces two	sets	of	

values	for	the	nonlinear	equations	of	the	voltage	magnitude	and	real	and	reactive	

power	injection/withdrawal: ሺiሻ	The	first	order	Taylor	series	approximations	pnh,	

qnh	and	vnh,	employed	in	the	linearization,	and	ሺiiሻ	the	actual	“nonlinear”	values	that	

the	decision	variables	give	rise	to	when	plugged	in	to	the	true,	nonlinear	equations.		

The	absolute	percentage	by	which	the	nonlinear	values	violate	the	constraints	ሺ7ሻ,	

ሺ9ሻ,	and	ሺ10ሻ	is	calculated	for	each	bus.		If	the	violated	p	or	q	constraint	ሺmin	or	

maxሻ	level	is	non‐zero,	the	percentage	violation	is	the	absolute	value	of	the	

difference	of	the	nonlinear	value	and	the	violated	constraint,	divided	by	the	

constraint	value.		If	the	right‐hand‐side	of	the	violated	constraint	is	zero	ሺwhich	can	

occur	for	p	and	q,	but	not	vሻ	since	the	previous	calculation	would	result	in	division	

by	zero,	the	percentage	violation	is	taken	to	be	the	nonlinear	power	

injection/withdrawal	divided	by	the	respective	real	or	imaginary	component	of	the	

power	flowing	through	the	node.		Then,	for	each	of	vn,	pn,	and	qn,	these	nodal	

percentage	constraint	violations	are	aggregated	across	nodes	into	two	metrics	of	

aggregate	violation:	the	maximum	percentage	violation	over	nodes,	and	the	sum	of	

percentage	violations	over	nodes.		For	each	of	the	two	aggregate	metrics,	the	three	

values	corresponding	to	violations	of	constraints	on	v,	p	and	q	are	added	together.		

After	some	experimentation,	we	settled	on	setting	the	threshold	for	the	sum	of	

summed	violations	to	be	five	times	the	threshold	for	sum	of	maximum	violations	

ሺhenceforth	“max	percentage	violations”ሻ.			When	these	feasibility	criteria	are	met,	

the	solution	is	determined	to	be	AC	feasible	and	the	procedure	is	terminated.		If	
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11

these	violation	criteria	are	not	met,	the	solution	is	determined	to	not	be	AC	feasible	

and	we	continue	the	iterative	procedure.		

	

	

6. Computational	Testing		

Problems.	The	test	problems	consistent	of	the	14,	30,	57,	and	118	bus	IEEE	test	

cases		ሺsee	Table	1ሻ	at	http://www.ee.washington.edu/research/pstca/index.html.		

The	quadratic	generator	costs	come	from	MATPOWER	ሺsee	Zimmerman	et	alሻ.		We	

formulate	the	20‐step	linear	approximation	to	each	quadratic	function.	Where	

there	are	multiple	transmission	lines	between	two	nodes,	the	lines	are	aggregated	

into	an	equivalent	single	line	between	the	two	nodes.		Each	test	problem	has	two	

levels	ሺtight	and	looseሻ	of	line	current	constraints	ሺsee	Lipka	et	al,	2013ሻ.		

Table	1:		IEEE	Test	Bus	System	Data	

Buses	 Lines	

Generators	 Total	

Demand

Best	Known	Value	

Tight	Current	

Limit

Best	Known	Value	

Loose	Current	

Limit

No.		 Capacity quadratic linear quadratic	 linear	

14	 20	 5 7.724 2.590 105.4 107.4 85.3 86.5

30	 41 6 326.80 42.42 5.89 6.10 5.79 5.98

57	 80 7 326.78 235.26 421.5 432.2 419.2 425.5

118	 186 54 99.66 42.42 1364.9 1388.4 1300.1 1315.5

Hardware	and	Software.		The	problems	were	solved	on	an	Intel	Xeon	E7458	server	

with	8	64‐bit	2.4GHz	processors	and	64	GB	memory.	However,	no	parallelization	

was	used	during	one	problem.		Minor	differences	in	solution	times	were	recorded	

when	the	problems	were	run	at	different	times,	but	the	differences	were	small	

enough	to	be	considered	background	noise.		The	problems	were	formulated	in	

GAMS	23.6.2.		The	nonlinear	programs	used	solver	IPOPT	version	3.8.		Linear	

programs	used	GUROBI	version	4.0.0	with	the	aggressive	presolve	option.		The	

implementation	was	simplistic	in	that	each	linear	program	was	solved	from	scratch	

at	each	major	iteration.	Starting	from	where	the	previous	linear	program	

terminated	was	not	an	option	in	the	GAMS	solver.	There	may	be	easily	gained	

speedups	by	not	starting	each	major	iteration	from	scratch.	

Optimization	Parameters	Settings.	For	IPOPT,	we	use	the	default	parameters.		

For	step‐size	constraints,	we	examine	bൌ1	ሺlinear	step‐sizeሻ	and	bൌ2	ሺquadratic	

step‐sizeሻ	and	a	ൌ	0.5	and	1.	We	choose	16‐	and	32‐sided	circumscribing	polygons	

based	on	the	testing	in	Pirnia	et	al	ሺ2012ሻ		

Initalization.		As	the	initial	starting	point,	we	use	a	flat	start,	vr n	ൌ	1	and	vi

nൌ	0,	for	

all	buses	n.	Other	starting	points,	for	example,	hot	starts,	random	starts	and	DC	

starts,	can	be	used	ሺsee	Castillo	et	al	2013ሻ.		In	commercial	practice,	a	hot	start	may	
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12

be	available	from	previous	solutions,	that	is,	the	solution	from	the	last	time	period	

or	similar	topology.		

Termination.		The	maximum	number	of	major	iterations	was	set	to	100.		We	

examine	two	sets	of	convergence	criteria:	ሺiሻ	0.1%	for	the	max	percentage	

violations	and	0.5%	for	the	summed	violations	ሺa	“tolerance”	of	.001ሻ,	as	well	as	

ሺiiሻ	0.5%	for	the	max	violations	and	2.5%	for	the	summed	violations	ሺa	tolerance	of	

.005ሻ.		The	maximum	CPU	time	was	never	reached.	

	

14‐bus	problem.	The	results	for	the	14‐bus	problem	are	shown	in	Tables	2	and	3.		

The	objective	function	values	were	within	.5	percent	of	the	best‐known	value.	The	

linear	approximations	with	step‐size	limits	solve	faster	than	the	linear	

approximation	without	step‐size	constraints	and	the	IPOPT	solver.	For	a	tolerance	

of	0.001	with	no	step‐size	limits,	for	both	16	and	32	cuts	at	the	loose	current	limits	

and	for	32	cuts	with	tight	current	limits,	the	procedure	was	terminated	at	the	major	

iteration	maximum.		However,	by	restricting	the	step‐size	for	the	voltages,	the	

program	converges	in	less	than	8	major	iterations.		It	takes	less	time	to	solve	with	

16	preprocessed	cuts	than	32.		For	the	loose	current	limits	with	step‐size	

restrictions	ሺsee	Table	2ሻ,	the	quadratic	step‐size	converged	to	similar	objective	

values	in	fewer	iterations	than	the	linear	step‐size.		The	tighter	tolerance	level	

takes	on	average	5.9%	more	iterations	ሺalthough	sometimes,	mysteriously,	

tightening	the	requirements	for	the	feasibility	check	actually	reduced	the	number	

of	major	iterationsሻ.		For	the	tight	current	limits	problem	ሺsee	Table	3ሻ	and	the	

tighter	tolerance	with	16	pre‐processed	cuts,	the	performance	was	almost	the	same	

with	and	without	step‐size	limits.	Overall,	the	performance	was	best	with	a	16	

preprocessed	cuts	and	a	quadratic	step‐size	with	a	ሺthe	step‐size	coefficientሻ	ൌ	.5.		

Table	2.	14‐Bus	Case	with	Loose	Current	Limits	

No	

Step‐size	

Constraints

Linear

Step‐size

Quadratic	

Step‐size	 IPOPT	

		 Cuts	 Tolerance	 a	ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	 results	

Objective	 16		 0.001	 86.101 86.098 86.098 86.341 86.137	 86.5	

Value	 16		 0.005	 85.957 86.098 85.942 86.211 86.153	 86.5	

		 32		 0.001	 86.101 86.019 86.102 86.202 86.146	 86.5	

		 32		 0.005	 85.959 86.097 85.959 86.202 86.624	 86.5	

CPU		 16		 0.001	 4.857 0.238 0.367 0.209 0.244	 0.91	

Time		 16		 0.005	 0.3 0.216 0.395 0.154 0.166	 0.91	

	seconds	 32		 0.001	 6.86 0.363 0.594 0.246 0.322	 0.91	

		 32		 0.005	 0.304 0.319 0.383 0.219 0.311	 0.91	

Number		 16		 0.001	 100 4 444 n/a	

of	Major	 16		 0.005	 7 4 7 3 3 n/a	

Iterations		 32		 0.001	 100 5 8 3 4	 n/a	

		 32		 0.005	 5 4 5 3 5	 n/a	
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Table	3	14‐Bus	Case	with	Tight	Current	Limits	

		 No	

Step‐size

Constraint

Linear

Step‐size

Quadratic

Step‐size

IPOPT	

Results	

		 Cuts	 Tolerance	 a	ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	 	

Objective.	 16		 0.001	 105.69 105.69 105.69 106.50 105.70	 107.4	

Value	 16		 0.005	 105.69 105.69 105.69 105.00 105.70	 107.4	

		 32		 0.001	 86.1 106.53 106.53 106.59 106.53	 107.4	

		 32		 0.005	 85.96 106.53 106.53 106.59 106.53	 107.4	

CPU		 16		 0.001	 0.299 0.3 0.271 0.301 0.279	 1.16	

	Time	 16		 0.005	 0.305 0.282 0.262 0.17 0.27	 1.16	

		seconds	 32		 0.001	 8.93 0.322 0.32 0.293 0.325	 1.16	

		 32		 0.005	 0.907 0.331 0.307 0.29 0.315	 1.16	

Number		 16		 0.001	 5 5 5 5 5	 n/a	

of	Major	 16		 0.005	 5 5 5 3 5	 n/a	

Iterations	 32		 0.001	 100 4 4 4 4	 n/a	

		 32		 0.005	 11 4 4 4 4	 n/a	

	

30‐bus	problem.	The	results	for	the	30‐bus	problem	are	shown	in	Tables	4	and	5.		

The	objective	function	values	were	within	1	percent	of	the	best‐known	value.		The	

linear	approximations	with	step‐size	limits	are	faster	than	the	linear	approximation	

without	step‐size	constraints	and	the	IPOPT	solver.		In	terms	of	CPU	time,	the	

approximation	with	16	preprocessed	cuts	outperforms	the	approximation	with	32.		

In	all	cases,	the	number	of	major	iterations	is	3	or	4.		For	the	.005	and	.001	

tolerance	and	the	loose	current	constraint,	the	quadratic	step	performs	best	with	

16	preprocessed	cuts.		For	the	.005	and	.001	tolerance	and	the	tight	current	

constraint,	there	is	no	clear	step‐size	winner.		Overall,	there	is	not	much	difference	

with	different	step‐sizes.			

	 	
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14

Table	4.	30‐Bus	Case	with	Loose	Current	Limits	

No

Step‐size

Constraints

Linear	

Step‐size

Quadratic	

Step‐size

IPOPT	

Results	

		 Cuts	 Tolerance a	ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	

Objective	 16		 0.001 5.98 5.98 5.98 5.98 5.98 5.98

Value	 16	 0.005 5.98 5.98 5.98 5.98 5.98 5.98

		 32	 0.001 5.98 5.98 5.98 5.98 5.98 5.98

		 32	 0.005 5.98 5.98 5.98 5.98 5.98	 5.98

CPU		 16	 0.001 0.709 0.664 0.88 0.666 0.738 2.88

	Time	 16	 0.005 0.751 0.647 0.721 0.508 7.52 2.88

		seconds	 32	 0.001 0.945 0.892 0.84 0.869 0.942 2.88

		 32	 0.005 0.938 0.882 0.911 0.926 0.904 2.88

Number		 16	 0.001 4 4 4 4 4 n/a

of	Major		 16	 0.005 4 4 4 3 4 n/a

Iterations	 32	 0.001 3 3 3 3 3 n/a

		 32	 0.005 3 3 3 3 3 n/a

Table	5.	30‐Bus	with	Tight	Current	Limits	

No

Step‐size

Constraints

Linear

Step‐size

Quadratic

Step‐size

IPOPT	

Results	

		 Cuts	 Tolerance a	ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	

Objective	 16		 0.001 6.1 6.09 6.1 6.1 6.1 6.10

Value	 16	 0.005 6.06 6.06 6.06 6.06 6.06 6.10

		 32	 0.001 6.1 6.1 6.1 6.08 6.1	 6.10

		 32	 0.005 6.08 6.08 6.08 6.08 6.08 6.10

CPU		 16	 0.001 0.817 0.67 0.617 0.773 0.697 1.55

Time		 16	 0.005 0.572 0.488 0.507 0.535 0.53 1.55

		seconds	 32	 0.001 1.287 1.21 1.161 0.854 1.21 1.55

		 32	 0.005 1.02 0.891 0.861 0.852 0.992 1.55

Number		 16	 0.001 4 4 4 4 4 n/a

of	Major		 16	 0.005 4 3 3 3 3	 n/a

Iterations		 32	 0.001 4 4 4 3 4 n/a	

		 32	 0.005 3 3 3 3 3 n/a

57‐bus	problem.		The	results	for	the	57‐bus	problem	are	shown	in	Tables	6	and	7.		

The	objective	function	values	were	within	2	percent	of	the	best‐known	value.		With	

loose	current	limit,	a	quadratic	step‐size	function	with	a	ൌ	.5	and	16	preprocessed	

cuts	performs	best.		With	the	tight	current	limit,	the	different	step‐size	constraint	

parameters	do	not	produce	a	clear	winner.		It	takes	less	time	to	solve	with	16	

preprocessed	cuts	than	preprocessed	32	cuts.	
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15

Table	6.	57‐Bus	Case	with	Loose	Current	Limits	

No	

Step‐size

Constraints

Linear	Step

Size	Rate

Quadratic		

Step‐size		

IPOPT		

Results	

		

		

Cuts	 Tolerance	 a	ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	

Objective	 16		 0.001	 422.6 422.6 422.6 424.3 423.27	 425.5	

Value	 16		 0.005	 422.6 422.6 422.6 422.84 423.27	 425.5	

		 32		 0.001	 422.42 422.96 422.42 424.41 423.47	 425.5	

		 32		 0.005	 422.42 422.8 422.42 423.9 423.47	 425.5	

CPU		 16		 0.001	 2.3 2.14 2.425 1.94 2.12	 12.05	

	Time	 16		 0.005	 2.317 1.93 2.205 1.53 2.1	 12.05	

	seconds	 32		 0.001	 5.27 6.38 4.3 3.68 3.69	 12.05	

		 32		 0.005	 4.98 5.3 4.67 2.88 3.73	 12.05	

Number		 16		 0.001	 4 4 4 4 4	 n/a	

of	Major		 16		 0.005	 4 4 4 3 4	 n/a	

Iterations		 32		 0.001	 5 7 5 4 4	 n/a	

		 32		 0.005	 5 6 5 3 4	 n/a	

Table	7.	57‐Bus	Case	with	Tight	Current	Limits	

No

Step‐size

Constraints

Linear	

Step‐size

Quadratic		

Step‐size		

IPOPT	

Results	

		 	Cuts	 Tolerance	 a	ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	

Objective	 16		 0.001	 422.66 422.66 422.66 423.62 424.42	 432.2	

Value	 16		 0.005	 423.43 423.43 423.4 423.62 424.4	 432.2	

		 32		 0.001	 422.55 422.55 422.55 425.13 423.69	 432.2	

		 32		 0.005	 422.71 422.71 422.71 424.01 423.69	 432.2	

CPU		 16		 0.001	 3.13 2.79 2.74 1.52 2.82	 8.42	

Time	 16		 0.005	 1.943 1.59 1.64 1.66 2.16	 8.42	

		seconds	 32		 0.001	 9.07 8.74 9.03 3.44 3.6	 8.42	

		 32		 0.005	 4.028 3.69 3.63 2.62 3.71	 8.42	

Number		 16		 0.001	 5 5 5 3 5	 n/a	

of	Major	 16		 0.005	 3 3 3 3 4	 n/a	

Iterations	 32		 0.001	 9 9 9 4 4	 n/a	

		 32		 0.005	 4 4 4 3 4	 n/a	

118‐bus	problem.	The	results	for	the	118‐bus	problem	are	shown	in	Tables	8	and	9.		

The	objective	function	values	were	within	.5	percent	of	the	best‐known	value.		For	

the	loose	current	limits	problem,	the	quadratic	step‐size	with	a	ൌ	.5	converged	

much	faster	than	the	linear	step‐size	and	the	IPOPT	solver.	With	a	tight	current	

limit,	the	linear	step‐size	performed	best,	although	it	converges	to	an	objective	
Page

16

value	about	.5%	below	both	the	IPOPT	optimum	the	quadradic	step‐size	optimum.		

It	takes	less	time	to	solve	with	16	preprocessed	cuts	than	with	preprocessed	32	

cuts.	

Table	8.	118‐Bus	Case	with	Loose	Current	Limits	

No

Step‐size

Constraint

Linear

Step‐size

Quadratic	Step	

Size	Rate

IPOPT	

Results	

		 Cuts	 Tolerance	 a ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	

Objective	 16		 0.001	 1315.2 1315.4 1315.2 1313.6 1313.3	 1315.5	

Value	 16		 0.005	 1307.7 1309.2 1307.7 1313.6 1310.6	 1315.5	

		 32		 0.001	 1314.3 1315.4 1314.3 1312 1314.6	 1315.5	

		 32		 0.005	 1298.1 1308.4 1314.3 1312 1314.6	 1315.5	

CPU		 16		 0.001	 243.27 190.08 348.91 6.83 12.08	 38.75	

Time		 16		 0.005	 29.31 11.3 25.84 6.98 9.7	 38.75	

		seconds	 32		 0.001	 167.31 221.8 148.75 12.31 28.16	 38.75	

		 32		 0.005	 403.95 23.92 148.97 12.23 26.17	 38.75	

Number		 16		 0.001	 100 100 100 3 5	 n/a	

of	Major	 16		 0.005	 11 5 11 3 4	 n/a	

Iterations		 32		 0.001	 41 69 41 3 7	 n/a	

		 32		 0.005	 100 6 41 3 7	 n/a	

Table	9.	118‐Bus	Case	with	Tight	Current	Limits	

No

Step‐size

Constraint

Linear

Step‐size

Quadratic	

Step‐size	

IPOPT	

Results	

		 Cuts	 Tolerance	 a	ൌ	.5 a	ൌ	1 a	ൌ	.5 a	ൌ	1	

Objective	 16	 0.001	 1381.15 1384.47 1384.47 1388.25 1388.31	 1388.4	

Value	 16	 0.005	 1381.15 1381.01 1381.01 1388.25 1388.31	 1388.4	

		 32	 0.001	 1388.36 1384.24 1384.24 1388.65 1388.31	 1388.4	

		 32	 0.005	 1380.39 1380.22 1380.22 1390.52 1388.31	 1388.4	

CPU		 16	 0.001	 12.3 10.04 10.56 11.77 18.47	 32.2	

	Time	 16	 0.005	 8.32 7.9 7.94 11.33 19.3	 32.2	

		seconds	 32	 0.001	 415.541 17.14 17.55 19.23 30.66	 32.2	

		 32	 0.005	 13.26 12.66 12.97 15.48 30.52	 32.2	

Number		 16	 0.001	 3 4 4 5 8	 n/a	

Of	Major		 16	 0.005	 3 3 3 5 8	 n/a	

Iterations	 32	 0.001	 100 4 4 5 8	 n/a	

		 32	 0.005	 3 3 3 4 8	 n/a	

	

	

	 	
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17

7. Conclusions	

Overall,	the	step‐size	constraints	improve	performance	of	the	linear	

procedure,	but	the	best	parameters	of	the	step‐size	constraint	are	problem‐

dependent.		Although	the	convergence	tests	are	different,	the	linear	procedure	with	

step‐size	constraints	is	considerably	faster	ሺup	to	six	timesሻ	the	nonlinear	solver	

and	without	a	step‐size	constraint.		The	tradeoff	between	the	iterative	linearization	

and	the	nonlinear	solver	was	speed	compared	to	greater	accuracy.	Increasing	the	

preprocessed	cuts	from	16	to	32	increases	the	solution	time.	As	the	problem	size	

gets	bigger,	we	see	diminishing	returns	to	the	number	of	preprocessed	constraints.		

The	tighter	tolerance	for	convergence	takes	longer,	but	does	not	seem	to	have	a	

major	impact	on	the	optimal	solution	value,	except	when	there	is	no	step‐size	

function,	where	a	more	restrictive	tolerance	sometimes	results	in	the	linear	

program	not	converging.		Nearly	uniformly,	the	ILIV‐ACOPF	objective	values	are	

slightly	lower	than	the	IPOPT	objective	values,	by	an	average	of	.9%.		Tightening	

the	tolerances	reduced	the	percentage	underestimate	of	cost	from	1.1%	to	.95%,	a	

consistent	but	essentially	negligible	pattern.			

	 	
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18

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