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Optimal	Power	Flow	Paper	8	

Staff	paper	by	

Mehrdad	Pirnia	

Richard	O’Neill	

Paula	Lipka		

Clay	Campaigne

June	2013

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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A	Computational	Study	of	Linear	Approximations	to	the	Convex	

Constraints	in	the	Iterative	Linear	IV‐ACOPF	Formulation	

	

Mehrdad	Pirnia,	Richard	O’Neill,	Paula	Lipka,	Clay	Campaigne	

mpirnia@uwaterloo.ca;	richard.oneill@ferc.gov;	plipka@berkeley.edu;	

claycampaigne@bekeley.edu	

June	2013	

	

Abstract	and	Executive	Summary

In	this	paper,	we	test	the	effect	of	preprocessed	and	iterative	linear	cuts	on	the	convex	constraints	

ሺmaximum	voltage	and	maximum	currentሻ	in	iterative	linear	approximation	of	the	current	voltage	

ሺIVሻ	formulation	of	the	ACOPF	ሺILIV‐ACOPFሻ	problem.	The	nonconvex	constraints	are	linearized	

using	iterative	first	order	Taylor	series	approximation.	The	ILIV‐ACOPF	is	solved	as	a	single	linear	

program	or	a	sequence	of	linear	programs.	The	ILIV‐ACOPF	model	is	tested	on	the	14‐bus,	30‐bus,	

57‐bus	and	118‐bus	IEEE	test	systems.	The	execution	time	and	the	quality	of	the	solution	obtained	

from	the	ILIV‐ACOPF	are	compared	for	different	test	systems	and	benchmarked	against	the	

equivalent	nonlinear	ACOPF	formulation.	The	results	show	execution	time	up	to	8	times	faster	and	

solutions	close	to	the	nonlinear	solver.	The	performance	of	different	algorithmic	parameters	varies	

depending	on	the	test	problem.	As	the	number	of	preprocessed	cuts	for	each	bus	and	line	increases,	

the	relative	error	decreases.	The	results	indicate	that	16	to	32	constraints	are	the	best	number	of	

preprocessed	constraints	in	a	tradeoff	between	accuracy	and	solution	time.	The	marginal	value	of	

more	than	32	and	maybe	16	preprocessed	constraints	in	this	setting	is	negative.	Nevertheless,	the	

number	of	preprocessed	constraints	can	be	set	based	on	the	performance	and	requirements	of	the	

specific	problem	being	solved.	Using	iterative	cuts	often	results	in	faster	convergence	to	a	feasible	

solution.	When	solving	without	using	iterative	cuts,	the	solution	is	within	18%	of	the	best‐known	

nonlinear	feasible	solution;	with	iterative	cuts,	the	solution	is	within	2.5%	of	the	best‐known	

nonlinear	solution.	At	16	or	32	preprocessed	constraints,	solution	time	is	up	to	about	8	times	faster	

than	the	nonlinear	solver	ሺIPOPTሻ.	

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.	Nomenclature	 	 	 	 	 	 	 	 		5	

3.	Formulations	 	 	 	 	 	 	 	 		6	

4.	Linear	Approximations	 	 	 	 	 	 	 		7	

5.	Computational	Results	 	 	 	 	 	 	 		9	

6.	Conclusion	 	 	 	 	 	 	 	 	 15	

References		 	 	 	 	 	 	 	 	 16	
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1.	Introduction	

Since	Carpentier	ሺ1962ሻ	introduced	AC	optimal	power	flow	ሺACOPFሻ	problem,	the	ACOPF	

has	received	considerable	attention.		It	is	at	the	heart	of	power	system	efficiency.		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,	bounds	on	

transmission	lines	flow,	bounds	on	real	and	reactive	power	injections,	and	system	contingencies.	

Since	the	ACOPF’s	introduction,	different	objective	functions	and	formulations	have	been	tried	ሺsee	

Cain	et	alሻ.		The	canonical	ACOPF	problem	uses	real	and	reactive	power	injections	and	the	polar	

representation	of	voltage	for	the	network	equations.		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	power	system’s	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	programming	ሺLPሻ	

methods	play	a	significant	role	in	solving	these	problems	with	more	robust	solutions	and	better	

execution	times.	One	common	linear	program	approach	is	the	distribution	factor	model.	In	another	

linear	program	approach,	the	real	part	of	the	admittance	matrix	is	considered	negligible,	reactive	

power	and	voltage	magnitude	are	dropped	from	the	formulation,	and	bus	voltage	angles	are	

assumed	to	be	near‐zero	ሺStott,	et	al.,	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.		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	variables	above	the	

threshold.	The	iterative	linear	program	approach	finds	a	near‐feasible	near‐optimal	in	almost	all	

problems	and	starting	points.		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	and	using	multiple	solvers.	
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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	test	the	iterative	linearization	ሺILሻ	approach	to	the	IV	ACOPF	formulation,	

proposed	in		ሺO'Neill,	et	al.,	2012ሻ	on	four	IEEE	standard	test	problems	ሺ14‐bus,	30‐bus,	57‐bus	and	

118‐bus	test	systemsሻ	with	line	current	constraints	added.	We	vary	the	number	of	preprocessed	

linear	cuts,	to	approximate	the	convex	quadratic	voltage	and	current	constraints	with	and	without	

iterative	constraints.	We	report	the	solution	time	and	accuracy.	

The	rest	of	the	paper	is	organized	as	follows.	Section	2	presents	the	notation	to	be	used	in	

following	sections.	Section	3	presents	a	brief	review	of	the	IV‐ACOPF	and	ILIV‐ACOPF	formulations.	

Section	4	presents	and	discusses	the	linear	approximation	techniques,	such	as	pre‐processed	cuts,	

iterative	cuts	and	non‐convex	constraint	linearizations,	which	are	used	to	approximate	the	IV‐

ACOPF	formulation.		Section	5	presents	numerical	results	for	the	test	systems	showing	the	

performance	of	this	method	compared	to	the	nonlinear	OPF	problem.	Finally,	Section	6	highlights	

the	main	conclusions	and	contributions	of	the	paper.	The	appendix	contains	the	numerical	results.	

	

2.	Notation	

Indices	and	Sets	

n,	m	 are	bus	ሺnodeሻ	indices;	n,	m	ε	ሼ1,	...,	Nሽ	where N	is	the	number	of	buses.	

k is	a	transmission	element.	 k	ε	ሼ1,	...,	Kሽ	where K is	the	number	of	transmission	elements.	Each

k has	a	pair	of	terminal	buses	n	and	m.

r	 real	part	of	the	complex	number	or	variable	ሺsuperscriptሻ

j	 imaginary	part	of	the	complex	number	or	variable	ሺsuperscriptሻ

j	 is	ሺ‐1ሻ1/2

s is	the	index	of	the	preprocessed	constraints	where s	ε S	ൌ	ሼ0,	1,	...,	S‐1ሽ.	

h indexes	a	major	iteration	ሺsolving	a	linear	programሻ of	the	algorithm

Variables	

pn is	the	real	power	injection	ሺpositiveሻ or	withdrawal	ሺnegativeሻ at	bus n	

qn		 is	the	reactive	power	injection	or	withdrawal	at	bus n

vr n		 real	part	of	the	voltage	at	bus	n

vj

n	 imaginary	part	of	the	voltage	at	bus	n

vn is	the	voltage	magnitude	at	bus	n.	vn ൌ	ሾሺvr n ሻ2൅ 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 is	the	current	magnitude	at	bus n	where		in ൌ	ሾሺi

r n ሻ2൅ ሺij

nሻ2ሿ1/2

i

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

i

j

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

inmk is	the	current	magnitude	into	k at	bus n	ሺto	bus	mሻ;	inmk ൌ	ሾሺi

r nmk ሻ2൅ ሺij

nmkሻ2ሿ1/2

ܸ	 the	complex	vector	of	bus	voltages;	ܸ ൌ ܸ௥ ൅ ࢐ܸ௝

ܫ the	complex	vector	of	bus	current	injections,	ܫ ൌ ܫ ൅ ௥ܫ࢐௝

ܲ the	vector	of	real	power	injections

ܳ the	vector	of	reactive	power	injections
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Parameters	and	Functions	

cp nሺpnሻ			cost	function	of	real	power	pn at	bus	n

cq nሺqnሻ	 cost	function	of	reactive	power	qn at	bus	n

cpl

nሺpnሻ			linear	cost	function	of	real	power	pn at	bus	n

cql

nሺqnሻ	 linear	cost	function	of	reactive	power	qn at	bus	n

bnmk imaginary	part	of	the	admittance	matrix	for	k between	bus	n	and	m

gnmk real	part	of	the	admittance	matrix	for	k between	bus	n	and	m

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 nh Coefficeint	for	real	component	of	bus	voltage	cut	at	n	from	iteration	h	

vj

nh Coefficeint	for	imaginary	component	of	bus	voltage	cut	at	n	from	iteration	h	

i

r nh real	part	of	the	current	at	bus	n at	the	previous	linear		program	solution	

i

j

nh imaginary	part	of	the	current	at	bus	n at	the	previous	linear		program	solution	

i

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

i

r nmkh coefficient	of	the	real	component	of	the	line	flow constraint on	line	k and	iteration h

i

j

nmkh coefficient	of	the	imaginary	component	of	the	line	flow constraint on	line	k and	iteration h

3.	Formulations	

In	2012,	O’Neill	et	al	ሺ2012ሻ	introduced	a	linearization	approach	to	the	IV	formulation.	In	

the	IV‐ACOPF	formulation,	the	network‐wide	nodal‐balance	constraints	are	linear	in	the	current	

and	voltage	ሺI	ൌYVሻ,	rather	than	the	nonconvex	power	flow	equations	based	on	power	and	voltage	

ሺS	ൌ	VI*ሻ.	The	objective	function	is	linearized	using	a	step	function	approximation.	

IV‐ACOPF	Formulation.		The	rectangular	form	of	the	nonlinear	IV‐ACOPF	is	shown	below.		

Min	 ∑n	ሾcpl

nሺpnሻ		൅	cqlnሺqnሻሿ ሺ1ሻ	

i

r n	ൌ	∑mkir nmk	 for	all	n ሺ2ሻ	

i

j

n	ൌ	∑mk i

j

nmk	 for	all	n ሺ3ሻ	

i

r nmk	ൌ	gnmkሺvr n ‐ vr mሻ	‐ bnmkሺvj

n ‐ vjmሻ for	all	k ሺ4ሻ	

	 ij

nmk	ൌ	bnmkሺvr n ‐ vr mሻ	൅ gnmkሺvjn ‐ vjmሻ for	all	k ሺ5ሻ	

pn	ൌ	vr ni

r n ൅	vj

ni

j

n			 for	all	n ሺ6ሻ	

pn	൑	pmax n		 	 	 for	all	n ሺ7ሻ	

pmin n	൑	pn 	 for	all	n ሺ8ሻ	

qn ൌ	vj

ni

r n ‐	vr nij

n	 for	all	n ሺ9ሻ	

qn	൑	qmax n	 	 for	all	n ሺ10ሻ	

qmin n ൑	qn	 	 for	all	n ሺ11ሻ	
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	 ሺvr nሻ2	൅ሺvj

nሻ2	൑	ሺvmax nሻ2 for	all	n ሺ12ሻ	

	 ሺvmin nሻ2	൑	ሺvr nሻ2	൅ሺvj

nሻ2 for	all	n ሺ13ሻ	

	 ሺi

r nmkሻ2	൅ሺij

nmkሻ2	൑	ሺi

max nmkሻ2 for	all	k ሺ14ሻ	

The	above	IV‐ACOPF	formulation	has	6N	variables	as		ሺP,	Q,	Vr ,	Vj,	Ir ,	Ijሻ.	As	shown	in	ሺ2ሻ	‐	ሺ5ሻ,	the	

current	flow	equations	are	linear.	These	equations	replace	the	quadratic	or	trigonometric	forms	of	

the	power	flow	equations	ሺrectangular	and	polar	formsሻ	in	traditional	ACOPF	models.	In	this	

formulation,	real	power	equation	ሺ6ሻ	and	reactive	power	equation	ሺ9ሻ	are	quadratic	nonconvex	

functions	of	current	and	voltage	with	bounds	ሺ7ሻ,	ሺ8ሻ,	ሺ10ሻ,	and	ሺ11ሻ.	Maximum	bus	voltage	

magnitude	and	current	flow	limits	ሺ12ሻ	and	ሺ14ሻ	are	quadratic	convex	constraints.	The	minimum	

voltage	magnitude	constraint	is	a	nonconvex	quadratic	constraint	ሺ13ሻ.		

4.	Linear	Approximations	

Iterative	Linear	Current	Voltage	ሺILIVሻ‐ACOPF.		To	form	the	ILIV‐ACOPF	from	the	above	ACOPF	

problem,	we	replace	the	convex	constraints	with	strong	linear	cuts.	Strong	cuts	are	hyperplanes	

that	are	tangent	to	the	constraint	sets	and	contain	the	constraints.	These	linear	constraints	form	a	

linear	relaxation	ሺouter	approximationሻ	of	nonlinear	convex	constraints.	The	nonconvex	

constraints	are	approximated	by	first	order	Taylor’s	series	approximations.		

Pre‐processed	cuts. Preprocessed	cuts	on	the	current	and	voltage	maximum	magnitude	

constraints	are	put	into	the	model	at	the	first	step.	The	strong	iterative	cuts	at	each	major	

iteration	are	added	when	a	linear	program	solution	violates	a	nonlinear	convex	

constraint.	These	cuts	are	added	and	kept	until	the	convergence	criteria	is	satisfied.	

The	maximum	voltage	and	current	magnitude	constraints	ሺ12ሻ	and	ሺ14ሻ	form	a	circle	with	

its	interior,	generically,	x2 	൅	y2 	൑	r2 .	We	add	strong	linear	cuts	to	create	a	circumscribing	

polygon,	as	shown	in	Figure	1.	For	any	angle	θ,	let	x	ൌ	rcosθ	and	y	ൌ	rsinθ.	Since ሺrcosθሻ2

൅ ሺrsinθሻ2	ൌ	r2 ,	ሺx,	yሻ	is	a	point	on	the	circle	and	xx	൅	yy	൑	r2 is	a	strong	cut	for	any	θ.	

Figure 1: Polygon circumscribing a circle 

Now,	let	θs	ൌ	2π/s,	xs	ൌ	rcosሺθsሻ	and	ys	ൌ		rsinሺθsሻ	where	s	ε	ሼ1,	...,	Sሽ	ൌS.	The	constraint	set	defined	

by S is	a	regular	polygon,	the	maximum	constraint	violation	of	the	circle	is	the	distance	from	a	

vertex	of	the	polygon,	x,	to	the	radius	r,	where	x	ൌ	r/cosሺ2π/Sሻ.	The	maximum	relative	error	of	the	

polygon	is	x/r	ൌ	1/cosሺ2π/Sሻ	as	s	increases	cosሺ2π/Sሻ	approaches	1.	Table	1	shows	x/r	when	S	ൌ	4,	

8,	16,	32	and	64	are	used	to	approximate	the	circle.		For	example,	increasing	the	cuts	from	4	to	16	
Page 8

cuts	increases	the	ratio	x/r	from	1.41	to	1.02.	Adding	32	cuts	provides	a	close	approximation	to	the	

actual	feasible	area	of	the	convex	constraints.		

Table	1:Maximum	relative	error	

Number of 

linear cuts 

Angle in radians 

(π/n) Cos (angle) 1/ Cos (angle) 

4 0.785 0.707 1.414 

8 0.393 0.924 1.082 

16 0.196 0.981 1.020 

32 0.098 0.995 1.005 

64 0.049 0.999 1.001 

Iterative	cuts.	Let	xh,	yh be	a	solution	at	the	major	iteration	h.	If	xh,	yh violates	a	maximum	

magnitude	constraint,	that	is,	zh

2 	ൌ	xh2൅	yh2 	൐	r2 .	Let	a	ൌ	r/zh,	x	ൌ	axh and	y	ൌ	ayh	,	then		

x2൅	y2	ൌ	ሺaxhሻ	2 ൅ ሺayhሻ2	ൌ	a2ሺxh

2 ൅	yh2ሻ	ൌ	a2ሺzh2ሻ	ൌ	r2	

For	voltage	constraints,	x,	y	and	r	are	replaced	with	vr nh,	vj

nh		and	vmax n.	For	the	current

constraints	x,	y	and	r	are	replaced	with	i

r nmh,	ir nmh		and	imax nmk .	These	new	iterative	constraints	

reduce	the	feasible	area	and	make	the	current	optimal	solution	infeasible.	At	each	iteration,	the	

iterative	constraints	are	added:

		vr nh	vr n ൅	vj

nh	vjn	൑	ሺvmax nሻ2

		ir nmkh i

r nmk ൅	ij

nmkh i

j

nmk	൑	ሺi

max nmkሻ2

The	minimum	voltage	constraint	is	addressed	by	an	active	set	approach,	that	is,	it	is	only	activated	

when	the	constraint	is	violated.			

Nonconvex	constraint	linearization.	The	nonconvex	constraint	approximation	is	a	first	order	Taylor	

series	approximation	and	is	updated	at	each	major	iteration.	The	minimum	voltage	constraint	is	

addressed	by	an	active	set	approach.		

Using	the	pre‐processed	iterative	cuts	and	first	order	Taylor	series	approximation	the	linear	

problem	at	major	iteration	h	is:	

LPh ൌ	Min ∑n	ሾcpl

nሺpnሻ		൅	cqlnሺqnሻሿ ሺ15ሻ

i

r n	ൌ	∑mkir nmk for	all	n	 ሺ16ሻ

i

j

n	ൌ	∑mk i

j

nmk for	all	n	 ሺ17ሻ

	 ir nmk	ൌ	gnmkሺvr n ‐ vr mሻ	‐	bnmkሺvj

n ‐ vjmሻ for	all	k	 ሺ18ሻ

	 ij

nmk	ൌ	bnmkሺvr n ‐ vr mሻ	൅	gnmkሺvjn ‐ vjmሻ for	all	k	 ሺ19ሻ

pn	ൌ	vr nhi

r n൅vj

nhi

j

n൅vr ni

r nh൅vj

ni

j

nh‐vr nhi

r nh‐vj

nhi

j

nh for	all	n	 ሺ20ሻ

pn	൑	pmax n		 	 	 for	all	n	 ሺ21ሻ

pmin n	൑	pn for	all	n	 ሺ22ሻ

qn ൌ	vj

nhi

r n ‐	vr nhij

n ൅	vjni

r nh ‐ vr nij

nh ‐ vjnhi

r nh ൅ vr nhij

nh for h	and	all	n	 ሺ23ሻ

qn	൑	qmax n	 	 for	all	n	 ሺ24ሻ

qminn ൑	qn	 	 for	all	n	 ሺ25ሻ

	 cosθsvr n	൅	sinθsvj

n൑	vmax n for	all	n,	s	 ሺ26ሻ
Page 9

vr nh	vr n ൅	vj

nh	vjn	൑	ሺvmax nሻ2 for	all	n,	h	 ሺ27ሻ

	 cosθsi

r nmk	൅sinθsij

nmk	൑	i

max nmk for	all	k, s	 ሺ28ሻ

i

r nmkh ir nmk ൅ ij

nmkh i

j

nmk	൑	ሺi

max nmkሻ2 for	all	k, h	 ሺ29ሻ

Equations	ሺ15ሻ‐ሺ19ሻ,	ሺ21ሻ,	ሺ22ሻ,	ሺ24ሻ	and	ሺ25ሻ	are	the	same	in	both	formulations.	Equations	ሺ26ሻ	

and	ሺ28ሻ	are	the	preprocessed	cuts.	The	minimum	and	maximum	limits	for	real	and	reactive	power	

are	linearized	at	each	iteration	using	first	order	Taylor	series	approximation	in	ሺ20ሻ	and	ሺ23ሻ.	

Finally	a	new	voltage	and	current	cut	for	each	infeasibility	in	ሺ27ሻ	and	ሺ29ሻ	are	added	in	each	

iteration.		

The	linearized	procedure	to	solve	the	above	ACOPF	problem	is:	

1. Initialization:	set	h	ൌ	0	and	vr n0	ൌ	1, vj

n0	ൌ	0	for	all	buses	n.	Compute	the	starting	values	

of	 i

r nmk and	 ij

nmk using	 equations	 ሺ2ሻ	 and	 ሺ3ሻ.	 Calculate	 and	 add	 the	 preprocessed

constraints.	

2. Set	h	ൌ	h൅1.	Solve	the	above	LPh	model.	

3. Compute	the	real	and	reactive	power	magnitude	using	the	nonlinear	equations.	Compute	

the	 voltage	 magnitude	 vnh	 ൌ	 ሾሺvr nhሻ2 ൅ሺvj

nhሻ2ሿ1/2.	 Calculate	 the	 average	 and	 maximum	

percent	violation	of	the	voltage,	real	power,	and	reactive	power	from	their	minimum	and	

maximum	limits.	If	the	sum	of	the	average	percent	violations	of	the	voltage,	real	power,	

and	 reactive	 power	 is	 under	 a	 certain	 threshold,	 and	 the	 maximum	 violations	 of	 the	

nodal	 voltage,	real	power,	and	reactive	power	is	also	under	a	 threshold,	 the	solution	is	

declared	 to	be	AC	 feasible.	The	solution	process	is	stopped	and	 the	answer	is	returned.	

Otherwise	continue.	

4. The	 optimal	 values	 from	 this	 iteration	 are	 used	 to	 calculate	 the	 iterative	 cuts	 and	 the	

Taylor’s	series	approximation	as	described	above.	Go	to	step	2.

Steps	2,	3	and	4	are	called	a	major	iteration.	

5.	Computational	Results	

We	examine	the	performance	of	the	ILIV‐ACOPF	problem	with	the	preprocessed	cuts	and	

with	iterative	cuts	ሺLP‐IterIVCutሻ	and	without	iterative	cuts	ሺLP‐NoIterIVCutሻ.	The	IEEE	14‐bus,	30‐

bus,	57‐bus	and	118‐bus	test	systems	with	tight	and	loose	line	current	limits	are	used	to	test	the	

performance	of	each	procedure.		Since	the	IEEE	test	problems	have	no	generic	line	current	limits,	

we	used	the	approach	in	ሺLipka,	O’Neill	and	Oren,	2013ሻ	to	establish	the	line	current	limits	shown	

in	Table	2.		The	results	are	then	benchmarked	against	a	nonlinear	IV‐ACOPF	problem	ሺNLP൅LP	

Objectiveሻ	with	the	same	linear	objective	function	as	the	linearized	problem.	For	the	14‐bus,	30‐bus	

and	57‐bus	test	systems,	the	convergence	criteria	for	the	sum	of	the	average	relative	ሺto	the	boundሻ	

violations	of	the	voltage,	real	power,	and	reactive	power	is	under	0.005	and	the	sum	of	the	

maximum	violations	of	the	voltage,	real	power,	and	reactive	power	is	under	0.001.	However,	

because	these	tolerance	levels	cause	convergence	difficulty	for	the	118‐bus	system,	they	are	

changed	to	0.05	and	0.01	respectively.		 	
Page

10

Table	2:	Ratios	used	to	calculate	current	limits	and	NLP	objective	value	and	Execution	time	

14	Bus 30	Bus 57	Bus	 118	Bus

Tight	Constraint	

%	of	Maximum	Optimal	Current 18.5 93.3 76.2 24.3

Max	Current	Level	 0.2264 0.3092 1.4027 0.9294

objective	function	value	 107.4 6.092 432.2 1388.4

Execution	time	ሺsecሻ	 1.12 5.23 20.28 87.59

Loose	Constraint	

%	of	Maximum	Optimal	Current 51.1 95.4 77.0 71.9

Max	Current	Level	 0.6246 0.3162 1.4168 2.7536

objective	function	value	 86.51 5.973 425.5 1315.5

Execution	time	ሺsecሻ	 0.89 5.81 30.92 70.72

The	problems	were	solved	on	an	Intel	Xeon	E7458	server	with	8	64‐bit	2.4GHz	processors	

and	64	GB	memory.	The	problems	were	formulated	in	GAMS	ሺRosenthal,	et	al.,	2007ሻ.	The	

nonlinear	solver	used	was	IPOPT	version	3.8	ሺWaechter,	2007ሻ.	The	linear	solver	was	GUROBI	5.0	

ሺRothberg,	et	al.,	2010ሻ.	

In	the	following	sections,	all	results	are	normalized	by	the	results	from	the	IPOPT	nonlinear	

solver	to	demonstrate	a	better	comparison	among	the	models	and	test	problems.	The	actual	values	

are	in	the	appendix.	

	

Effect	of	preprocessed	cuts.		As	shown	in	figures	2‐5,	for	both	loose	and	tight	current	constraints,	as	

the	number	of	preprocessed	cuts	increases,	the	objective	function	value	in	most	cases	gets	closer	to	

the	nonlinear	objective	function	with	iterative	and	without	iterative	cuts.		However,	as	the	number	

of	preprocessed	cuts	increases	beyond	32,	the	execution	time	increases	and	there	is	little,	if	any,	

improvement	in	the	objective	function	value.	The	best	performance	measures	for	the	execution	

time	and	the	objective	function	value	are	obtained	with	16	or	32	preprocessed	cuts.		For	the	16	and	

32	preprocessed	cuts,	in	all	but	one	case	ሺ14‐bus	test	system	with	loose	current	limits	and	iterative	

cutsሻ	the	ILIV	solves	faster	than	the	IPOPT.	In	most	cases,	the	ILIV	takes	less	than	half	the	CPU	time.		

Effect	of	iterative	cuts.		Iterative	cuts	are	important	in	improving	the	accuracy	of	the	objective	

function	value.	In	all	cases,	when	the	iterative	cuts	are	implemented,	the	objective	function	value	is	

significantly	improved	and	is	very	close	to	the	NLP	objective	function	value.	In	addition,	they	can	

reduce	the	execution	time,	as	the	linear	model	converges	much	faster.	For	all	the	test	systems	the	

solution	converged	in	less	than	5	major	iterations.	

	
Page

11

	 	

Figure	2.	Objective	Function	Percent	Difference	and	CPU	Time	Relative	to	the	Nonlinear	Solver	for	

Loose	Current	Limit	and	No	Iterative	Cuts	

Figure	3.	Objective	Function	Percent	Difference	and	Execution	Time	Relative	to	the	Nonlinear	Solver	

for	Loose	Current	Limit	and	Iterative	Cuts

	

Figure	4.	Objective	Function	Percent	Difference	and	Execution	Time	Relative	to	the	Nonlinear	Solver	

for	Tight	Current	Limit	and	No	Iterative	Cuts

0

0.2

0.4

0.6

0.8

1

1.2

1.4

4 16 32 128 360

Percent difference

Number of Cuts

14-bus

30-bus

57-bus

118-bus
Page

12

	 	

Figure	5.	Objective	Function	Percent	Difference	and	Execution	Time	Relative	to	the	Nonlinear	

Solver	for	Tight	Current	Limit	and	Iterative	Cuts	

	

6.	Conclusion	

As	the	number	of	preprocessed	constraints	for	each	bus	and	line	increases,	the	relative	

error	decreases.	For	the	problems	tested,	the	results	indicate	that	16	to	32	constraints	are	

the	best	number	of	preprocessed	constraints	in	a	tradeoff	between	accuracy	and	solution	

time.	The	marginal	value	of	more	than	32	and	maybe	16	preprocessed	constraints	in	this	

setting	is	negative.	Nevertheless,	the	number	of	preprocessed	constraints	can	be	set	based	

on	the	performance	and	requirements	of	the	specific	problem	being	solved.	Using	iterative	

cuts	often	results	in	faster	convergence	to	a	feasible	solution.	When	solving	without	using	

iterative	cuts,	the	solution	is	within	18%	of	the	best‐known	nonlinear	feasible	solution;	with	

iterative	cuts,	the	solution	is	within	2.5%	of	the	best‐known	nonlinear	solution.	At	16	or	32	

preprocessed	constraints,	solution	time	is	up	to	about	8	times	faster	than	the	nonlinear	

solver	ሺIPOPTሻ.	This	approach	may	be	useful	in	solving	optimal	transmission	topology	and	

unit	commitment	problems	because	mixed‐integer	linear	solvers	are	much	faster	than	

mixed‐linear	nonlinear	solvers.	 	
Page

13

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Formulations,”	FERC	Staff	Technical	Paper,	December	2012,	

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Page

15

APPENDIX:		NUMERICAL	RESULTS	

14‐BUS	SYSTEM

Table	A1:		Objective	Function	Values:	14‐bus	system,	tight	current	constraint	

	 Number of	cuts

	 4 16 32 128 360	

LP‐NoIterIVCut	 89.2100 99.9100 105.3400 106.6800 107.0600	

LP‐IterIVCut	 106.5500 105.6900 106.5300 106.9400 107.0700	

NLP൅LP	Obj	 107.3686 107.3686 107.3686 107.3686 107.3686	

Table	A2:	Execution	Time:		14‐bus	system,	tight	current	constraint	

Number of	cuts

4 16 32 128 360

LP‐NoIterIVCut 1.940 0.220 1.070 1.170 2.660

LP‐IterIVCut 0.140 0.300 0.310 1.280 2.580

NLP൅LP	Obj 1.125 1.125 1.125 1.125 1.125

Table	A3:		Objective	Function	Values:		14‐bus	system,	loose	current	constraint	

	 Number of	cuts

	 4 16 32 128 360	

LP‐NoIterIVCut	 79.7700 85.8100 86.0500 86.1600 86.1700	

LP‐IterIVCut	 86.0900 85.9600 85.9600 86.1600 86.1700	

NLP൅LP	Obj	 86.5062 86.5062 86.5062 86.5062 86.5062	

Table	A4:		Execution	Time:		14‐bus	system,	loose	current	constraint	

	 Number of	cuts

	 4 16 32 128 360	

LP‐NoIterIVCut	 1.690 0.170 0.230 0.830 2.420	

LP‐IterIVCut	 0.190 0.790 1.030 1.490 4.760	

NLP൅LP	Obj	 0.891 0.891 0.891 0.891 0.891	

30‐BUS	SYSTEM

Table	A5:	Objective	Function	Values:	30‐bus	system,	tight	current	limit	

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 5.8200 5.9600 6.0300 6.0900 6.1000	

LP‐IterIVCut	 6.0100 6.0600 6.0800 6.0900 6.1000	

NLP൅LP	Obj	 6.0920 6.0920 6.0920 6.0920 6.0920	

Table	A6:	Execution	Time:		30‐bus	system,	tight	current	limit	

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 4.540 0.950 0.990 3.870 10.520	

LP‐IterIVCut	 0.270 0.630 1.310 4.010 10.090	

NLP൅LP	Obj	 5.234 5.234 5.234 5.234 5.234	
Page

16

Table	A7:		Objective	Function	Values:	30‐bus	system,	loose	current	limit	

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 5.8200 5.9300 5.9700 5.9700 5.9700	

LP‐IterIVCut	 5.9500 5.9800 5.9800 5.9700 5.9700	

NLP൅LP	Obj	 5.9738 5.9738 5.9738 5.9738 5.9738	

Table	3:		execution	Time:		30‐bus	system,	loose	current	limit	

Number	of	cuts		 4 16 32 128 360	

LP‐NoIterIVCut	 16.870 1.490 1.420 3.310 10.600	

LP‐IterIVCut	 0.480 1.630 2.400 5.340 15.330	

NLP൅LP	Obj	 5.812 5.812 5.812 5.812 5.812	

57‐BUS	SYSTEM

Table	A9:		Objective	Function	Values:	57‐bus	system,	tight	current	limit	

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 416.9117 422.9959 422.8063 422.4807 422.5419	

LP‐IterIVCut	 422.6372 423.4306 422.7098 422.4812 422.5419	

NLP൅LP	Obj	 432.1900 432.1900 432.1900 432.1900 432.1900	

Table	A10:		execution	Time:		57‐bus	system,	tight	current	limit

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 19.709 2.436 5.780 27.088 87.331	

LP‐IterIVCut	 1.127 2.041 4.832 21.268 59.018	

NLP൅LP	Obj	 20.281 20.281 20.281 20.281 20.281	

Table	A11:		Objective	Function	Values:	57‐bus	system,	loose	current	limit	

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 416.7944 424.7595 422.4737 422.3770 422.4237	

LP‐IterIVCut	 422.0834 422.5992 422.4155 422.3798 422.4236	

NLP൅LP	Obj	 425.4805 425.4805 425.4805 425.4805 425.4805	

Table	A12:		Execution	Time:		57‐bus	system,	loose	current	limit

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 16.144 1.969 4.545 21.344 76.306	

LP‐IterIVCut	 0.989 2.874 6.614 24.608 60.344	

NLP൅LP	Obj	 30.922 30.922 30.922 30.922 30.922	
Page

17

118‐BUS	SYSTEM

Table	A13:		Objective	Function	Values:	118‐bus	system,	tight	current	limit	

Number	of	cuts	 4 16 32 128 360	

LP‐ 1357.1281 1383.0689 1379.9237 1379.1194 1379.0062	

LP‐IterIVCut	 1382.1567 1381.0145 1380.2231 1379.1220 1379.0067	

NLP൅LP	Obj	 1388.4251 1388.4251 1388.4251 1388.4251 1388.4251	

Table	A14:		Execution	Time:	118‐bus	system,	tight	current	limit

Number	of	cuts	 4 16 32 128 360	

LP‐NoIterIVCut	 93.134 14.373 18.289 73.726 124.514	

LP‐IterIVCut	 38.001 8.472 15.817 53.749 124.637	

NLP൅LP	Obj	 87.594 87.594 87.594 87.594 87.594	

Table	4:		Objective	Function	Values:	118‐bus	system,	loose	current	limit	

Number	of	cuts	 4 16 32 128 360	

LP‐ 1268.3334 1284.7513 1295.2132 1295.8839 1291.0350	

LP‐IterIVCut	 1287.1712 1287.5420 1298.0719 1291.0000 1296.2755	

NLP൅LP	Obj	 1315.4988 1315.4988 1315.4988 1315.4988 1315.4988	

Table	5:		Execution	Time:	118‐bus	system,	loose	current	limit	

Number	of	cuts		 4 16 32 128 360	

LP‐NoIterIVCut	 68.232 259.395 446.980 1885.862 3869.237	

LP‐IterIVCut	 130.440 256.572 438.003 1497.221 3983.677	

NLP൅LP	Obj	 70.719 70.719 70.719 70.719 70.719