Page 1

Optimal	Power	Flow	Paper	7	

Staff	paper	by	

Paula	A.	Lipka		

Richard	P.	O’Neill	

Shmuel	Oren

April	2013

	 	
Page 2

	

	

Developing	Line	Current	magnitude	Constraints	for		

IEEE	Test	Problems	

Optimal	Power	Flow	Paper	7	

Paula	A.	Lipka,	Richard	P.	O’Neill,	and	Shmuel	Oren	

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

April	2013			

Abstract	and	Executive	Summary	

All	power	system	operators	ensure	their	systems	adhere	to	thermal	limits	on	

transmission	lines	in	order	to	avoid	line	deformation.	Also,	thermal	limits	are	used	

as	surrogates	for	voltage	stability.	The	IEEE	test	problems	do	not	include	data	on	

these	limits.		The	purpose	of	this	paper	is	to	present	a	simple	method	for	

constructing	current	magnitude	constraints	and	to	report	on	the	computational	

properties	in	solving	the	resulting	problems.		This	paper	finds	limits	on	the	

maximum	allowable	current	magnitude	that	result	in	a	feasible	solution	for	the	14‐

bus,	30‐bus,	57‐bus,	and	118‐bus	IEEE	test	problems.		For	each	test	problem,	one	

single	limit	is	applied	to	all	lines	that	makes	the	optimal	solution	without	these	

limits	infeasible.	For	each	problem	we	develop	a	‘tight’	and	a	‘loose’	constraint.		We	

solve	the	resulting	problem	using	the	current	voltage	formulation.	Different	test	

problems	exhibit	different	characteristics.	Including	these	constraints	in	the	ACOPF	

increases	the	solution	time	between	2	to	20	times	and	costs	ሺobjective	functionሻ	up	

to	25	percent.	

	

	

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

Federal	Energy	Regulatory	Commission	or	any	of	its	Commissioners.	 	

	 	
Page 3

Table	of	Contents	

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

2.	Notation	............................................................................................................	4	

3.	Current‐Voltage	ACOPF	Model	................................................................	5	

4.	Methods	............................................................................................................	6	

5.	Computational	Analysis	..............................................................................	.7	

6.	Conclusions	......................................................................................................	17	

References	............................................................................................................	18	

Appendix	...............................................................................................................	21	

	 	
Page 4

1. Introduction	

The	amount	of	current	that	can	flow	through	power	system	transmission	

assets	ሺreferred	to	here	as	linesሻ	is	limited	by	thermal	restrictions.		The	thermal	

ratings	of	the	transmission	are	functions	of	the	materials	that	compose	the	assets.	

The	excessive	heat	caused	by	the	line	current	can	deform	and	degrade	transmission	

lines	and	cause	them	to	sag.	Heat	losses	are	proportional	to	current	magnitude	

squared.		In	addition,	current	magnitude	constraints	are	often	used	as	surrogates	for	

other	constraints	such	as	voltage	stability.		However,	most	IEEE	test	problems	do	

not	include	current	magnitude	limits	on	the	transmission	lines	even	though	they	are	

an	important	aspect	in	model	testing.			

In	the	absence	of	these	constraints,	one	approach	is	to	put	in	constraints	

based	on	physical	characteristics.	Often,	there	is	little	information	about	the	lines.	It	

takes	considerable	time	to	develop	constraints	based	on	physical	characteristics,	

and	the	result	may	not	be	binding	constraints.		

The	purpose	of	this	paper	is	to	develop	a	methodology	for	creating	line	

current	magnitude	constraints	using	a	set	of	the	IEEE	test	problems.		We	are	

interested	in	creating	binding	constraints	based	on	maximum	current	magnitude	

rather	than	on	apparent	power	on	the	lines	or	on	voltage	angle	differences.	The	

approach	we	employ	is	to	create	constraints	from	the	optimal	solution	without	

these	constraints.		With	these	constraints,	the	resulting	power	flow	problem	has	a	

feasible	solution.	Subsequent	testing	helps	to	understand	how	constraining	the	line	

flows	affects	the	resulting	power	flow	solution	and	solution	time.		

2. Notation	

Variables	and	parameters	are	indexed	over	buses	denoted	by	subscripts	n

and	m.	Transmission	lines	are	indexed	by	terminal	buses	n	and	m	and	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.	

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

vj

n	 imaginary	part	of	the	voltage	at	bus	n

vn	 the	voltage	magnitude	at	bus	n

i

r n	 real	part	of	the	current	injected	at	bus	n

i

j

n	 imaginary	part	of	the	current	injected	at	bus	n

in	 the	current	magnitude	of injection at	bus	n

i

r nmk real	part	of	the	current	on	line	k at	bus	n to	bus	m
Page 5

i

j

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

inmk the	current	magnitude	on	line	k at	bus	n to	bus	m

Parameters

cp nሺpnሻ cost	function	of	real	power	injected	by	a	generator	at	bus	n

cq nሺqnሻ cost	function	of	reactive	power	injected	by	a	generator	at	bus	n

cpl nሺpnሻ linear	cost	function	of	real	power	by	a	generator	at	bus	n

cql nሺqnሻ linear	cost	function	of	reactive	power	by	a	generator	at	bus	n

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

gnmk real	part	of	the	admittance	matrix	for	line	k between	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

i

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

3. Current‐Voltage	ACOPF	Model	

The	current‐voltage	ሺIVሻ	ACOPF	formulation	is	used	to	find	a	set	of	voltages	and	

currents	at	each	bus	and	currents	on	each	transmission	line	that	minimize	the	

objective	function	in	terms	of	real	and	reactive	power.		More	detail	can	be	found	

in	Cain	et	al	ሺ2012ሻ	and	O’Neill	et	al	ሺ2012ሻ.		The	IV‐ACOPF	model	is:	

Minimize	∑n cp n ሺpnሻ൅cq n ሺqnሻ	

Subject	to	

i

r nmk	ൌ	gnmkሺvr n	‐ vr mሻ	‐	bnmkሺvj

n	‐ vjmሻ

i

j

nmk	ൌ	bnmkሺvr n	‐ vr mሻ	൅	gnmkሺvjn	‐ vjmሻ

i

r n	ൌ	∑mki	r nmk	

i

j

n	ൌ	∑mki	jnmk	

pn	ൌ	vr ni

r n	൅	vj

ni

j

n		

pmin n ൑	pn ൑	pmax n

qn	ൌ	vj

ni

r n	‐	vr nij

n		 		 	 	 	 	 	

qmin n ൑	qn ൑	qmax n		 	 	

ሺvr nሻ2	൅	ሺvj

nሻ2 ൑ ሺvmax nሻ2		 	 	 	

ሺvmin nሻ2	൑ ሺvr nሻ2	൅	ሺvj

nሻ2			

ሺi

r nmkሻ2	൅	ሺij

nmkሻ2 ൑ ሺi

max nmkሻ2	
Page 6

The	objective	function	for	test	problems	is	generally	presented	as	a	quadratic	

function,	but	many	applications	require	linear	bid	functions.		The	IV‐ACOPF	with	

linearized	objective	function	program	has	the	same	constraints	as	the	nonlinear	

model,	but	the	objective	function	is	a	step	function	approximation	of	the	nonlinear	

objective	function.		In	solving	the	ACOPF	models,	we	allow	for	an	infeasible	answer	

that	is	penalized	by	the	amount	the	system	is	infeasible.	We	add	cPenalty ሺvr ,	vj,	ir ,	ij,	p,	

qሻ		to	the	linearized	objective	function,	where	the	quantity	x൅	is	equal	to	maxሺx,	0ሻ,	

that	is,	if	x	is	positive,	x൅ ൌ	x;	if	x	is	negative,	then	x൅ ൌ	0.		For	example,	if	the	real	

power	is	greater	than	the	max,	the	objective	function	is	penalized	by	that	quantity	

times	some	cost;	if	it	is	less	than	or	equal	to	the	maximum,	there	is	no	penalty.	

					cPenalty ሺvr ,	vj,	ir ,	ij,	p,	qሻ	ൌ	∑nmk cpennሺvn	‐	vmin nሻ൅	 ൅	cpennሺvmax n		‐	vሻ൅൅	

															cpennሺinmk	‐	imax nmkሻ൅			൅	cpennሺpn	‐	pmax nሻ൅	൅	cpennሺpmin n	‐	pnሻ൅	൅		

															cpennሺqmin n	‐	qnሻ൅	൅	cpennሺqn	‐	qmax nሻ൅	

The	system	infeasibilities	could	possibly	occur	because	the	voltage	is	above	the	

maximum	or	below	the	minimum	levels,	because	the	current	is	above	the	maximum	

level,	or	because	real	or	reactive	power	violate	maximum	or	minimum	limits,	as	

detailed	in	the	penalty	cost.		Here	we	set	cpenn	ൌ	105.	A	problem	is	declared	

infeasible	if	the	solution	sets	cPenalty to	be	greater	than	10‐3.

4. Methods	

We	considered	two	methods	for	creating	line	current	magnitude	constraints.		

In	both	these	methods,	we	solved	the	alternating	current	optimal	power	flow	model	

without	line	constraints	and	extracted	the	optimal	current	magnitudes	of	each	line	

from	the	optimal	solution.			

In	the	first	method,	we	constrained	the	current	magnitude	on	each	line	to	be	a	

fraction	ሺless	than	1ሻ	of	its	optimal	current	magnitude	in	the	unconstrained	

problem.		However,	this	method	did	not	always	return	feasible	solutions.		This	is	

likely	because	the	unconstrained	optimization	minimizes	losses	by	lowering	current	

magnitudes	subject	to	all	other	constraints	in	the	model.		Therefore,	it	may	not	be	

feasible	for	all	line	currents	to	be	restricted	to	something	lower	than	in	the	

unconstrained	solution.		

In	the	second	method,	we	constrained	the	current	magnitude	on	each	

transmission	line	to	some	fraction	of	the	highest	optimal	current	magnitude	over	all	

lines	in	the	unconstrained	problem.		This	method	returned	local	optimal	solutions	

ሺsince	the	problem	is	non‐convexሻ	for	some	limits,	but,	if	the	current	magnitude	

limit	was	too	low,	the	solver	could	not	find	a	feasible	point.		This	current	magnitude	

limit	level	varied	widely	depending	on	the	test	problem.	

Procedure:	
Page 7

1. Solve	the	ACOPF	without	any	thermal	line	constraints	

2. Extract	the	optimal	line	current	magnitudes	inmk* for	current	magnitude	

for	line	k	at	bus	n	to	m.

3. Let	i

max* 	ൌ	maximum	ሼinmk* ሽ	over	all	n,	m,	and	k.	

4. Solve	the	thermally	constrained	problem	by	including	the	constraint		

					inmk ൑	f	*i

max* 	for	all	combinations	of	n,	m,	and	k	where	f	is	a	parameter	

with	 0	൏	f	൏	1.	

5. Computational	Analysis	

The	input	model	data	used	is	from	the	14,	30,	57,	and	118‐bus	IEEE	test	system	

data	at	http://www.ee.washington.edu/research/pstca/index.html.		The	generator	

costs	come	from	MATPOWER	ሺsee	Zimmerman	et.	al.	2011ሻ.		The	quadratic	cost	

parameters	are	shown	in	the	appendix.		Where	there	are	multiple	transmission	lines	

between	two	nodes,	the	lines	are	aggregated	into	an	equivalent	single	line	between	

the	two	nodes.		The	current	magnitude	measurement	is	per	unit.	Table	1	

summarizes	the	test	problem	characteristics.	

Table	1:	IEEE	Test	System	Data	

Buses	 Lines	 Generators Demand Vmax Vmin Best		

Number	 Total	Capacity Known	Value*

14	 20	 5	 7.72 2.59 1.06 0.94 80.81

30	 41	 6	 326.80 42.42 1.10 0.94 5.745

57	 80	 7	 326.78 235.26 1.06 0.94 417.4

118	 186	 54	 99.66 42.42 1.06 0.94 1297

*Quadratic	Objective,	Nonlinear	Constraints	

Hardware.	The	problems	were	solved	on	an	Intel	Xeon	E7458	server	with	8	64‐bit	

2.4GHz	processors	and	64	GB	memory.	Minor	differences	in	solution	times	were	

recorded	when	the	same	problems	were	run	at	different	times,	but	the	differences	

were	small	enough	to	be	considered	background	noise.		

Software.	The	procedures	were	formulated	in	GAMS	23.6.2.	The	nonlinear	solver	

used	was	IPOPT	version	3.8.			

Approach.	The	IV‐ACOPF	formulation,	as	described	above,	was	used	to	solve	the	test	

cases	shown	below.		For	each	test	problem,	a	figure	shows	the	impact	on	the	line	

current	magnitudes	on	test	systems	were	examined	with	the	current	magnitude	

level	at	75%,	80%,	90%,	and	100%	of	the	maximum	current	magnitude	level	in	the	

test	problem	without	line	current	magnitude	constraints.		The	optimal	solution	may	

be	infeasible,	that	is,	have	a	penalty	function	greater	than	zero.		A	second	figure	

shows	the	line	index	versus	the	line	current	magnitude	where	the	line	index	is	

sorted	from	lowest	to	highest	current	magnitude	level.		The	red	solid	line	in	each	

graph	is	the	chosen	tight	current	magnitude	limit	and	the	dashed	green	line	is	the	
Page 8

chosen	loose	current	magnitude	limit.		A	cross	walk	from	line	number	to	the	buses	

the	line	is	connecting	is	in	the	appendix.	

Testing	Current	Magnitude	Constraints.		From	the	testing	described	in	the	previous	

sections,	the	uniform	rule	restricting	current	magnitude	to	the	same	percentage	of	

maximum	current	magnitude	has	a	different	impact	on	the	different	test	cases.		

Therefore,	we	picked	different	percentages	of	current	magnitudes	for	each	case	and	

created	two	sets	of	constrained	problems:		tight	and	loose.		The	tight	current	

magnitude	level	is	near	the	current	magnitude	level	where	the	problem	becomes	

infeasible.		The	loose	current	magnitude	level	is	a	level	that	constrains	the	problem	

but	is	farther	away	from	the	point	of	infeasibility.	These	levels	are	shown	in	Table	2.	

Table	2:		Tight	and	loose	testing	levels	of	current	magnitude	constraints	

	 14‐bus 30‐bus 57‐bus	 118‐bus

Low	Level	ሺTight	Constraintሻ:		

%	of	Maximum	Optimal	Current	magnitude: 18.5 93.3 76.2	 24.3

High	Level	ሺLoose	Constraintሻ:		

%	of	Maximum	Optimal	Current	magnitude: 51.1 95.4 77.0	 71.9

Low	ሺTight	Constraintሻ	Current	magnitude	Level 0.2264 0.3092 1.4027	 0.9294

High	ሺLoose	Constraintሻ	Current	magnitude	Level 0.6246 0.3162 1.4168	 2.7536

	 	
Page 9

14‐Bus	Problem.	In	the	14‐bus	system	when	current	is	not	constrained,	one	line	has	

nearly	twice	the	current	magnitude	as	the	next	highest	current	line.		As	the	current	

magnitude	constraint	is	decreased,	the	effect	on	other	lines	is	shown	in	figures	1	and	

2.		The	current	magnitudes	that	change	the	least	under	restriction	are	the	lines	with	

comparatively	lower	current	magnitudes.		For	the	loose	constraint,	only	one	current	

has	a	binding	constraint.	For	the	tight	constraint,	four	constraints	are	binding	or	

near	binding.		

	

Figure	1.	‘Optimal’	14‐Bus	Current	Magnitudes	per	Line	under	Percent	Restrictions	

		

Figure	2.	14‐Bus	Optimal	Current	Magnitudes	under	the	Loose	and	Tight	Constraints.		

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 2 4 6 8 10 12 14 16 18 20

Line	Current	Magnitude	ሺPer	Unitሻ

Line	Index

14‐bus	Line	Current	Magnitude	as	a	Function	of	

%	Restriction	on	the	Line

100%	Max	Current

90%	Max	Current

80%	Max	Current

75%	Max	Current

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 5 10 15 20

Line	Current	Magnitude	ሺPer	Unitሻ

Line	Index

14‐bus	Line	Current	Magnitude	with	no,	loosely,	and	

tightly	constricting	current	limits

No	Limit

Loose	Limit

Tight	Limit

Loose	Constraint

Tight	Constraint
Page

10

30‐Bus	Problem.	For	the	30‐bus	system,	the	current	level	could	not	be	restricted	

much	lower	than	the	maximum	optimal	current	level	ሺ0.3314ሻ	before	the	system	

became	infeasible.		Restricting	the	current	shifts	the	line	current	magnitudes	more	

than	in	the	14‐bus	problem;	there	are	many	lines	where	current	actually	increases.		

While	the	difference	between	the	tight	ሺ0.3092ሻ	and	loose	ሺ0.3162ሻ	constraints	is	

small,	these	restrictions	do	have	a	large	impact	on	the	resulting	current	through	

some	of	the	lines.	In	both	cases,	only	two	lines	have	binding	constraints	

	

Figure	3.	‘Optimal’	30‐Bus	Current	Magnitudes	per	Line	under	Percent	Restrictions		

	

	

Figure	4.	30‐Bus	Optimal	Current	Magnitudes	under	the	Loose	And	Tight	Constraints.	

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42

Current	Magnitude

Line	Index

30‐bus	Current	per	Line	as	a	Function	of	%	Restriction	

on	the	Line

ሺ75%,	80%,	and	90%	are	Infeasibleሻ

100%	Max	Current

90%	Max	Current

80%	Max	Current

75%	Max	Current

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 3 6 9 12 15 18 21 24 27 30 33 36 39 42

Line	Current	Magnitude	ሺPer	Unitሻ

Line	Index

30‐bus	Line	Current	Magnitude	with	no,	loosely,	and	tightly	

constricting	current	limits

No	Limit

Loose	Limit

Tight	Limit

Loose	Constraint

Tight	Constraint
Page

11

57‐Bus	Problem.		For	the	57‐bus	system,	the	optimal	solution	has	most	of	current	

magnitudes	of	half	or	less	the	highest	current	magnitude,	similar	to	the	14‐bus	

problem.		While	about	half	of	the	lines	exhibit	minimal	change,	the	other	half	have	

currents	that	change	significantly	with	the	restriction.		Like	the	30‐bus	problem,	the	

two	current	restrictions	are	close	together,	with	1.4027	for	the	loose	limit	and	

1.4168	for	the	tight	limit.		In	both	cases,	only	two	lines	have	binding	constraints.		

However,	even	this	small	difference	in	limits	greatly	impacts	the	current	on	some	

lines.

Figure	5.	‘Optimal’	57‐Bus	Current	Magnitudes	per	Line	under	Percent	Restrictions

Figure	6.	57‐Bus	Optimal	Current	Magnitudes	under	the	Loose	and	Tight	Constraints.	

0

0.5

1

1.5

2

0 20 40 60 80

Line	Current	Magnitude	ሺPer	Unitሻ

Line	Index

57‐bus	Current	per	Line	as	a	Function	of	%	Restriction	

on	the	Line

ሺ75%	is	Infeasibleሻ

100%	Max	Current

90%	Max	Current

80%	Max	Current

75%	Max	Current

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

0 20 40 60 80

Line	Current	Magnitude	ሺPer	Unitሻ

Line	Index

57‐bus	Line	Current	Magnitude	with	no,	loosely,	and	tightly	

constricting	current	limits

No	Limit

Loose	Limit

Tight	Limit

Loose	Constraint

Tight	Constraint
Page

12

118‐Bus	Problem.			The	118‐bus	system	has	several	lines	with	higher	currents,	

similar	to	the	30‐bus	system.		The	current	can	be	restricted	greatly	before	the	

problem	becomes	infeasible.		Reducing	current	by	up	to	75%	of	its	original	optimal	

maximum	does	not	have	much	impact	on	line	currents	except	the	highest	ones,	as	

seen	in	Figure	7.		Constraining	the	current	with	the	loose	limit	impacts	the	currents	

slightly;	however,	constraining	the	current	with	the	tight	limit	has	a	major	impact	on	

the	current	magnitudes,	as	in	Figure	8.		Under	the	loose	constraint,	only	one	line	

constraint	is	binding.		Under	the	tight	current	magnitudes,	22	of	the	lines	have	the	

highest	permissible	current	under	this	restriction,	and	even	lines	with	lower	

currents	exhibit	big	differences	from	the	unrestricted	and	loose	current	cases,	both	

higher	and	lower	than	before.

Figure	7.	‘Optimal’	118‐Bus	Current	Magnitudes	per	Line	under	Percent	Restrictions		

	

Figure	8.	118‐Bus	Optimal	Current	Magnitudes	under	Loose	and	Tight	Constraints.	

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150

Line	Current	Magnitude	ሺPer	Unitሻ

Line	Index

118‐bus	Current	per	Line	as	a	Function	of	%	Restriction	on	

the	Line

100%	Max	Current

90%	Max	Current

80%	Max	Current

75%	Max	Current

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

0 50 100 150

Line	Current	Magnitude	ሺPer	

Unitሻ

Line	Index

118‐bus	Line	Current	Magnitude	with	no,	loosely,	and	

tightly	constricting	current	limits

No	Limit

Loose	Limit

Tight	Limit

Loose	Constraint

Tight	Constraint
Page

13

Normalized	Objective	Function	Value	as	a	Function	of	the	Current	Magnitude	Limit	

When	the	current	magnitude	is	restricted,	this	limit	is	binding	on	at	least	one	

line.		As	shown	in	Figure	9,	the	different	IEEE	test	cases	exhibit	infeasibility	

according	to	the	solver	at	different	percentages	of	their	maximum	current	

magnitude.		The	denominator	of	the	normalized	objective	function	is	the	objective	

function	without	constraints.			

Figure	9.	Normalized	Objective	Function	versus	Current	Magnitude	Limit	

	

Normalized	Solution	Time	as	a	Function	of	the	Current	Magnitude	Limit	

As	shown	in	Figure	10,	the	computational	time	to	solve	ሺfind	a	local	optimal	

solution	or	declare	the	problem	infeasibleሻ	the	constrained	problem	only	seems	to	

grow	greatly	when	the	current	magnitude	is	restricted	to	5‐15%	of	its	original	value.		

When	it	is	restricted	to	20%	or	more	of	its	original	value,	the	added	constraints	do	

not	seem	to	increase	or	decrease	the	computational	time	predictably.			

1.00

1.05

1.10

1.15

1.20

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Fraction Restriction on Maximum Optimal Line Current

Normalized Objective Function

14 Bus System

30 Bus System

57 bus System

118 Bus System
Page

14

	

Figure	10:	Impact	of	Current	magnitude	Constraints	on	CPU	Time	

Effect	of	Current	Magnitude	Constraints	on	the	Optimal	Value.		When	binding	current	

magnitude	constraints	are	added	to	the	model,	the	feasible	region	shrinks	and	the	

objective	function	value	increases,	as	shown	in	Figures	11	and	12	and	Table	3.		The	

objective	function	value	of	the	14‐bus	problem	with	the	tight	constraint	is	30	

percent	greater.	For	other	problems,	the	increases	in	objective	function	value	with	

current	magnitude	constraints	were	6	percent	or	less.	

Figure	11.	Value	of	the	quadratic	objective	function	with	current	magnitude	

constraints		

0.1

1

10

100

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

Fraction Restriction on Maximum Optimal Line Current

Normalized CPU Time

14 Bus

30 Bus

57 Bus

118 Bus

Effect on Objective Value of Current Line Constraints on 

Fully Nonlinear IV ACOPF

1.00

1.04

1.08

1.12

1.16

1.20

1.24

1.28

1.32

14 Bus 30 Bus 57 Bus 118 Bus

Ratio of Objective Function to Unconstrained 

objective Function

Tight Constraint

Loose Constraint
Page

15

Figure	12.		Value	of	the	linear	objective	function	with	current	magnitude	constraints	

Table	3:	Objective	function	value	for	each	test	system	and	constraint	type	

Problem	Type	 Constraint	type 14‐bus 30‐bus 57‐bus	 118‐bus

Quadratic	Objective	

Nonlinear	Constraints	

Tight	Constraint 105.4 5.89 421.5	 1364.9

Loose	Constraint 85.3 5.79 419.2	 1300.1

Unconstrained 80.8 5.75 417.4	 1296.6

Linear	Objective	

Nonlinear	Constraints	

Tight	Constraint 107.4 6.10 432.2	 1388.4

Loose	Constraint 86.5 5.98 425.5	 1315.5

Unconstrained 82.8 5.92 423.8	 1311.5

Effect	of	Current	Magnitude	Constraints	on	the	Solution	Time.		Current	magnitude	

constraints	also	impact	the	solution	time	of	each	test	system	as	shown	in	Figures	13	

and	14	and	Table	4.		Adding	constraints	at	least	doubles	the	solution	time	of	the	

problem.		For	the	30‐bus	and	118‐bus	problems,	the	more	tightly	constrained	

problem	solves	faster	than	the	less	constrained	problem.		For	the	14‐bus	and	118‐

bus	problems,	the	less	constrained	problem	solves	faster	than	the	more	constrained	

problem.		The	57‐bus	and	118‐bus	solution	time	took	significantly	longer	than	the	

14‐bus	and	30‐bus	systems,	but	solution	time	is	not	solely	based	on	the	number	of	

buses.		The	linear	objective	function	required	considerably	more	time	to	solve.	

Effect on Objective Value of Current Line Constraints on 

Nonlinear IV ACOPF with Linear Objective Function

1.00

1.05

1.10

1.15

1.20

1.25

1.30

14 Bus 30 Bus 57 Bus 118 Bus

Ratio of Objective Function to Unconstrained 

Objective Function

Tight Constraint

Loose Constraint
Page

16

Figure	13.	CPU	Time	with	current	magnitude	constraints	of	quadratic	objective	

Figure	14.		CPU	Time	of	the	current	magnitude	constrained	linear	objective	

	

Table	4.	CPU	Time	for	all	bus	systems,	constraint	types,	and	model	types	

Problem	Type	 Constraint	type	 14‐bus 30‐bus 57‐bus	 118‐bus

Quadratic	Objective	

Nonlinear	Constraints	

Tight	Constraint 0.72 1.80 8.34	 28.95

Loose	Constraint 0.58 1.94 7.77	 33.33

Unconstrained 0.28 0.78 1.95	 8.13

Linear	Objective	

Nonlinear	Constraints	

Tight	Constraint 1.16 1.55 8.42	 32.20

Loose	Constraint 0.91 2.88 12.05	 38.75

Unconstrained 0.31 0.48 0.70	 2.61

Effect on CPU Time of Current Line Constraints on the Fully 

Nonlinear IV ACOPF

1

1.5

2

2.5

3

3.5

4

4.5

14 Bus 30 Bus 57 Bus 118 Bus

Ratio of CPU Time to Unconstrained CPU Time

Tight Constraint

Loose Constraint

Effect on CPU Time of Current Line Constraints on the 

Nonlinear IV ACOPF with Linear Objective Function

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

14 Bus 30 Bus 57 Bus 118 Bus

Ratio of CPU Time to Unconstrained CPU Time

Tight Constraint

Loose Constraint
Page

17

6. Conclusions

For	each	test	problem,	one	single	limit	is	applied	to	all	lines	that	makes	the	

optimal	solution	without	these	limits	infeasible.	We	solve	the	resulting	problem	

using	the	IV‐ACOPF	formulation.	For	each	problem	we	develop	a	‘tight’	and	a	‘loose’	

constraint.			

For	the	14,	30,	57	and	118‐bus	problems,	creating	line	current	magnitude	

constraints	for	the	ACOPF	problem	can	result	in	infeasible	problems.		As	one	

tightens	the	current	magnitude	constraints,	the	objective	function	increases	

gradually	at	first,	then	increases	exponentially	near	the	point	of	infeasibility.		

Different	test	problems	exhibit	different	characteristics	in	the	line	current	

magnitude	distribution	and	at	what	current	magnitude	level	constraint	the	problem	

becomes	infeasible.			

The	current	magnitude	constraints	also	increase	the	solution	time,	although	

stricter	constraints	do	not	necessarily	increase	the	solution	time	more	than	looser	

constraints.	For	problems	like	this	case	where	the	problem	becomes	infeasible	

quickly,	it	may	work	well	to	restrict	only	some	of	the	line	current	magnitudes	rather	

than	all	of	them.			

Including	these	constraints	in	the	ACOPF	increases	the	solution	time	between	

2	to	20	times	and	objective	function	up	to	25	percent.	

	 	
Page

18

	

References	

Mary	B.	Cain,	Richard	P.	O’Neill,	Anya	Castillo,	“History	of	Optimal	Power	Flow	and	

Formulations,”	FERC	Staff	Technical	Paper,	December	2012	

http://www.ferc.gov/industries/electric/indus‐act/market‐planning/opf‐papers/acopf‐1‐

history‐formulation‐testing.pdf

Richard	P	O’Neill,	Anya	Castillo	and	Mary	Cain,	“The	IV	Formulation	and	Linear	

Approximations	of	the	AC	Optimal	Power	Flow	Problem,”	FERC	Staff	Technical	

Paper,	December	2012		

http://www.ferc.gov/industries/electric/indus‐act/market‐planning/opf‐papers/acopf‐2‐

iv‐linearization.pdf

University	of	Washington,	Dept.	of	Electrical	Engineering,	“Power	System	Test	Case	

Archive,”	May	2010.		Available:		

http://www.ee.washington.edu/research/pstca/index.html.	

R.D.	Zimmerman,	C.E.	Murillo‐Sánchez,	and	R.J.	Thomas,	“MATPOWR	Stead‐State	

Operations,	Planning	and	Analysis	Tools	for	Power	Systems	Research	and	

Education,”	Power	Systems,	IEEE	Transactions	on,	vol.	26,	no.	1,	pp.	12‐19,	Feb	

2011.	
Page

19

Appendix	

	 The	generator	costs	take	on	the	form	cost	ൌ	aP൅bP2൅.0001*|Q|,	where	P	is	the	

real	power	and	|Q|	is	the	magnitude	of	reactive	power	q.	We	list	all	generator	costs	

used	for	each	test	system.	

Table	A1:		Generator	Cost	Coefficients	for	the	14,	30	and	57‐bus	Problems	

14‐bus	 Cost	Coefficient	 30‐bus Cost	Coefficient 57‐bus Cost	Coefficient

Generator	 a	 b	 Generator a b Generator a	 b

1	 0.04303	 20	 1 0.02000 2.00 1 0.07758	 20

2	 0.25000	 20	 2 0.01750 1.75 2 0.01000	 40

3	 0.01000	 40	 13 0.02500 3.00 3 0.25000	 20

6	 0.01000	 40	 22 0.06250 1.00 6 0.01000	 40

8	 0.01000	 40	 23 0.02500 3.00 8 0.02222	 20

	 	 	 27 0.00834 3.25 9 0.01000	 40

	 	 	 12 0.03226	 20

	

Table	A2:		118‐bus	Generator	Costs	

Generator	 Cost		

Coefficient	

Generator Cost	

Coefficient

Generator	 Cost		

Coefficient	

	 a	 b	 a b r a	 b	

1	 0.01000	 40	 46 0.52632 20 87 2.50000	 20	

4	 0.01000	 40	 49 0.04902 20 89 0.01647	 20	

6	 0.01000	 40	 54 0.20833 20 90 0.01000	 40	

8	 0.01000	 40	 55 0.01000 40 91 0.01000	 40	

10	 0.02222	 20	 56 0.01000 40 92 0.01000	 40	

12	 0.11765	 20	 59 0.06452 20 99 0.01000	 40	

15	 0.01000	 40	 61 0.06250 20 100 0.03968	 20	

18	 0.01000	 40	 62 0.01000 40 103 0.25000	 20	

19	 0.01000	 40	 65 0.02558 20 104 0.01000	 40	

24	 0.01000	 40	 66 0.02551 20 105 0.01000	 40	

25	 0.04545	 20	 69 0.01936 20 107 0.01000	 40	

26	 0.03185	 20	 70 0.01000 40 110 0.01000	 40	

27	 0.01000	 40	 72 0.01000 40 111 0.27778	 20	

31	 1.42857	 20	 73 0.01000 40 112 0.01000	 40	

32	 0.01000	 40	 74 0.01000 40 113 0.01000	 40	

34	 0.01000	 40	 76 0.01000 40 116 0.01000	 40	

36	 0.01000	 40	 77 0.01000 40 	

40	 0.01000	 40	 80 0.02096 20 	

42	 0.01000	 40	 85 0.01000 40 	

	 	
Page

20

Table	A3:	14	and	30‐bus	Line	Index	Mapping	

14‐bus	 30‐bus

Line	

Index	

Buses	Connected Line	

Index

Buses	Connected	

To	 From To From	

1	 1	 2 1 6	 8

2	 1	 5 2 12	 13

3	 2	 3 3 21	 22

4	 4	 5 4 2	 6

5	 2	 4 5 4	 6

6	 5	 6 6 1	 2

7	 2	 5 7 1	 3

8	 7	 9 8 2	 4

9	 4	 7 9 3	 4

10	 6	 13 10 5	 7

11	 4	 9 11 2	 5

12	 3	 4 12 4	 12

13	 9	 14 13 27	 28

14	 7	 8 14 25	 27

15	 6	 12 15 15	 23

16	 6	 11 16 6	 7

17	 9	 10 17 9	 10

18	 13	 14 18 6	 9

19	 10	 11 19 8	 28

20	 12	 13 20 10	 20

21 15	 18

22 10	 21

23 10	 17

24 12	 15

25 24	 25

26 10	 22

27 12	 16

28 27	 30

29 27	 29

30 19	 20

31 6	 28

32 12	 14

33 6	 10

34 22	 24

35 23	 24

36 18	 19

37 25	 26

38 29	 30

39 16	 17

40 14	 15

41 9	 11

35 23	 24

36 18	 19

37 25	 26

38 29	 30

39 16	 17

40 14	 15

41 9	 11
Page

21

Table	A4:		57‐bus	Line	Index	Mapping	

57‐bus	 57‐bus

Line	Index	

Buses	Connected	

Line	Index

Buses	Connected

To	 From	 To From

1	 8	 9	 40 50 51

2	 14	 46	 41 38 44

3	 10	 51	 42 35 36

4	 24	 26	 43 24 25

5	 7	 8	 44 4 5

6	 7	 29	 45 6 7

7	 15	 45	 46 41 43

8	 1	 15	 47 52 53

9	 9	 55	 48 54 55

10	 3	 15	 49 11 41

11	 1	 2	 50 41 42

12	 1	 17	 51 38 49

13	 13	 49	 52 12 16

14	 9	 11	 53 3 4

15	 46	 47	 54 25 30

16	 14	 15	 55 49 50

17	 6	 8	 56 34 35

18	 2	 3	 57 53 54

19	 9	 10	 58 32 34

20	 9	 13	 59 41 56

21	 1	 16	 60 10 12

22	 4	 18	 61 12 17

23	 28	 29	 62 18 19

24	 11	 43	 63 48 49

25	 27	 28	 64 30 31

26	 5	 6	 65 36 40

27	 44	 45	 66 20 21

28	 4	 6	 67 32 33

29	 12	 13	 68 37 39

30	 37	 38	 69 22 38

31	 38	 48	 70 22 23

32	 11	 13	 71 56 57

33	 13	 14	 72 23 24

34	 9	 12	 73 39 57

35	 36	 37	 74 42 56

36	 26	 27	 75 40 56

37	 29	 52	 76 19 20

38	 47	 48	 77 31 32

39	 13	 15	 78 21 22

	 	
Page

22

Table	A5:		118‐bus	Line	Index	Mapping	

118‐bus	 118‐bus	 118‐bus 118‐bus	

Line	

Index	

Buses	 Line	

Index	

Buses	 Line	

Index

Buses	 Line	

Index	

Buses	

To	 From	 To	 From To From To	 From

1	 9	 10	 46	 77	 78 91 51 52 136	 51	 58

2	 8	 9	 47	 66	 67 92 104 105 137	 105	 106

3	 5	 8	 48	 45	 49 93 93 94 138	 100	 101

4	 37	 38	 49	 76	 77 94 2 12 139	 34	 43

5	 17	 30	 50	 22	 23 95 103 104 140	 105	 107

6	 49	 66	 51	 59	 61 96 75 77 141	 29	 31

7	 26	 30	 52	 49	 50 97 55 59 142	 32	 114

8	 89	 92	 53	 47	 69 98 20 21 143	 70	 71

9	 68	 116	 54	 100	 106 99 34 36 144	 19	 20

10	 68	 69	 55	 56	 59 100 82 83 145	 70	 75

11	 64	 65	 56	 94	 95 101 37 40 146	 27	 32

12	 59	 63	 57	 100	 104 102 99 100 147	 43	 44

13	 89	 90	 58	 75	 118 103 62 67 148	 106	 107

14	 23	 25	 59	 85	 88 104 68 81 149	 7	 12

15	 80	 81	 60	 74	 75 105 1 3 150	 60	 62

16	 63	 64	 61	 94	 100 106 54 59 151	 40	 42

17	 38	 65	 62	 92	 93 107 101 102 152	 52	 53

18	 65	 66	 63	 59	 60 108 40 41 153	 3	 12

19	 25	 26	 64	 11	 12 109 78 79 154	 90	 91

20	 25	 27	 65	 21	 22 110 12 117 155	 12	 16

21	 77	 80	 66	 62	 66 111 23 24 156	 15	 33

22	 4	 5	 67	 48	 49 112 17 31 157	 32	 113

23	 60	 61	 68	 80	 97 113 27 115 158	 105	 108

24	 61	 64	 69	 110	 111 114 80 99 159	 56	 58

25	 69	 75	 70	 49	 69 115 96 97 160	 24	 72

26	 69	 70	 71	 37	 39 116 53 54 161	 86	 87

27	 34	 37	 72	 92	 94 117 16 17 162	 55	 56

28	 88	 89	 73	 45	 46 118 41 42 163	 71	 73

29	 15	 17	 74	 11	 13 119 47 49 164	 98	 100

30	 42	 49	 75	 110	 112 120 91 92 165	 39	 40

31	 5	 6	 76	 103	 110 121 82 96 166	 13	 15

32	 23	 32	 77	 27	 28 122 77 82 167	 1	 2

33	 100	 103	 78	 103	 105 123 85 86 168	 108	 109

34	 69	 77	 79	 35	 37 124 70 74 169	 71	 72

35	 65	 68	 80	 31	 32 125 83 84 170	 114	 115

36	 8	 30	 81	 80	 98 126 56 57 171	 15	 19

37	 5	 11	 82	 50	 57 127 33 37 172	 54	 56

38	 30	 38	 83	 6	 7 128 95 96 173	 35	 36

39	 49	 54	 84	 61	 62 129 17 113 174	 19	 34

40	 17	 18	 85	 44	 45 130 94 96 175	 109	 110

41	 79	 80	 86	 83	 85 131 18 19 176	 76	 118

42	 85	 89	 87	 80	 96 132 92 100 177	 24	 70

43	 4	 11	 88	 92	 102 133 12 14 178	 54	 55

44	 49	 51	 89	 46	 47 134 46 48 179	 14	 15

45	 3	 5	 90	 84	 85 135 28 29