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November	2013

	

Optimal	Power	Flow	Paper	10	

Paula	A.	Lipka,	

Richard	P.	O’Neill,	

Shmuel	S.	Oren,	

Anya	Castillo,	

Mehrdad	Pirnia,	

Clay	Campaigne
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Optimal	Transmission	Switching	Using		

the	IV‐ACOPF	Linearization		

Optimal	Power	Flow	Papers	Paper	10.	

Paula	A.	Lipka,	Richard	P.	O’Neill,	Shmuel	S.	Oren,	Anya	Castillo,		

Mehrdad	Pirnia,	Clay	Campaigne	

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

anya.castillo@ferc.gov;	mpirnia@uwaterloo.ca;	clay.campaigne@gmail.com	

	

November	2013			

Abstract	

In	this	paper,	we	seek	to	investigate	the	performance	of	transmission	

switching	using	the	iterative	linear	program	approximation	to	the	Current	Voltage	

AC	Optimal	Power	Flow	ሺIV‐ACOPFሻ.			Several	different	methods	of	using	this	

switching	are	investigated	to	find	a	method	that	addresses	the	MIP	challenges	and	is	

both	fast	and	accurate.		We	consider	opening	only	one	line,	opening	up	to	five	lines,	

and	progressively	opening	one	line	at	a	time.		The	linear	method	with	switching	is	

much	faster	than	the	nonlinear	ACOPF	and	generally	finds	solutions	within	1%	of	

the	nonlinear	ACOPF.	

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.	Notation	............................................................................................................	6	

3.	Nonlinear	IV‐ACOPF	Transmission	Switching	Model	...................	7	

4.	Iterative	Linear	IV‐ACOPF	Transmission	Switching	Model	.......	8	

5.	Computational	Analysis	.............................................................................	9	

6.	Summary	..........................................................................................................	38	

7.	Conclusions	.....................................................................................................	38	

References	............................................................................................................	40	

Appendix	...............................................................................................................	42	
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1. Introduction	

In	a	given	electric	power	network,	some	transmission	lines	are	often	out	of	

service	due	to	outages	or	maintenance.		However,	of	the	lines	that	remain	operable,	

removing	additional	lines	may	reduce	the	total	power	generation	cost.		Power	

companies	use	this	practice	in	times	of	low	load,	but	often	use	experience	and	

intuition	with	reliability	testing;	that	is,	without	explicitly	searching	for	the	optimal	

topology.		The	benefit	of	transmission	switching	has	been	demonstrated	by	Fisher	

et.	al.	ሺ2008ሻ	and	Hedman	et.	al.	ሺ2008ሻ	by	solving	a	‘DC’	approximation	of	the	

alternating	current	optimal	power	ሺACOPFሻ	flow	system.	Hedman	et.	al.	ሺ2008ሻ	

achieved	savings	of	over	20%	by	switching	lines	off	versus	the	original	network	on	

the	IEEE	118	bus	test	case.	Even	when	the	considering	network	contingencies,	

transmission	switching	ሺagain	using	the	‘DC’	approximationሻ	can	still	provide	

substantial	benefits.	Hedman	et.	al.	ሺ2008ሻ	show	that	opening	five	lines	reduces	

costs	by	8%	in	the	N‐1	compliant	RTS‐96	system.			Potluri	and	Hedman	ሺ2012ሻ	

suggest	that	switching	has	significant	benefits	over	maintaining	a	static	network	

when	solving	the	full	ACOPF.		They	also	demonstrate	that	an	improvement	in	the	

system	cost	using	the	DC	solution	procedure	does	not	always	give	a	lower	cost	in	the	

original	AC	system.		In	fact,	the	DC	solution	with	line	switching	is	AC	infeasible	or	

increases	the	system	cost	over	no	switching	in	several	instances.		

The	nonlinear	ACOPF	transmission	switching	problem	is	rarely	solved	due	to	

its	high	computational	intensity.		In	practice,	the	OPF	is	solved	using	the	DC	

approximation	and	iterations	that	try	to	obtain	AC	feasibility	and	N‐1	reliability.		In	

order	to	solve	the	nonlinear	ACOPF	with	binary	variables,	the	solver	could	possibly	

branch	on	every	combination	of	opened/closed	lines	in	the	network	and	solve	a	

nonlinear	program	ሺNLPሻ	for	each	different	instance,	which	is	prohibitively	

computationally	expensive	at	the	current	time.		If	there	are	N	lines,	this	could	

require	solving	2N		nonlinear	programs.		In	addition,	the	nonlinear	program	is	

nonconvex.		

Mixed	integer	linear	programs	ሺMIPsሻ	generally	solve	much	faster	than	

mixed	integer	nonlinear	programs.		However,	even	solving	a	MIP	can	also	be	quite	

time‐intensive,	as	one	may	need	to	solve	the	linear	program	ሺLPሻ	2N	times;	however,	

techniques	for	solving	MIPs	have	significantly	reduced	this	upper	bound.			

One	way	to	limit	the	number	of	branches	is	to	constrain	the	number	of	lines	

allowed	to	be	open.		Although	not	necessarily	optimal,	the	approach	finds	a	better	

solution	than	the	status	quo	ሺSee	Hedman	et	al,	Fuller	et	al,	and	Ruiz	et	al,	2012ሻ	and	

is	often	very	close	to	the	optimal	solution.		Our	work	in	this	paper	also	shows	that	

opening	more	lines	past	a	certain	number	has	no	or	little	benefit.	

If	the	system	is	forced	to	open	S	lines,	then	the	number	of	branches	for	LPs	in	

the	MIP	is	N	choose	S.		However,	in	the	DC	model,	it	is	never	optimal	to	open	lines	

that	island	parts	of	the	network	ሺsee	Ostrowski	et	al,	2012ሻ.		In	a	connected	
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network,	you	can	travel	from	any	bus	of	the	network	to	any	other	bus	via	

transmission	lines	ሺalthough	one	may	have	to	travel	through	other	busesሻ.		

“Islanding”	means	that	there	are	pairs	of	buses	that	are	not	connected	by	any	path.		

Aka,	a	connected	network	would	be	one	circle,	and	a	network	with	islanding	would	

be	two	circles;	you	could	never	travel	from	a	bus	in	the	first	circle	to	the	bus	in	the	

second	circle.		Therefore,	one	can	reduce	the	MIP	branch	and	bound	tree	by	

removing	any	paths	that	result	in	islanding.	To	a	first	order	approximation,	we	

believe	this	is	true	for	AC	networks.	In	this	paper,	we	have	not	implemented	this	

procedure.		

An	additional	issue	is	that	our	linear	program	is	not	static;	rather,	it	is	based	

on	the	last	iteration	ሺsee	O’Neill	et	al,	2012	and	Campaigne	et	al,	2013ሻ.		Our	

linearization	depends	on	the	previous	optimal	point.		However,	we	may	then	switch	

the	network	based	on	a	revised	linearization	in	a	later	iteration.				

This	paper	seeks	to	address	the	following	questions:	

 Does	the	linearized	ACOPF	select	lines	to	switch	in	accordance	with	the	nonlinear	

ACOPF?	

 How	much	faster	is	the	iterative	linearized	ACOPF	compared	to	the	nonlinear	

ACOPF?	

 How	well	do	the	proposed	heuristics	perform?	

	 	
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2. Notation.	

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

n,	 m	ε	N.	Here	we	refer	to	transmission	assets	as	lines.	Transmission	lines	are	

indexed	by	k,	k	ε	K,		with	one	terminal	of	line	k	being	n	and	the	other	terminal	being	

m.			Set	Wሺnሻ	is	the	set	of	lines	k	that	connect	to	bus	n.		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

vj

n		 imaginary	part	of	the	voltage	at	bus	n	

i

r n		 real	part	of	the	current	injection	at	bus	n

i

j

n		 imaginary	part	of	the	current	injectionat	bus	n

i

r k		 real	part	of	the	current	on	line	k ij

k		imaginary	part	of	the	current	on	

line	k		

zk binary	variable	that	is	1,	if	the	switch	at	n	for	line	k	is	open,	and	0,	if	

closed	

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	

bk	 susceptance	of	line	k

bn0				 self	susceptance	of	node	n	ሺto	groundሻ

gk	 conductance	of	line	k gn0						 self	conductance	of	node	n	ሺto	groundሻ	

yk	ൌ	gk൅jbk	 admittance	of	line	k

yn0 admittance	from	bus	n	to	ground	

pn

d real	power	demand	at	bus	n

qn

d	 reactive	power	demand	at	bus	n	

pmin n		 minimum	required	real	power	generation	at	bus	n	

pmax n		 maximum	allowed	real	power	generation	at	bus	n

qmin n	 minimum	required	reactive	power	generation	at	bus	n	

qmax n	 maximum	allowed	reactive	power	generation	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		
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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 k				 	maximum	current	magnitude	on	line	k	ir k		the	real	current	value	on	

line	k	from	the	previous	linear		program	solution	

i

r k		 imaginary	current	value	on	line	k	from	the	previous	linear	program	

solution	

i

j

k		 imaginary	current	value	on	line	k	from	the	previous	linear	program	

solution	

Mk	 a	large	constant,	used	to	create	an	either‐or	constraint	

3. Nonlinear	IV‐ACOPF	Transmission	Switching	Model		

	 The	nonlinear	current‐voltage	ሺIVሻ‐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

Subject	to	

i

r k	ൌ	gkሺvr n	‐ vr mሻ	‐	bkሺvj

n	‐ vjmሻ		 	 for all	k	 	 ሺ2ሻ

i

j

k	ൌ	bkሺvr n	‐ vr mሻ ൅	gkሺvjn	‐ vjmሻ	 	 for all	k	 	 ሺ3ሻ	

i

r n	ൌ	∑k∊Wሺnሻ		i	r k	–	gn0vr n	൅bn0	vj

n	 	 for all	n	 	 ሺ4ሻ

i

j

n	ൌ	∑	k∊Wሺnሻ			i	jk	–gn0vjn	–	bn0	vr n	 	 for all	n	 	 ሺ5ሻ

pn	ൌ	vr ni

r n ൅	vj

ni

j

n	൅	pn

D		 	 	 for all	n	 	 ሺ6ሻ

pmin n ൑	pn ൑	pmax n		 	 	 	 for all	n	 	 ሺ7ሻ

qn

G	ൌ	vj

ni

r n	‐	vr nij

n	൅	qn

D	 		 	 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 kሻ2 ൅ ሺij

kሻ2 ൑ ሺi

max kሻ2				 for all	k	 	 ሺ12ሻ

The	network	flow	equations,	ሺ2ሻ‐ሺ5ሻ	are	linear.		The	upper	bounds	on	the	line	

current	magnitudes	ሺ12ሻ	and	voltage	magnitudes	at	buses	ሺ10ሻ	are	circles	with	their	

interiors	ሺconvexሻ,	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.		The	IV‐ACOPF	with	transmission	switching	is	formed	by	

adding	binary	decision	variable	zk		and	replacing	equations	ሺ2ሻ,	ሺ3ሻ,	and	ሺ12ሻ	with	

equations	ሺ2.1ሻ,	ሺ2.2ሻ,	ሺ3.1ሻ,	ሺ3.2ሻ,	and	ሺ12.1ሻ.		zkൌ1	if	line	k	is	disconnected	and	

zkൌ0 if	line	k	is	connected.			

i

r k ൑	gkሺvr n	‐ vr mሻ	‐	bkሺvj

n	‐ vjmሻ ൅	zkM		 	 	 ሺ2.1ሻ
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i

r k	൒	gkሺvr n	‐ vr mሻ	‐	bkሺvj

n	‐ vjmሻ ‐		zkM		 	 	 ሺ2.2ሻ

i

j

k		൒bkሺvr n	‐ vr mሻ	൅	gkሺvjn	‐ vjmሻ ൅	zkM		 	 	 ሺ3.1ሻ

i

j

k		൑	bkሺvr n	‐ vr mሻ	൅	gkሺvjn	‐ vjmሻ	–	zkM	 	 	 ሺ3.2ሻ

	ሺi

r kሻ2	൅	ሺij

kሻ2 ൑ ሺ1‐zkሻሺi

max kሻ2		 	 	 	 ሺ12.1ሻ	

	

4. Iterative	Linear	IV‐ACOPF	Transmission	Switching	Model	

We	approximate	the	quadratic	constraint	equations	ሺwhich	express	real	and	

reactive	power	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 k ൑ gkሺvr n	‐ vr mሻ ‐ bkሺvj

n	‐ vjmሻ ൅	zkM for	all	k	 ሺ2.1ሻ

to	 i

r k	൒	gkሺvr n	‐ vr mሻ ‐ bkሺvj

n	‐ vjmሻ ‐ zkM ሺ2.2ሻ

i

j

k		൒bkሺvr n	‐ vr mሻ	൅	gkሺvjn	‐ vjmሻ ൅	zkM ሺ3.1ሻ

i

j

k		൑ bkሺvr n	‐ vr mሻ	൅	gkሺvjn	‐ vjmሻ – zkM ሺ3.2ሻ

i

r n	ൌ	∑k∊Wሺnሻ		i	r k	–	gn0vr n൅bn0 vj

n for	all	n ሺ4ሻ

i

j

n	ൌ	∑ k∊Wሺnሻ			i	k –gn0vjn	– bn0 vr n for	all	n ሺ5ሻ

pn

G ൌ	vr nir n ൅	vj

ni

j

n ൅	vr ni

r n ൅ vj

ni

j

n ‐ ሺvr ni

r n ൅ vj

ni

j

nሻ൅pn

D for all	n	 ሺ26ሻ

pn

min ൑ pnG ൑ pnmax 	 for all	n	 ሺ27ሻ

qn

G ൌ	vj

ni

r n ‐	vr nij

n ‐	vr ni

j

n ൅ vjni

r n ‐ ሺvj

ni

r n ‐ vr nij

nሻ൅qn

D for all	n	 ሺ28ሻ

qn

min ൑ qnG ൑ qnmin	 for all	n	 ሺ29ሻ

vrf

nvr n ൅	vjf nvjn	൑ ሺvmax nሻ2 for	f	ൌ	0	1,	...,	f

max

and all	n

ሺ30ሻ

vrd nvr n ൅	vjd nvj

n	൑ ሺvmax nሻ2 for	d	ൌ	0,	...,	h‐1	and

all	n

ሺ31ሻ

i

rf

ki

r k ൅ ijf kij

k൑ ሺ1‐zkሻ	ሺi

max kሻ for	f	ൌ	1,	...,	fmax and

all	k

ሺ32ሻ

i

rd kir k ൅ ijd kijb k൑ ሺimax kሻ2 for	d	ൌ	0,	...,	h‐1	and	

all	k

ሺ33ሻ

	 ‐2vr n	ሺa/hbሻ ൑	vr n‐vr n ൑ 2vr nሺa/hbሻ for all	n	 ሺ34ሻ

‐2vj

n	ሺa/hbሻ ൑	vjn‐vjn ൑ 2vjn ሺa/hbሻ for all	n	 ሺ35ሻ

where	f

max is	the	number	of	sides	of	the	preprocessed	circumscribing	polygons	

and	d indexes	the	iterative	tight	cuts.		

To	limit	the	number	of	lines	to	be	opened,	we	add	the	constraint:	

∑k	zk ൑	k’		 	 	 	 	 	 	 	 	 ሺ36ሻ	

The	network	flow	equations,	ሺ22ሻ‐ሺ25ሻ	are	linear	and	unchanged.		For	the	voltage	

magnitudes,	the	preprocessed	upper	bounds	are	in	ሺ30ሻ	and	the	iterative	tight	cuts	

are	in	ሺ31ሻ.		For	the	line	current	magnitudes,	the	preprocessed	upper	bounds	are	in	

ሺ32ሻ	and	the	iterative	tight	cuts	are	in	ሺ33ሻ.			
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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.			

Here	we	use	the	formulation	that	removes	the	entire	transmission	element	

from	the	network;	no	current	can	flow	on	the	line.		If	we	open	only	one	of	the	

switches,	the	line	becomes	a	capacitor.		

The	iterative	linearization	method	used	to	solve	the	problem	is	as	follows:	

1ሻ	Set	h	ൌ	0.	Choose	a	starting	point	vr0 n		and	vi0 n.		Add	a	circumscribing	polygon	

for	each	maximum	voltage	magnitude	and	maximum	current	magnitude	constraint.		

Approximate	P	and	Q	with	hyperplanes	that	are	tangent	to	the	constraint	

hypersurfaces.	

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

the	current	and	voltage,	vrh n,	vjh n,	irh n,	ijh n

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

nonlinear	P	and	Q	equations	ሺ6ሻ	through	ሺ12ሻ.	If	within	tolerance,	stop;	otherwise	

continue.	

4ሻ	Add	another	set	of	tight	voltage	and	current	constraints	at	the	optimal	

voltage	solution	to	further	cut	off	infeasible	voltage	and	current	solutions.	

Relinearize	the	p	and	q	approximation	using	the	current	answer	and	adjust	the	

stepsize	range	of	the	bus	voltage.	Go	to	step	2.	

The	convergence	criteria	is	as	follows:	if	the	percent	violations	of	real	power	

ሺ7ሻ,	reactive	power	ሺ9ሻ,	the	voltage	ሺ10ሻ,	and	the	current	ሺ12ሻ	constraints	is	under	a	

certain	threshold,	and	the	sum	of	the	average	percent	violations	of	the	voltage,	real	

power,	and	reactive	power	is	also	under	a	threshold,	the	solution	is	determined	to	

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

to	be	AC	feasible.			

5. 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.		

Single‐line	diagrams	of	the	14	and	30	bus	cases	are	shown	in	Figures	1	and	2.		The	

quadratic	generator	costs	come	from	MATPOWER	ሺZimmerman	et	al,	2011ሻ.		We	

formulate	the	20‐step	linear	approximation	to	the	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.		
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10

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

	

	

Figure	1:		14	Bus	Diagram	
Page

11

	

Figure	2:		30	Bus	Diagram	

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,	all	problems	only	

used	one	processor	at	a	time.		Minor	differences	in	solution	times	were	recorded	

when	the	problems	were	run	at	different	times	of	day,	but	the	differences	were	

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

GAMS	23.6.2.		The	nonlinear	mixed	integer	program	used	was	KNITRO.		The	

nonlinear	programs	without	integer	variables	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	the	problem	was	solved	from	scratch	at	each	

major	iteration.	Starting	from	the	previous	linear	program	was	not	an	option	in	the	

GAMS	solver.	There	may	be	easily	gained	speedups	by	not	starting	each	major	

iteration	from	scratch. The	allowable	MIP	gap	was	set	to	0.1%.		

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	in	ሺ34ሻ	and	ሺ35ሻ.	We	choose	16	and	32	sided	circumscribing	

polygons	based	on	the	testing	in	Pirnia	et	al	ሺ2012ሻ.		

The	maximum	number	of	major	iterations	ሺlinear	programሻ	was	set	to	20.		

Earlier	testing	revealed	that	the	solution	usually	converged	to	a	feasible	solution	
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12

ሺwithin	toleranceሻ	in	20	iterations	or	less,	and	problems	that	ran	for	more	than	20	

iterations	would	run	to	the	maximum	number	of	allowed	iterations.		The	linear	

problem	is	considered	feasible	within	tolerance	when	the	average	deviation	of	

voltages,	real	power,	and	reactive	power	is	less	than	0.1%	and	the	maximum	

deviation	of	all	individual	nodes	is	less	than	0.5%.		

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

nൌ	0,	for	

all	buses	n.	Other	starting	points	can	be	used:		these	include	hot	starts,	random	

starts,	and	DC	starts	ሺsee	Castillo	et	al	2013ሻ.		In	commercial	practice,	a	hot	start	

may	be	available	from	previous	solutions	since	each	dispatch	is	solved	in	sequential	

time	steps.	

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

convergence	tolerances	were	set	at	0.001	or	0.005.	The	maximum	CPU	time	was	

never	reached.	

	 We	examine	five	different	approaches:		brute	force	ሺtotal	enumerationሻ	one‐

line	optimization,	MIP	one‐line	optimization,	iterative	linearized	ACOPF	with	a	MIP	

at	every	iteration	ሺopening	up	to	5	linesሻ,	linearized	ACOPF	with	the	MIP	at	the	first	

iteration	ሺopening	up	to	5	linesሻ,	and	a	progressive	approach	where	one	additional	

line	is	allowed	to	be	open	at	each	stage	ሺup	to	5	linesሻ.		

Brute‐Force	One‐Line	Approach.		First,	we	examine	the	results	of	the	linearized	

ACOPF	versus	the	nonlinear	ACOPF	when	the	network	is	fixed	to	one	line	removed	

using	the	following	procedure,	where	N	subproblems	are	solved.	

The	procedure	in	steps	1‐5	is	repeated	for	4	different	approaches:		16	and	32	

preprocessed	voltage	cuts	with	linear	and	quadratic	voltage	step	size	reduction.	

1ሻ		For	each	line	k		in	the	transmission	system	with	K	lines:	

2ሻ		In	the	model	described	in	section	4,	z	is	a	parameter	instead	of	a	variable.		

Let	all	lines	be	in	service	ሺaka,	z	ൌ0ሻ	except	for	line	k	ሺset	z	ൌ1ሻ.		Note	that	since	

z	is	a	parameter	instead	of	a	variable,	an	LP	instead	of	a	MIP	is	solved.	

3ሻ		Use	the	iterative	linearization	method	to	solve	the	problem.	

4ሻ		Set	line	k	to	be	line	k൅1.		Go	to	step	2	if	k	is	less	than	or	equal	to	K.	

5ሻ		When	all	of	the	results	have	been	tabulated	for	each	line	taken	out	ሺkൌK൅1ሻ,	

the	subproblem	with	the	lowest	cost	is	considered	to	be	the	optimal	

configuration.	

6ሻ			To	obtain	a	nonlinear	comparison,	steps	1‐5	are	repeated	except	in	step	2,	

the	nonlinear	model	described	in	section	3	is	used		and	step	3	is	replaced	with	

the	solver’s	nonlinear	solution	methods.	

MIP	One‐Line	Approach.		We	also	compare	the	results	of	the	linearized	ACOPF	with	a	

MIP	at	every	iteration	to	the	results	of	brute	force	linearized	ACOPF	in	order	to	

demonstrate	whether	the	MIP	at	each	iteration	works	appropriately.	
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13

The	procedure	in	steps	1‐2	is	repeated	for	4	different	approaches:		16	and	32	

preprocessed	voltage	cuts	with	linear	ሺb	ൌ	1ሻ	and	quadratic	ሺb	ൌ	2ሻ	voltage	step	

size	reduction.	

1ሻ		Use	the	model	described	in	section	4	with	the	added	constraint	∑k	zk ൌ	1.		

This	means	that	in	any	answer,	one	transmission	line	will	be	open.	

2ሻ		Use	the	iterative	linearization	method	to	solve	the	problem.	At	every	

iteration,	a	MIP	will	be	solved.	

3ሻ		Then,	to	obtain	a	nonlinear	comparison,	steps	1	and	2	are	repeated	except	in	

step	1,	the	nonlinear	model	ሺKNITROሻ	described	in	section	3	is	used	and	step	2	

is	replaced	with	the	solver’s	nonlinear	solution	methods.	

Iterative	Linearized	ACOPF	with	a	MIP	at	every	iteration opening	up	to	5	lines	More	

extensively,	the	ACOPF	was	run	for	opening	up	to	5	lines.		This	limit	on	the	number	

of	lines	opened	was	placed	to	achieve	a	reasonable	run	time.		Additionally,	allowing	

the	program	open	up	to	ten	lines	did	not	result	in	significant	reduction	of	the	total	

power	production	cost	versus	allowing	it	to	open	5	lines.	

The	procedure	is	repeated	for	4	different	approaches:		16	and	32	preprocessed	

voltage	cuts	with	linear	and	quadratic	voltage	step	size	reduction.	

1ሻ		Use	the	model	described	in	section	4	with	the	added	constraint		

∑k	zk ൑	k’	where	k’	ൌ	5.		This	means	that	in	any	answer,	up	to	5	transmission	

lines	can	be	open.	

2ሻ		Use	the	iterative	linearization	method	to	solve	the	problem.			At	every	

iteration,	a	MIP	will	be	solved.	

3ሻ		When	the	iterative	linearization	method	has	either	converged	or	exceeded	

the	iteration	limit,	solve	the	nonlinear	ACOPF	with	transmission	switching	as	in	

section	3	but	again	the	line	configuration	ሺzkሻ	variables	are	set	as	parameters	

ሺaccording	to	the	solution	in	step	2ሻ,	again	to	evaluate	whether	the	LIV‐ACOPF	

solution	was	truly	AC	feasible.			

	Linearized	ACOPF	with	the	MIP	at	the	first	iteration	opening	up	to	5	lines.			

To	reduce	the	computational	time	of	solving	the	ACOPF,	a	second	approach	is	to	run	

a	MIP	only	on	the	first	iteration	of	the	iterative	linearization	method,	then	run	the	

other	LP	iterations	on	the	fixed	network	that	was	the	optimal	answer	from	the	MIP	

repeatedly.	The	steps	used	are	as	follows:	

The	procedure	is	repeated	for	4	different	approaches:		16	and	32	preprocessed	

voltage	cuts	with	linear	and	quadratic	voltage	step	size	reduction.	

1ሻ Use	the	model	described	in	section	4	with	the	added	constraint	

∑k	zk ൑	k’	where	k’	ൌ	5.		This	means	that	in	any	answer,	up	to	5	transmission	

lines	can	be	open.	
Page

14

2ሻ		Do	one	iteration	of	the	iterative	linearization	method	to	solve	the	problem.			

This	iteration	will	solve	a	MIP.	

3ሻ		In	the	model	described	in	section	4,	replace	the	variables	z	with	parameters.		

The	line	configuration	will	be	set	to	the	optimal	solution	found	in	step	2.		For	

example,	if	step	2	found	that	all	lines	are	in	the	network	except	the	lines	from	

bus	1	to	4	and	from	bus	2	to	5,	then	z141	ൌ	1,	z251	ൌ	1,	and	all	other	z	ൌ0.	

4ሻ		Solve	the	model	in	step	3	using	the	iterative	linearization	method.			

5ሻ		When	the	iterative	linearization	method	has	either	converged	or	exceeded	

the	iteration	limit,	solve	the	nonlinear	ACOPF	with	transmission	switching	as	in	

section	3,	but	again	the	line	configuration	ሺzkሻ	variables	are	set	as	parameters	

ሺaccording	to	the	solution	in	step	2ሻ,	again	to	evaluate	whether	the	LIV‐ACOPF	

solution	was	truly	AC	feasible.			

It	is	expected	that	this	model	will	take	less	time	than	the	previous	model,	since	less	

MIPs	will	be	run	during	the	process.	

Progressive	MIP.		The	purpose	of	this	method	is	to	reduce	the	number	of	branches	

the	MIP	has	to	consider	in	the	case	of	opening	up	to	5	transmission	lines.		The	

previous	approach	has	ሺN	choose	5ሻ൅ሺN	choose	4ሻ൅ሺN	choose	3ሻ൅ሺN	choose	2ሻ൅ሺN

choose	1ሻ	potential	branches.		In	this	approach,	at	each	iteration	k,	with	the	first	

iteration	as	kൌ0,	the	algorithm	decides	which	of	N‐k	lines	to	open	or	whether	to	not	

open	any	additional	lines,	with	a	total	of	N‐k൅1	possibilities.		This	approach	has	up	

to	ሺN൅1ሻ൅ሺNሻ൅ሺN‐1ሻ൅ሺN‐2ሻ൅ሺN‐3ሻൌ5N‐5	potential	branches,	which	is	many	fewer	

than	the	previous	approach.		Table	2	shows	this	significant	reduction.	

Table	2.		Number	of	Potential	MIP	branches:		Normal	vs.	Progressive	

	 	 Number	of	Potential	Branches

Buses	 Lines	 Normal	MIP Progressive	MIP

14	 20	 21,700	 95	

30	 41	 862,190	 200	

57	 86	 37,055,939	 425	

118	 186	 1,806,641,034	 925	

The	following	procedure	is	repeated	for	4	different	approaches:		16	and	32	

preprocessed	voltage	cuts	with	linear	and	quadratic	voltage	step	size	reduction.	

1ሻ		Use	the	model	described	in	section	4	with	the	added	constraint	∑k	zk ൑	k’	

where	k’	ൌ	1.		This	means	that	in	any	answer,	only	1	transmission	line	can	be	

open.	

2ሻ		Use	the	iterative	linearization	method	to	solve	the	problem.			This	iteration	

will	solve	a	MIP	with	N൅1branches.	
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15

3ሻ		If	a	line	is	opened,	fix	it	as	a	parameter	ሺaka,	zൌ1ሻ.		Now,	let	k’ൌ2.		Note	that	

since	one	line	is	open,	only	one	more	line	can	possibly	be	opened.	

4ሻ		Again,	use	the	iterative	linearization	method	to	solve	the	problem.			

5ሻ		Fix	all	open	lines	as	parameters,	and	increment	k’ൌk’൅1.	

6ሻ		Repeat	steps	4	and	5	until	the	problem	where	k’ൌ5	has	been	solved.			

7ሻ		When	the	iterative	linearization	method	has	either	converged	or	exceeded	

the	iteration	limit,	solve	the	nonlinear	ACOPF	with	transmission	switching	as	in	

section	3,	but	again	the	line	configuration	ሺzkሻ	variables	are	set	as	parameters	

ሺaccording	to	the	solution	in	step	2ሻ.		This	is	done	to	evaluate	whether	the	LIV‐

ACOPF	solution	was	truly	AC	feasible.		

		

5.1		BRUTE‐FORCE	VERSUS	MIP	ONE	LINE	APPROACH	RESULTS	

To	benchmark	the	linearized	ACOPF	program,	both	the	linearized	and	

nonlinear	ACOPF	programs	were	solved	on	a	fixed	configuration	of	open/closed	

lines	for	all	possible	configurations	of	one	line	open	and	all	others	closed	ሺthe	Brute	

Force	Methodሻ.		The	objective	function	values	of	all	these	configurations	are	

compared,	and	the	‘best’	answer	is	the	problem	with	the	lowest	objective	function	

value.		This	Brute	Force	approach	is	compared	to	solving	the	problem	with	the	

linearized	ACOPF	and	allowing	the	mixed	integer	solver	to	find	the	best	network	

configuration	with	one	line	out	ሺMIPሻ.		The	nonlinear	“brute	force”	method	uses	

IPOPT	to	solve	every	possible	network		configuration	with	one	line	out,	and	the	

objective	value	given	is	the	minimum	cost	of	all	of	these	configurations.		The	

nonlinear	“MIP”	method	uses	KNITRO	to	solve	the	nonlinear	MIP	with	one	open	line.	

	

	 	
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14‐bus	problem.	The	results	for	the	14	bus	problem	are	shown	in	Tables	3	and	4.		

The	objective	function	values	were	within	2%	of	the	best‐known	value	ሺthe	

nonlinear	solutionሻ	for	both	the	MIP	and	Brute	Force	methods	in	both	the	tight	and	

loose	current	constrained	cases.	In	addition,	all	of	the	linear	approaches	were	at	

least	4	times	as	fast	as	the	nonlinear	case.		In	the	tight	current	constrained	case,	the	

same	line	ሺ4‐7ሻ	was	switched	out	by	all	but	one	of	the	8	approaches.			The	linear	

case	objective	values	are	all	very	close	to	each	other	–	the	biggest	difference	is	0.1%.		

Removing	line	4‐7	was	highly	beneficial	in	the	tightly	constrained	case;	the	total	

generation	cost	was	reduced	by	nearly	10%.		In	the	loose	current	constrained	case,	

again	the	same	line	ሺ2‐5ሻ	was	selected	to	be	switched	by	all	but	one	of	the	8	

approaches.		However,	the	objective	values	of	the	linear	approaches	differ	more	

than	in	the	tight	current	case;	the	biggest	difference	is	3.2%.		While	the	nonlinear	

case	shows	a	small	improvement	in	the	total	generation	cost	by	switching	out	a	line,	

some	of	the	linear	approaches	show	an	improvement	while	some	do	not.		In	the	

tightly	constrained	case,	the	MIP	runs	faster	than	the	brute	force	method.		

Surprisingly,	in	the	loosely	constrained	case,	the	MIP	runs	slower	than	the	brute	

force	method;	this	is	likely	not	significant	due	to	run‐to‐run	timing	variability.	

Table	3.	One	Line	Results:	14	Bus,	Tight	Current	Constraint	

Step	Size	Function	 		 Linear Quadratic	

Number	of	Cuts	 		 16 32 16 32	 Nonlinear

Without	Opening	Lines	 Obj	Value	 105.69 106.53 105.69 106.53	 107.36

MIP	 Obj	Value	 95.05 95.04 95.05 95.04	 93.77

Opening	One	Line	 Line	Switched 4‐7 4‐7 4‐7 4‐7	 4‐7

		 CPU	Time	 6.25 6.38 5.18 9.32	 54.30

Brute	Force	 Obj	Value	 95.04 94.99 95.00 94.97	 94.09

Opening	One	Line	 Line	Switched 4‐7 2‐5 4‐7 4‐7	 4‐7

		 CPU	Time	 6.81 9.44 10.85 14.59	 192.12

Table	4.	One	Line	Results:	14	Bus,	Loose	Current	Constraint

Step	Size	Function	 		 Linear Quadratic	

Number	of	Cuts	 		 16 32 16 32	 Nonlinear

Without	Opening	

Lines	 Obj	Value	 85.94 85.63 85.15 86.13	 86.51

MIP	 Obj	Value	 82.80 85.26 85.54 85.55	 84.07

Opening	One	Line	 Line	Switched 2‐5 2‐3 2‐5 2‐5	 2‐5

		 CPU	Time	 10.89 22.06 5.80 9.80	 100.78

Brute	Force	 Obj	Value	 84.94 84.85 85.28 85.29	 84.42

Opening	One	Line	 Line	Switched 2‐5 2‐5 2‐5 2‐5	 2‐5

		 CPU	Time	 8.94 9.47 5.11 6.97	 63.52
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30	bus	problem.	The	results	for	the	30	bus	problem	are	shown	in	Tables	5	and	6.			

In	both	the	tightly	and	loosely	constrained	cases,	the	biggest	difference	between	

the	objective	values	in	the	linear	approaches	is	2%.	In	both	tight	and	loose	

current	constraint	cases,	it	appears	that	removing	one	line	does	improve	the	

objective	value,	although	this	improvement	is	very	small.		The	MIP	and	Brute	

Force	methods	choose	different	lines	to	open	in	the	same	problem,	although	

each	method	chooses	the	same	line	within	the	linear	approaches.		However,	

choosing	the	different	line	does	not	seem	to	affect	the	value	of	the	objective	

function.		Here,	the	MIP	has	a	clear	advantage	over	the	Brute	Force	method	in	

the	solution	time.		The	MIP	runs	more	than	twice	as	fast	as	the	Brute	Force	

method	in	all	approaches	except	for	the	loosely	constrained	case	with	quadratic	

step	size	and	32	cuts.		In	the	brute	force	method,	linear	approaches	solved	at	

least	4	times	as	fast	as	the	nonlinear	approach.		Since	there	were	different	

results	on	different	simulations	for	the	MIP	with	the	nonlinear	case,	the	

simulation	result	of	the	last	run	is	captured	here.	

Table	5.	One	Line	Results:	30	Bus,	Tight	Current	Constraint	

Step	Size	Function	 		 Linear Quadratic

Number	of	Cuts	 		 16 32 16 32	 Nonlinear

Without	Opening	

Lines	 Obj	Value	 6.02 6.08 6.02 6.08	 6.10

MIP	 Obj	Value	 5.82 5.91 5.87 5.91	 5.74

Opening	One	Line	 Line	Switched 25‐27 25‐27 25‐27 25‐27	 25‐27

		 CPU	Time	 37.05 40.49 20.71 34.85	 536.4

Brute	Force	 Obj	Value	 5.93 5.93 5.92 5.93	 5.79

Opening	One	Line	 Line	Switched 6‐28 6‐28 6‐28 6‐28	 6‐28

		 CPU	Time	 124.8 117.2 93.19 113.6	 554.5

Table	6.	One	Line	Results:	30	Bus,	Loose	Current	Constraint	

Step	Size	Function	 		 Linear Quadratic

Number	of	Cuts	 		 16 32 16 32	 Nonlinear

Without	Opening	

Lines	 Obj	Value	 5.96 5.98 5.96 5.98	 6.00

MIP	 Obj	Value	 5.81 5.81 5.81 5.89	 5.74

Opening	One	Line	 Line	Switched 25‐27 25‐27 25‐27 25‐27	 24‐25

		 CPU	Time	 30.81 66.29 21.67 29.90	 633.7

Brute	Force	 Obj	Value	 5.92 5.92 5.92 5.92	 5.78

Opening	One	Line	 Line	Switched 6‐28 6‐28 6‐28 6‐28	 24‐25

		 CPU	Time	 75.39 156.0 89.76 53.73	 683.1

	
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18

57	bus	problem.		The	results	for	the	57‐bus	problem	are	shown	in	Tables	7	and	8.		

8The	lines	chosen	to	open	are	not	very	consistent	between	the	different	approaches,	

suggesting	that	the	objective	function	has	several	topologies	with	similar	objective	

values.		The	objective	function	values	were	within	0.2%	of	the	best‐known	value	

with	the	brute	force	method.		Here,	the	advantage	of	the	MIP	over	the	Brute	Force	

solution	time	increases;	the	MIP	solves	more	than	25	times	faster	than	the	brute	

force	method.		However,	the	Brute	Force	method	appears	to	find	better	solutions;	all	

solutions	of	the	brute	force	method	are	an	improvement	over	not	switching	any	

lines,	and	all	MIP	solutions	have	greater	generator	cost	than	the	Brute	Force	

solutions.

Table	7.		One	Line	Results:		57	Bus,	Tight	Current	Constraint	

Step	Size	Function	 		 Linear Quadratic

Number	of	Cuts	 		 16 32 16 32 Nonlinear

Without	Opening	Lines	 Obj	Value	 424.02 422.71 424.07 423.69	 433.85

MIP	 Obj	Value	 424.66 426.98 426.99 426.75	 419.08

Opening	One	Line	 Line	Switched 56‐57 38‐48 2‐3 2‐3	 9‐13

		 CPU	Time	 34.82 54.85 12.15 21.10	 3000.1

Brute	Force	 Obj	Value	 421.16 421.65 421.49 421.70	 420.71

Opening	One	Line	 Line	Switched 12‐16 38‐49 9‐11 9‐13	 48‐49

		 CPU	Time	 1,298.15	 1,708.27	 815.8 1,034.99		 10,966.5	

Table	8.		One	Line	Results:		57	Bus,	Loose	Current	Constraint	

Step	Size	Function	 		 Linear Quadratic

Number	of	Cuts	 		 16 32 16 32 Nonlinear

Without	Opening	Lines	 Obj	Value	 422.60	 422.42	 423.27	 423.47		 	425.48	

MIP	 Obj	Value	 426.33	 426.98	 429.01	 426.75		 418.47

Opening	One	Line	 Line	Switched 38‐48	 38‐48	 1‐2	 2‐3		 	3‐4

		 CPU	Time	 34.09	 54.85	 9.62	 21.10		 2982.6

Brute	Force	 Obj	Value	 421.16	 421.65	 421.49	 421.70		 	420.71	

Opening	One	Line	 Line	Switched 12‐16	 38‐49	 9‐11	 9‐13		 	48‐49	

		 CPU	Time	 1,292.2 1,668.6 768.5 964.6		 	9,287.3

	

	 	
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19

	

118	bus	problem.	The	results	for	the	118‐bus	problem	are	shown	in	Tables	9	and	10.			

In	the	tight	current	constraint	case,	the	linear	approaches	choose	either	line	77‐80	

or	line	77‐82	to	be	removed.		In	the	loose	current	case,	there	is	more	deviation	in	

which	line	is	taken	out.		The	MIP	is	again	much	faster	than	the	brute	force	method	

ሺin	both	tightly	and	loosely	constrained	casesሻ.		Objective	function	values	are	within	

3.2%		of	the	nonlinear	values	in	the	tight	case	and	within	1.5%	of	the	loose	case.		In	

the	tight	current	constraint	case,	the	objective	values	found	by	the	MIP	and	the	

brute	force	method	are	very	similar.		In	the	loose	current	constraint,	the	brute	force	

method	appears	to	find	better	solutions;	all	the	objective	values	in	the	brute	force	

method	are	an	advantage	over	not	switching	any	lines;	most	of	the	solutions	found	

by	the	MIP	are	more	costly	than	not	switching	any	lines.		

Table	9:		One	Line	Results:		118	Bus,	Tight	Current	Constraint	

Step	Size	Function	 		 Linear Quadratic

Number	of	Cuts	 		 16 32 16 32 Nonlinear

Without	Opening	Lines	 Obj	Value	 1,381.15 1,380.4 1,388.3 1,388.3	 1,364.9

MIP	 Obj	Value	 1,379.12	 1,378.2 1,385.7 1,385.8		 	1,344.3

Opening	One	Line	 Line	Switched 77‐82	 77‐82	 77‐82	 77‐82		 	17‐31	

		 CPU	Time	 213.1 307.8 256.6 330.1		 	30,000

Brute	Force	 Obj	Value	 1,379.2 1,378.4 1,384.6 1,384.1		 	1,368.9

Opening	One	Line	 Line	Switched 77‐80	 77‐82	 77‐82	 77‐80		 	77‐82	

		 CPU	Time	 12,303 14,631 6,534.6 16,175		 	26,359

	

Table	10:		One	Line	Results:		118	Bus,	Loose	Current	Constraint

Step	Size	Function	 		 Linear Quadratic

Number	of	Cuts	 		 16 32 16 32 Nonlinear

Without	Opening	Lines	 Obj	Value	 1,307.7 1,311.8 1,310.7 1,314.6		 	1,300.1	

MIP	 Obj	Value	 1,314.1 1,315.2 1,314.4 1,314.4		 	1,296.8

Opening	One	Line	 Line	Switched 25‐26	 25‐26	 25‐26	 25‐26		 	24‐70	

		 CPU	Time	 123.1 186.0 34.56	 61.64		 	322.6

Brute	Force	 Obj	Value	 1,305.3 1,304.9 1,305.9 1,307.6		 	1,304.6

Opening	One	Line	 Line	Switched 46‐48	 100‐104	 45‐46	 40‐41		 	19‐20

		 CPU	Time	 22,780 34,747 11,874	 16,091		 	21,236

	

	
Page

20

	

COMPARISON:		ONE	MIP,	REPEATED	MIP,	AND	PROGRESSIVE	MIP	

14	Bus,	Tightly	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 All	three	methods	of	solving	the	MIP	reported	very	similar	objective	function	

values,	except	for	the	progressive	MIP	with	a	linear	step	size	function	and	32	cuts,	

where	the	answer	was	infeasible	ሺsee	Table	11ሻ.	

Table	11.		14	Bus,	Tightly	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 95.02 95.04 95.10 95.04	

One	MIP	 95.25 95.04 95.25 95.04	

Progressive	MIP	 95.19 INF 95.42 95.04	

No	lines	open	 105.69 106.53 105.69 106.53	

	

	

	

LP	vs.	NLP	Objective	Function	Value	Differences	

	 The	LP	vs.	NLP	Objective	Value	Difference	is	computed	as	the	objective	values	of	the	

ሺNLP‐LPሻ/LP.			As	a	reminder,	the	NLP	Objective	Value	is	the	answer	to	the	problem	where	

the	transmission	lines	found	to	be	taken	out	in	the	LP	are	fixed	using	the	solver	IPOPT,	and	

the	ACOPF	is	solved	with	the	original,	nonlinear	constraints	ሺbut	with	linear	objective	

functionሻ.		In	this	case,	as	shown	in	tables	12	and	13,	all	the	nonlinear	answers	are	close	to	

the	linear	objective	value,	except	for	the	progressive	MIP	with	32	cuts	and	a	linear	step	size.		

This	large	discrepancy	occurs	because	the	linear	method	finds	an	infeasible	solution	and	the	

nonlinear	method	finds	a	feasible	solution.	

92

93

94

95

96

97

98

99

100

16 32 16 32

Linear Quadratic

Objective Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

21

Table	12.	14	Bus,	Tightly	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 95.06	 95.07 95.06 95.06	

One	MIP	 95.41	 95.06 95.42 95.06	

Progressive	MIP	 95.43	 94.95 95.26 95.06	

Table	13.	14	Bus,	Tightly	Constrained:	%	Difference	between	LP	and	NLP	Objective	Value	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 0.04 0.04 ‐0.05 0.02	

One	MIP	 0.17 0.02 0.18 0.02	

Progressive	MIP	 0.26 INF ‐0.17 0.02	

	

CPU	Time	

	 Table	14	shows	that	using	only	one	MIP	at	the	beginning	dramatically	reduces	the	

time	to	solve	the	switching	problem.		The	progressive	MIP	has	a	slight	advantage	over	using	

one	MIP.	

Table	14.	14	Bus,	Tightly	Constrained:		CPU	Time	ሺsecondsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 34.68 101.03 26.75 59.09	

One	MIP	 8.36 20.60 15.83 22.56	

Progressive	MIP	 7.11 15.18 6.24 14.15	

	

	

‐3%

‐2%

‐1%

0%

1%

2%

3%

16 32 16 32

Linear Quadratic

LP vs. NLP

% Difference

Repeated MIP

One MIP

Progressive MIP
Page

22

	

	

14	Bus,	Loosely	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 The	linear	objective	values	found	by	all	methods	are	relatively	close,	as	seen	in	

Table	15,	except	the	repeated	MIP	with	linear	step	size	function	appears	to	deviate	from	all	

the	other	methods,	although	it	still	finds	a	feasible	solution.

Table	15.	14	Bus,	Loosely	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 82.53 82.10 84.65 84.38	

One	MIP	 84.49 85.08 84.49 85.08	

Progressive	MIP	 85.01 85.21 85.09 85.36	

No	lines	open	 85.94 85.63 85.15 86.13	

	

	

	

	 	

0.00

20.00

40.00

60.00

80.00

100.00

16 32 16 32

Linear Quadratic

CPU Time (s)

Repeated MIP

One MIP

Progressive MIP

81.00

81.50

82.00

82.50

83.00

83.50

84.00

84.50

85.00

85.50

16 32 16 32

Linear Quadratic

Objective Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

23

LP	vs.	NLP	Objective	Function	Value	Differences	

	 Tables	16	and	17	display	that	the	nonlinear	and	linear	objective	values	are	within	

1%	except	for	the	repeated	MIP	method	with	linear	step	size	reductions.		However,	the	

nonlinear	objective	value	for	this	case	is	close	to	85	rather	than	82,	which	is	what	the	one	

MIP	and	progressive	MIP	methods	found.	

Table	16.	14	Bus,	Loosely	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 85.20 85.43 85.21 85.20	

One	MIP	 85.11 85.12 85.11 85.11	

Progressive	MIP	 85.02 85.36 85.10 85.23	

Table	17.	14	Bus,	Loosely	Constrained:	%	Difference	between	LP	and	NLP	Objective	Value	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 3.13 3.90 0.66 0.97	

One	MIP	 0.73 0.05 0.73 0.04	

Progressive	MIP	 0.02 0.18 0.01 ‐0.15	

	

	

	

	 	

‐1%

0%

1%

1%

2%

2%

3%

3%

4%

4%

5%

16 32 16 32

Linear Quadratic

LP vs NLP:

% Difference

Repeated MIP

One MIP

Progressive MIP
Page

24

CPU	Time	

	 Table	18	shows	that	the	one	MIP	method	solved	significantly	faster	than	the	

repeated	MIP	method,	and	the	progressive	MIP	method	solved	significantly	faster	than	the	

progressive	MIP	method.		However,	there	is	not	a	clear	pattern	which	number	of	cuts	or	

step	size	function	is	faster.	

	

Table	18.	14	Bus,	Loosely	Constrained:		CPU	Time	ሺsecondsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 258.65 152.38 126.00 165.26	

One	MIP	 22.08 59.40 23.12 40.16	

Progressive	MIP	 4.21 5.15 9.77 5.14	

	

	

	

	

30	Bus,	Tightly	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 The	results	of	solving	the	30	bus	system	with	tight	current	constraints	are	given	in	

Table	19.		All	methods	found	a	feasible	answer	for	this	problem.		Like	the	previous	problem,	

the	repeated	MIP	method	with	linear	step	size	function	seemed	to	find	answers	that	

deviated	from	all	the	other	methods.	

Table	19.	30	Bus,	Tightly	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 5.83	 5.83 5.96 5.93	

One	MIP	 5.94	 5.97 5.96 5.97	

Progressive	MIP	 5.89	 5.90 5.93 5.93	

No	lines	open	 6.02	 6.08 6.02 6.08	

	

	

0

30

60

90

120

150

180

210

240

270

16 32 16 32

Linear Quadratic

CPU Time (s)

Repeated MIP

One MIP

Progressive MIP
Page

25

	

	

LP	vs.	NLP	Objective	Function	Value	Differences	

	 Tables	20	and	21	reveal	that	the	NLP	solution	was	very	close	to	the	LP	solution	in	

the	One	MIP	and	Progressive	MIP	approaches;	however,	the	network	configuration	found	by	

the	LP	in	the	repeated	MIP	cases	ሺexcept	for	the	linear	step	size	with	16	cutsሻ	was	infeasible	

once	all	the	nonlinearities	were	considered.	

Table	20.	30	Bus,	Tightly	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 5.97 INF INF INF	

One	MIP	 5.99 5.98 5.99 6.01	

Progressive	MIP	 5.94 5.98 5.95 6.07	

	

Table	21.	30	Bus,	Tightly	Constrained:	%	Difference	between	LP	and	NLP	Objective	Value	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 2.30	 INF INF INF		

One	MIP	 0.81	 0.25 0.54 0.67	

Progressive	MIP	 0.80	 1.31 0.40 2.20	

	

5.75

5.80

5.85

5.90

5.95

6.00

16 32 16 32

Linear Quadratic

Objective Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

26

	

	

CPU	Time	

	

From	Table	22,	we	see	that	the	progressive	MIP	approach	solved	significantly	faster	

than	the	one	MIP	approach	that	solved	the	problem	much	faster	than	the	repeated	

MIP	approach.	

Table	22.		30	Bus,	Tightly	Constrained:		CPU	Time	ሺsecondsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 3908.01	 6886.65 1718.02 2409.33	

One	MIP	 763.19	 1001.37 806.55 1000.99	

Progressive	MIP	 24.85	 34.15 20.71 41.44	

	

	

	

	

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP: %Difference

Repeated MIP

One MIP

Progressive MIP

0

1000

2000

3000

4000

5000

6000

7000

16 32 16 32

Linear Quadratic

CPU Time (s)

Repeated MIP

One MIP

Progressive MIP
Page

27

30	Bus,	Loosely	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 Table	23	shows	that	the	repeated	MIP	approach	ሺexcept	for	the	16	cuts,	

quadratic	step	size	caseሻ	seemed	to	find	significantly	different	answers	than	those	

found	by	using	One	MIP	and	the	Progressive	MIP.	

	

Table	23.		30	Bus,	Loosely	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 5.82	 5.83 5.93 5.81	

One	MIP	 5.94	 5.89 5.95 5.90	

Progressive	MIP	 5.89	 5.90 5.90 5.91	

	

	

	

LP	vs.	NLP	Objective	Function	Value	Differences	

	 There	is	only	one	case	where	the	network	configuration	found	by	the	linear	

approach	was	not	feasible	–	the	repeated	MIP,	quadratic	step	size,	16	cuts	case	as	

seen	in	Tables	24	and	25.		Otherwise,	the	difference	between	the	answers	from	the	

NLP	and	the	LP	are	3.1%	or	less.	

	

Table	24.	30	Bus,	Loosely	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 5.94	 5.97 INF 6.00	

One	MIP	 5.98	 5.94 5.99 5.94	

Progressive	MIP	 5.98 5.95 5.95 5.98	

No	lines	open	 5.96 5.98 5.96 5.98	

5.70

5.75

5.80

5.85

5.90

5.95

6.00

16 32 16 32

Linear Quadratic

Objectie Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

28

	

	

Table	25.	30	Bus,	Loosely	Constrained:	%	Difference	between	LP	and	NLP	Objective	Value	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32	

Repeated	MIP	 2.01	 2.36 INF 3.09	

One	MIP	 0.67	 0.82 0.77 0.70	

Progressive	MIP	 1.43	 0.87 0.80 1.16	

	

	

	 	

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP: %Difference

Repeated MIP

One MIP

Progressive MIP
Page

29

CPU	Time	

	 Again,	as	seen	in	Table	26,	the	progressive	MIP	was	much	faster	than	one	

MIP	that	was	much	faster	than	the	progressive	MIP.		The	repeated	MIP	seemed	to	

run	much	faster	when	a	quadratic	step	size	was	used.	

	

Table	26.		30	Bus,	Loosely	Constrained:		CPU	Time	ሺsecondsሻ	 	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 8834.50	 7083.96 2871.75 1831.51	

One	MIP	 1000.54	 1002.15 1000.51 1000.98	

Progressive	MIP	 25.99	 29.75 20.17 28.86	

	

	

	

57	Bus,	Tightly	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 In	this	case,	Table	27	shows	that	5	out	of	the	12	approaches	found	infeasible	

solutions.		Only	the	repeated	MIP	consistently	found	a	feasible	solution.		The	optimal	

objective	value	found	by	the	repeated	MIP	differed	greatly	between	the	linear	and	

quadratic	case.	

	

Table	27.	57	Bus,	Tightly	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 417.41	 417.39 426.67 425.75	

One	MIP	 INF	 INF INF INF	

Progressive	MIP	 INF	 428.56 426.15 427.96	

No	lines	open	 424.02	 422.71 424.1 423.7	

	

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

16 32 16 32

Linear Quadratic

CPU Time (s)

Repeated MIP

One MIP

Progressive MIP
Page

30

	

	

LP	vs.	NLP	Objective	Function	Value	Differences	

	 The	configurations	found	by	the	LP	solution	were	infeasible	in	9	out	of	the	12	

different	approaches	as	displayed	in	Tables	28	and	29.		This	suggests	that	the	57	bus	

problem	with	a	tight	current	constraint	is	likely	to	become	infeasible	during	

switching.		A	constraint	enforcing	the	voltage	minimum	would	need	to	be	added	to	

avoid	these	infeasibilities.	

	

Table	28.	57	Bus,	Tightly	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 426.93 427.25 INF INF	

One	MIP	 INF INF INF INF	

Progressive	MIP	 INF INF 427.62 INF	

Table	29.	57	Bus,	Tightly	Constrained:	%	Difference	between	LP	and	NLP	Objective	Value	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 2.23 2.31 99.98 99.99	

One	MIP	 99.93 99.68 95.29 99.77	

Progressive	MIP	 99.95 85.03 0.34 100.00	

	

360

380

400

420

440

460

480

500

16 32 16 32

Linear Quadratic

Objective Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

31

	

CPU	Time	

	 Table	30	reveals	that	once	again,	the	progressive	MIP	was	much	faster	than	one	

MIP	was	much	faster	than	the	repeated	MIP	approach.	

Table	30.		57	Bus,	Tightly	Constrained:		CPU	Time	ሺsecondsሻ

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 4001.22	 6003.74 4528.23 3115.72	

One	MIP	 1015.75	 1020.67 1009.96 1011.68	

Progressive	MIP	 138.03	 182.29 80.39 139.19	

	

	

	

	

	 	

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP: %Difference

Repeated MIP

One MIP

Progressive MIP

0

800

1600

2400

3200

4000

4800

5600

6400

16 32 16 32

Linear Quadratic

CPU Time (s)

Repeated MIP

One MIP

Progressive MIP
Page

32

57	Bus,	Loosely	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 Table	31	shows	that	6	of	the	12	approaches	could	not	find	a	feasible	solution	

to	the	loosely	constrained	57	bus	problem.	

	

Table	31.	57	Bus,	Loosely	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32	

Repeated	MIP	 419.36 416.56 INF 426.51	

One	MIP	 INF INF INF INF	

Progressive	MIP	 INF 426.41 425.79 424.63	

No	lines	open	 422.6 422.4 423.3 423.5	

	

	

	

LP	vs.	NLP	Objective	Function	Value	Differences	

	 As	seen	in	Tables	32	and	33,	9	of	the	12	approaches	could	not	find	a	feasible	

solution	with	the	57	bus	problem	with	loose	current	constraint.

Table	32.	57	Bus,	Loosely	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ		

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 INF	 427.47 INF INF	

One	MIP	 INF	 INF INF INF	

Progressive	MIP	 INF	 INF 428.93 430.18	

	

	 	

0

50

100

150

200

250

300

350

400

450

500

16 32 16 32

Linear Quadratic

Objective Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

33

Table	33.	57	Bus,	Loosely	Constrained:	%	Difference	between	LP	and	NLP	Objective	Value	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 n/a	 2.55 n/a n/a

One	MIP	 n/a	 n/a n/a n/a

Progressive	MIP	 n/a	 n/a 0.73 1.29	

	

	

	

	

CPU	Time	

	 Table	34	shows	that	the	progressive	MIP	is	significantly	faster	than	the	one	MIP	

approach,	which	is	usually	much	faster	than	doing	a	repeated	MIP.		In	this	case,	using	the	

progressive	MIP	is	both	the	fastest	method	and	the	most	likely	to	find	a	feasible	solution.	

Table	34.	57	Bus,	Loosely	Constrained:		CPU	Time	ሺsecondsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 20007.06 7005.45 1031.57 4388.51	

One	MIP	 1016.32 1019.00 1010.00 1010.06	

Progressive	MIP	 138.00 182.26 71.79 122.51	

	

	

‐20%

‐15%

‐10%

‐5%

0%

5%

10%

15%

20%

1234

LP vs. NLP: %Difference

Repeated MIP

One MIP

Progressive MIP

0

5000

10000

15000

20000

16 32 16 32

Linear Quadratic

CPU TIme (s)

Repeated MIP

One MIP

Progressive MIP
Page

34

118	Bus,	Tightly	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 As	seen	in	Table	35,	there	is	only	one	approach	ሺone	MIP,	linear	step	size,	16	

cutsሻ	where	the	linear	solver	cannot	find	a	feasible	solution.	

	

Table	35.	118	Bus,	Tightly	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 1381.32 1377.03 1385.04 1380.17	

One	MIP	 INF 1389.79 1385.13 1390.99	

Progressive	MIP	 1377.49 1377.63 1381.14 1382.52	

No	lines	open	 1381.2 1380.4 1388.3 1388.3	

	

	

	

	

	

LP	vs.	NLP	Objective	Function	Value	Differences	

	 Tables	36	and	37	show	that	the	NLP	was	able	to	find	a	feasible	solution	to	the	

same	configuration	of	lines	as	the	LP	where	the	LP	approach	could	not	find	a	

feasible	solution.		In	all	other	cases,	the	nonlinear	solution	was	within	1%	of	the	

linear	solution.	

	

Table	36.	118	Bus,	Tightly	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 1388.74	 1386.37 1387.64 1384.41	

One	MIP	 1385.19	 1391.23 1385.19 1391.23	

1350

1355

1360

1365

1370

1375

1380

1385

1390

1395

1400

16 32 16 32

Linear Quadratic

Objective Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

35

Progressive	MIP	 1383.35	 1383.37 1382.15 1382.82	

	

Table	37.	118	Bus,	Tightly	Constrained:	%	Difference	between	LP	and	NLP	Objective	Value	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 0.53 0.67 0.19 0.31	

One	MIP	 INF 0.10 0.00 0.02	

Progressive	MIP	 0.42 0.42 0.07 0.02	

	

	

	

	

CPU	Time	

	 While	the	progressive	MIP	was	faster	than	one	MIP	which	was	faster	than	the	

repeated	MIP,	the	progressive	MIP	approach	was	comparatively	much	faster	in	the	

57	bus	cases	than	this	case.		Table	38	shows	the	progressive	MIP	speeding	up	

solution	times	between	1.2	and	2.4x;	in	57	bus	case,	the	progressive	MIP	sped	up	

solution	times	by	at	least	5x	versus	the	one	MIP	approach.	

	

Table	38.	118	Bus,	Tightly	Constrained:		CPU	Time	ሺsecondsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 7008.08	 3010.86 4089.92 4014.06	

One	MIP	 1106.94	 1149.24 1033.84 1043.54	

Progressive	MIP	 591.78	 936.41 422.64 652.28	

	

	

‐20%

‐15%

‐10%

‐5%

0%

5%

10%

15%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP:

% Difference

Repeated MIP

One MIP

Progressive MIP
Page

36

	

	

118	Bus,	Loosely	Constrained	

	

Objective	Value	Function	ሺLPሻ	

	 As	seen	in	Table	39,	the	118	bus	solutions	mainly	differ	in	the	linear	step	size,	

repeated	MIP	approach	versus	all	of	the	other	approaches.	

	

Table	39.	118	Bus,	Loosely	Constrained:		Linear	Objective	Value	ሺlinear	objective,	linear	

constraintsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 1291.18	 1301.09 1314.15 1316.88	

One	MIP	 1313.85	 1311.76 1313.47 1314.60	

Progressive	MIP	 1306.34	 1307.86 1314.62 1315.33	

No	lines	open	 1307.7	 1311.8 1310.7 1314.6	

	

	

	

0

1000

2000

3000

4000

5000

6000

7000

16 32 16 32

Linear Quadratic

Repeated MIP

One MIP

Progressive MIP

1280

1285

1290

1295

1300

1305

1310

1315

1320

16 32 16 32

Linear Quadratic

Objective Value (LP)

Repeated MIP

One MIP

Progressive MIP
Page

37

LP	vs.	NLP	Objective	Function	Value	Differences	

	 There	was	one	line	configuration	out	the	12	approaches	where	the	nonlinear	

solver	could	not	find	a	feasible	solution,	seen	in	Tables	40	and	41.		Otherwise,	the	

nonlinear	answers	were	within	2%	or	less	of	the	linear	answers.		Like	in	some	of	the	

other	cases,	the	nonlinear	solution	for	the	configuration	found	with	the	repeated	

MIP,	linear	step	size	is	more	consistent	with	the	other	approaches	than	the	linear	

solution.		If	we	exclude	the	linear	step	size,	repeated	MIP	approach	and	the	

infeasible	answer,	the	nonlinear	solution	is	within	1%	of	the	linear	solution.	

Table	40.	118	Bus,	Loosely	Constrained:	NLP	Objective	Value	ሺnonlinear	constraints,	linear	

objectiveሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 1317.14	 1319.32 1322.53 INF	

One	MIP	 1318.49	 1315.89 1318.77 1315.55	

Progressive	MIP	 1317.29 1317.29 1325.20 1322.33	

	

	

Table	41.	118	Bus,	Loosely	Constrained:	%	Difference	between	LP	and	NLP	Objective	Values	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 1.97	 1.38 0.63 99.99	

One	MIP	 0.35	 0.31 0.40 0.07	

Progressive	MIP	 0.83	 0.72 0.80 0.53	

	

	

CPU	Time	

	 As	displayed	in	Table	42,	both	the	one	MIP	and	progressive	MIP	approaches	

run	much	faster	than	the	repeated	MIP	approaches;	However,	the	progressive	MIP	is	

slower	than	the	one	MIP	approach	when	the	linear	step	size	is	used.	

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

16 32 16 32

Linear Quadratic

LP vs. NLP:

% Difference

Repeated MIP

One MIP

Progressive MIP
Page

38

	

Table	42.	118	Bus,	Loosely	Constrained:		CPU	Time	ሺsecondsሻ	

Step	Size	Function	 Linear Quadratic

Number	of	Cuts	 16	 32 16 32

Repeated	MIP	 13009.89	 13033.29 5003.83 6012.06	

One	MIP	 1075.83	 1134.47 1014.72 1038.70	

Progressive	MIP	 1159.56	 2642.57 666.49 818.48	

	

	

	

	

	

	 	

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

11000

12000

13000

16 32 16 32

Linear Quadratic

CPU Time (s)

Repeated MIP

One MIP

Progressive MIP
Page

39

6. Summary	

From	this	study,	we	see	that	transmission	switching	provides	cost	benefits	in	

AC	systems.		Tables	43	and	44	show	that	the	magnitude	of	this	cost	savings	is	very	

dependent	on	the	current	constraints;	one	obtains	more	savings	with	lower	line	

limits.		We	find	that	the	linear	approximation	generally	achieves	about	the	same	

amount	of	savings	as	the	original	nonlinear	program;	however,	there	are	cases	

where	the	linear	program	finds	a	better	answer	ሺlike	the	14	bus	case	with	loose	

current	limitsሻ	or	a	worse	answerሺlike	the	14	bus	case	with	tight	current	limitsሻ.		In	

this	case,	opening	the	first	line	provides	the	most	of	the	benefits.		The	conflation	of	

the	MIP	plus	the	changing	fixed	points	occasionally	leads	to	a	better	answer	with	

allowing	one	line	open	rather	than	five	lines	open.		This	is	often	due	to	the	five	lines	

open	problem	choosing	a	network	configuration	that	is	better	in	the	first	iteration	

than	opening	one	line	but	worse	when	the	problem	has	finally	converged.	

Table	43.		%	Savings	with	up	to	one	line	open	

Current	Constraint	 Method	 14 30 57 118

Loose	 Linear	 3% 3% 0% 0%

		 IPOPT	 2% 4% 1% 0%

Tight	 Linear	 11% 3% 0% 0%

		 IPOPT	 13% 5% 3% 0%

	

Table	44.	%	Savings	with	up	to	five	lines	open	

Current	Constraint	 Method	 14 30 57 118

Loose	 Linear	 4% 2% 1% 1%

		 IPOPT	 2% 1% 0% 0%

Tight	 Linear	 10% 3% 1% 0%

		 IPOPT	 12% 3% 1% 0%

	

7. Conclusions	

The	one	MIP	approach	solves	the	14	and	30	bus	problems	quicker	than	the	

repeated	MIP	approach	and	within	4%	or	less	of	the	repeated	MIP	objective	value.		

However,	while	the	one	MIP	approach	has	fast	performance	in	the	57	and	118	bus	

problems,	both	the	one	and	repeated	MIP	approaches	find	worse	solutions	than	not	

switching	any	lines	in	many	of	the	cases.		In	the	larger	networks,	it	appears	that	

there	are	either	multiple	optimal	solutions	or	multiple	network	configurations	that	

have	a	very	similar	cost.	

	 The	linearized	ACOPF	is	much	faster	than	the	unmodified	ACOPF	and	usually	

finds	solutions	within	1%	of	the	ACOPF.		The	progressive	and	one	MIP	approaches	

are	much	faster	than	the	repeated	MIP	approach	and	generally	give	answers	within	

1%	of	the	repeated	MIP	approach.		However,	there	are	instances	where	the	
Page

40

linearized	ACOPF	finds	an	optimal	network	configuration	that	is	not	feasible	in	the	

unmodified	ACOPF,	due	to	the	linearized	ACOPF	finding	a	solution	with	voltages	that	

are	below	the	minimum	threshold.	

	 	
Page

41

REFERENCES		

M.B.	Cain,	R.P.	O’Neill,	A.	Castillo.	“History	of	AC	Optimal	Power	Flow	Problems,”	

FERC	Staff	Paper,	2012.	

C.	Campaigne,	P.	A.	Lipka,	R.	P.	O’Neill,	and	S.S.	Oren.	“Testing	Step	Size	Limits	for	

Solving	the	Linearized	Current	Voltage	AC	Optimal	Power	Flow,”	FERC	Staff	Paper,	

2013.	

E.B.	Fisher,	R.P.	O’Neill,	and	M.C.	Ferris.		“Optimal	transmission	switching,”	IEEE	

Trans.	Power	Syst,	vol.	23,	no.	3,	pp.	1346‐1355,	Aug.	2008.	

J.	D.	Fuller,	R.	Ramasra,	and	A.	Cha,	“Fast	heuristics	for	transmission	line	switching,”	

IEEE	Trans.	Power	Syst.,	vol.	27,	no.	3,	pp.	1377–1386,	Aug.	2012.	

K.W.	Hedman,	R.P.	O’Neill,	E.B.	Fisher,	and	S.S.	Oren,	“Optimal	transmission	

switching	–	sensitivity	analysis	and	extensions,”	IEEE	Trans.	Power	Syst,	vol.	23,	no.	

3,	pp.	1469‐1479,	Aug.	2008.	

K.W.	Hedman,	R.P.	O’Neill,	E.B.	Fisher,	and	S.S.	Oren,	“Optimal	transmission	

switching	with	contingency	analysis,”	IEEE	Trans.	Power	Syst,	vol.	24,	no.	3,	pp.	

1577‐1586,	Aug.	2009.	

P.	A.	Lipka,	R.	P.	O’Neill,	and	Shmuel	Oren,		“Developing	Line	Current	magnitude	

Constraints	for	IEEE	Test	Problems,”	FERC	Staff	Paper,	2012.			

R.P.	O’Neill,	Anya	Castillo,	M.B.	Cain.		“Formulation	to	the	AC	Optimal	Power	Flow	

Problem	and	Linearizations,”	FERC	Staff	Paper,	2012.	

J.	Ostrowski,	J.	Wang,	and	C.	Liu,	“Anti‐Islanding	in	Transmission	Switching”,	2012	

ሺin	reviewሻ.	

Exploiting	symmetry	in	transmission	lines	for	transmission	switching,”	IEEE	Trans.	

Power	Syst.,	vol.	27,	no.	3,	pp.	1708–1709,	Aug.	2012.	

Mehrdad	Pirnia,	Richard	P.	O’Neill,	Paula	A.	Lipka,	Clay	Campaigne,	“A	

Computational	Study	of	LP	Approximation	to	the	IV	ACOPF	Formulation,”	FERC	Staff	

Paper,	2012.	
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T.	Potluri	and	K.	W.	Hedman,	“Impacts	of	Topology	Control	on	the	ACOPF,”	in	IEEE	

PES	General	Meeting	2011,	San	Diego,	CA,	July	2012.	

Ruiz,	P.A.	,	Foster,	J.M.	;		Rudkevich,	A.	;		Caramanis,	M.C.,	“Tractable	Transmission	

Topology	Control	Using	Sensitivity	Analysis,”	IEEE	Trans.	Power	Syst.,	vol.	27,	No.	3,	

pp.	1550‐1559,	Aug.	2012	

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,”	IEEE	Trans.	Power	Syst.,	vol.	26,	no.	1,	pp.	12‐19,	Feb.	2011.	

	

	 	
Page

43

Appendix:		Detailed	Numerical	Results	

	

A.1	FULL	REPEATED	MIP	VERSUS	MIP	ONLY	ON	FIRST	PASS	RESULTS	

	

14	bus	problem.	

Tables	A.1	and	A.2	show	how	the	repeated	MIP	compares	to	the	approach	of	using	

only	one	MIP.		In	the	tightly	constrained	case,	the	objective	values	are	very	similar	

between	both	approaches,	differing	at	the	most	by	0.25%.		Using	one	MIP	has	a	

significant	time	advantage.		The	lines	opened	differ	slightly,	with	the	repeated	MIP	

choosing	to	open	two	lines	in	some	chases	while	the	approach	using	one	MIP	only	

results	in	one	line	being	opened.	

The	loosely	constrained	case	displays	more	variability	between	the	approaches	

using	one	MIP	and	the	repeated	MIP,	although	there	is	also	more	variability	between	

step	size/number	of	cuts	approaches.		However,	the	biggest	difference	in	objective	

function	values	is	3.5%.	The	approach	using	one	MIP	is	at	least	twice	as	fast	as	the	

repeated	MIP	approach.		The	lines	open	vary	between	one	MIP	and	repeated	MIP	

with	the	repeated	MIP	approach	opening	line	3‐4	but	never	opening	line	6‐12,	which	

the	one	MIP	approach	opens	in	every	case.		However,	both	cases	open	line	2‐5,	and	

the	lines	opened	in	the	one	MIP	approach	except	for	line	6‐12	are	opened	in	one	of	

the	repeated	MIP	approaches.

	

Table	A.1.		Repeated	vs.	One	MIP:		14	Bus,	Tight	Current	Constraint

	 		 Step	Size	Function Linear Quadratic	

		 Number	of	Cuts 16 32 16	 32

		 		 ൏ൌ	#	Lines	Open 	 		

Objective	Value		

No	Lines	Open	 0 105.69 106.53 105.69	 106.53

Repeated	MIP	 5 95.02 95.04 95.10	 95.04

		 One	MIP	 5 95.25 95.04 95.25	 95.04

CPU	Time	 Repeated	MIP	 5 34.7 101.0 26.7	 59.1

		 One	MIP	 5 8.4 20.6 15.8	 22.6

No.	of	Lines	Opened	

Repeated	MIP	 5 1 1 1	 1

One	MIP	 5 2 1 2	 1

Lines	Opened	 Repeated	MIP	 5 4‐7 4‐7 4‐7	 4‐7

		 		 		

		

		

		

		 One	MIP	 5 2‐5 4‐7 2‐5	 4‐7

		 		 4‐7 4‐7	
Page

44

Table	A.2:		Repeated	vs.	One	MIP:		14	Bus,	Loose	Current	Constraint	

	 		 Step	Size	Function Linear Quadratic	

		 Number	of	Cuts 16 32 16	 32	

		 		 ൏ൌ	#	Lines	Open 		

Objective	Value		

No	Open	Lines 0 85.94 85.63 85.15	 86.13	

Repeated	MIP 5 82.53 82.10 84.65	 84.38	

		 One	MIP	 5 84.49 85.08 84.49	 85.08	

CPU	Time	 Repeated	MIP 5 258.6 152.4 126.0	 165.3	

		 One	MIP	 5 22.1 59.4 23.1	 40.2	

No.	of	Lines	Opened		

Repeated	MIP 5 2 5 5	 5	

One	MIP	 5 5 5 5	 5	

Lines	Opened		

Repeated	MIP 5 2‐5 2‐5 2‐5	 2‐5	

		 3‐4 3‐4 3‐4	 3‐4	

		 		 4‐9 4‐5	 4‐5	

		 		 9‐14 4‐7	 4‐7	

		 		 12‐13 4‐9	 4‐9	

One	MIP	 2‐5 2‐5 2‐5	 2‐5	

		 		 4‐5 4‐5 4‐5	 4‐5	

		 		 4‐7 4‐7 4‐7	 4‐7	

		 		 4‐9 4‐9 4‐9	 4‐9	

		 		 6‐12 6‐12 6‐12	 6‐12	

30	bus	problem.	

Table	A.3	shows	that	the	tight	current	constraint	case	has	up	to	a	2.3%	gap	between	

the	one	and	repeated	MIP	approaches.		There	is	a	clear	advantage	to	opening	lines.		

Using	one	MIP	is	at	least	twice	as	fast	as	using	a	MIP	at	every	step.	In	both	cases,	line	

6‐28	is	opened,	but	the	other	four	lines	are	completely	different	in	the	case	of	linear	

and	quadratic	step	size	with16	cuts;	3	lines	are	different	for	the	linear	and	quadratic	

step	size	with	32	cuts.		

In	Table	A.4,	the	loose	current	constraint	case	has	up	to	a	2%	gap	between	the	one	

and	repeated	MIP	approaches.			However,	the	one	MIP	approach	always	finds	a	more	

costly	answer	than	the	repeated	MIP	approach;	in	some	cases,	opening	lines	has	a	

minor	impact.		Again,	the	one	MIP	approach	is	much	faster	than	the	repeated	MIP	

approach.		In	both	tight	and	loose	cases,	the	quadratic	step	size	with	repeated	MIP	

appears	to	run	much	faster	than	the	linear	step	size.		All	cases	open	the	line	

connecting	buses	6	and	28;	the	other	lines	opened	vary	greatly	between	the	one	and	

repeated	MIP	approaches	as	well	as	the	different	step	size	and	number	of	cuts	

approaches.	

	

	 	
Page

45

Table	A.3:		Repeated	vs.	One	MIP:		30	Bus,	Tight	Current	Constraint	

	 		 Step	Size	Function Linear Quadratic

	 		 Number	of	Cuts 16 32 16	 32

		 		 ൏ൌ	#	Lines	Open 		 	

Objective	Value	

Without	Opening	

Lines	 0 6.02 6.08	 6.02	 6.08

		 Repeated	MIP	 5 5.83 5.83	 5.96	 5.93

		 One	MIP	 5 5.94 5.97	 5.96	 5.97

CPU	Time	 Repeated	MIP	 5 3908.0 6886.7	 1718.0	 2409.3

		 One	MIP	 5 763.2 1001.4	 806.5	 1001.0

#	of	Lines	Opened	 Repeated	MIP	 5 5 5	 5	 3

		 One	MIP	 5 5 5	 5	 5

Lines	Opened	 Repeated	MIP	 5 6‐28 6‐28	 6‐28	 6‐28

		 		 9‐11 9‐11	 9‐11	 9‐11

		 		 10‐21 10‐21	 10‐21	 25‐27

		 		 10‐22 10‐22	 10‐22	

		 		 23‐24 16‐17	 23‐24	

		 One	MIP	 5 6‐28 6‐28	 6‐28	 6‐28

		 		 10‐20 10‐20	 10‐20	 10‐20

		 		 12‐15 12‐15	 12‐15	 12‐15

		 		 14‐15 16‐17	 14‐15	 16‐17

		 		 25‐27 25‐27	 25‐27	 25‐27

Table	A.4.		Repeated	vs.	One	MIP:		30	Bus,	Loose	Current	Constraint	

	 		 Step	Size	Function Linear Quadratic

		 Number	of	Cuts 16 32 16	 32

		 		 ൏ൌ	#	Lines	Open 		 	

Objective	Value	

Without	Opening	

Lines	 0 5.96 5.98	 5.96	 5.98

		 Repeated	MIP	 5 5.82 5.83	 5.93	 5.81

		 One	MIP	 5 5.94 5.89	 5.95	 5.90

CPU	Time	 Repeated	MIP	 5 8834.5 7084.0	 2871.7	 1831.5

		 One	MIP	 5 1000.5 1002.1	 1000.5	 1001.0

#	of	Lines	Opened	 Repeated	MIP	 5 5 5	 5	 4

		 One	MIP	 5 5 4	 5	 4

Lines	Opened	 Repeated	MIP	 5 4‐12 6‐28	 6‐9	 6‐9

		 		 6‐28 9‐11	 6‐28	 6‐28

		 		 16‐17 10‐21	 9‐10	 25‐27

		 		 18‐19 10‐22	 9‐11	 27‐19

		 		 23‐24 23‐24	 25‐27	

		 One	MIP	 5 6‐28 6‐28	 6‐28	 6‐28

		 		 12‐15 10‐21	 12‐15	 10‐21

		 		 14‐15 14‐15	 14‐15	 14‐15

		 		 19‐20 15‐18	 19‐20	 15‐18

		 		 25‐27 		 25‐27	
Page

46

57	bus	problem.	

In	the	tightly	constrained	case,	the	repeated	MIP	finds	a	feasible	solution	while	the	

one	MIP	approach	does	not,	displayed	in	Table	A.5.		The	lines	chosen	to	open	are	

very	different	between	the	two	cases.		In	the	both	the	tightly	and	loosely	constrained	

cases	ሺTable	A.6ሻ,	the	one	MIP	approach	does	not	find	a	feasible	solution,	while	the	

repeated	MIP	finds	a	feasible	solution	in	3	of	the	4	approaches.		The	reason	the	one	

MIP	approach	finds	infeasible	solutions	is	that	the	network	configuration	it	chooses	

in	the	first	iteration	is	infeasible,	but	in	the	first	stage,	more	infeasible	area	is	in	the	

linear	program	that	is	cut	off	by	iterative	voltage	constraints	later.	

Table	A.5.		Repeated	vs.	One	MIP:		57	Bus,	Tight	Current	Constraint	

	 		 Step	Size	Function Linear Quadratic

		 Number	of	Cuts 16 32	 16	 32

		 		 ൏ൌ	#	Lines	Open 		

Objective	Value	 Without	Opening	Lines 0 424.02 422.71	 424.1	 423.7

		 Repeated	MIP	 5 417.4 417.4	 426.7	 425.8

		 One	MIP	 5 INF1 INF	 INF	 INF

CPU	Time	 Repeated	MIP	 5 4001 6003	 4528	 3116

		 One	MIP	 5 10167 1021	 1010	 1012

#	of	Lines	Opened	 Repeated	MIP	 5 5 4	 3	 5

		 One	MIP	 5 5 5	 5	 5

Lines	Opened	 Repeated	MIP	 5 1‐15 3‐4	 9‐10	 2‐3

		 		 2‐3 3‐15	 9‐13	 3‐4

		 		 11‐13 22‐38	 11‐13	 3‐15

		 		 13‐15 53‐54	 		 22‐38

		 		 53‐54 		 		 54‐55

		 One	MIP	 5 9‐10 9‐13	 9‐10	 9‐13

		 		 9‐55 10‐51	 9‐55	 10‐51

		 		 11‐43 13‐49	 11‐43	 13‐49

		 		 14‐46 14‐46	 14‐46	 14‐46

		 		 35‐36 38‐44	 35‐36	 38‐44
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Table	A.6.		Repeated	vs.	One	MIP:		57	Bus,	Loose	Current	Constraint	

	 		 Step	Size	Function Linear Quadratic	

		 Number	of	Cuts 16 32 16	 32	

		 		 ൏ൌ	#	Lines	Open 		

Objective	Value	

		

No	Lines	Open	 0 422.6 422.4 423.3	 423.5	

Repeated	MIP 5 419.4 416.6 INF	 426.5	

		 One	MIP	 5 INF INF INF	 INF	

CPU	Time	 Repeated	MIP 5 20007 7006 1032	 4389	

		 One	MIP	 5 1016 1019 1010	 1010	

No.	of	Lines	Opened		 Repeated	MIP 5 5 5 5	 5	

One	MIP	 5 5 5 5	 5	

Lines	Opened		 Repeated	MIP 5 3‐4 6‐7 8‐9	 1‐2	

		 14‐15 9‐11 13‐49	 1‐15	

		 		 19‐20 9‐13 14‐46	 11‐41	

		 		 34‐35 12‐13 15‐45	 12‐17	

		 		 49‐50 12‐16 49‐50	 54‐55	

		 One	MIP	 5 9‐10 9‐10 9‐10	 9‐10	

		 		 9‐55 9‐13 9‐55	 9‐13	

		 		 14‐46 10‐51 14‐46	 10‐51	

		 		 25‐30 11‐13 25‐30	 11‐13	

		 		 44‐45 14‐46 44‐45	 14‐46	

	 	
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48

118	bus	problem.	

In	the	tightly	constrained	case	ሺTable	A.7ሻ,	the	repeated	and	one	MIP	approaches	

find	solutions	within	1%	of	each	other	if	the	case	where	the	one	MIP	approach	finds	

an	infeasible	solution	is	not	included.		The	one	MIP	approach	is	more	than	2.5	times	

faster	than	the	repeated	MIP	approach.		There	is	a	large	variance	in	which	lines	are	

selected	to	be	open	between	the	repeated	and	one	MIP	approaches,	and	between	the	

different	step	size	and	number	of	cuts	approaches.		The	one	MIP	approach	finds	

higher	costs	solutions	that	the	repeated	MIP	approach,	and	in	two	cases	finds	worse	

solutions	than	when	no	lines	are	open.		In	the	loosely	constrained	case	ሺTable	A.8ሻ,	

feasible	solutions	are	found	in	both	repeated	and	one	MIP	approaches,	and	the	

answers	are	within	2%	of	each	other.		However,	both	the	repeated	MIP	and	one	MIP	

approaches	find	some	solutions	that	are	worse	than	not	opening	any	lines.		The	one	

MIP	approach	runs	five	times	faster	or	more	than	the	repeated	MIP	approach.		The	

lines	opened	vary	greatly	between	the	different	approaches,	although	some	same	

lines	show	up	in	several	different	approaches.	

Table	A.7.		Repeated	vs.	One	MIP:		118	Bus,	Tight	Current	Constraint	

	 		

Step	Size	

Function Linear Quadratic

		 Number	of	Cuts 16 32 16	 32

		 		

൏ൌ	#	Lines	

Open 		 	

Objective	Value		 Without	Opening	

Lines	 0 1381.2 1380.4	 1388.3	 1388.3

Repeated	MIP 5 1381.3 1377.0	 1385.0	 1380.2

		 One	MIP	 5 INF 1389.8	 1385.1	 1391

CPU	Time	 Repeated	MIP 5 7008 3011	 4090	 4014

		 One	MIP	 5 1107 1149	 1034	 1044

No.	of	Lines	Opened		 Repeated	MIP 5 5 5	 5	 5

One	MIP	 5 5 5	 5	 5

Lines	Opened		 Repeated	MIP 5 60‐61 5‐6	 34‐36	 42‐49

		 61‐62 6‐7	 35‐36	 60‐62

		 		 61‐64 47‐49	 42‐49	 61‐62

		 		 75‐77 48‐49	 77‐82	 92‐94

		 		 77‐82 77‐80	 82‐83	 93‐94

		 One	MIP	 5 32‐113 8‐30	 32‐113	 8‐30

		 		 77‐82 47‐49	 77‐82	 47‐49

		 		 92‐94 48‐49	 92‐94	 48‐49

		 		 92‐100 92‐93	 92‐100	 92‐93

		 		 109‐110 94‐100	 109‐110	 94‐100
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Table	A.8.		Repeated	vs.	One	MIP:		118	Bus,	Loose	Current	Constraint	

	 		

Step	Size	

Function Linear Quadratic

		 Number	of	Cuts 16 32 16	 32

		 		

൏ൌ	#	Lines	

Open 		 		

Objective	Value	

Without	Opening	

Lines	 0 1307.7 1311.8 1310.7	 1314.6

		 Repeated	MIP	 5 1291.2 1301.1 1314.2	 1316.9

		 One	MIP	 5 1313.9 1311.8 1313.5	 1314.6

CPU	Time	 Repeated	MIP	 5 13010 13033 5004	 6012

		 One	MIP	 5 1075.8 1134.5 1014.7	 1038.7

#	of	Lines	Opened	 Repeated	MIP	 5 5 5 5	 4

		 One	MIP	 5 5 0 5	 0

Lines	Opened	 Repeated	MIP	 5 1‐2 1‐2 41‐42	 65‐68

		 		 8‐30 54‐55 42‐49	 82‐83

		 		 11‐12 64‐65 49‐66	 99‐100

		 		 65‐68 76‐118 68‐81	 110‐112

		 		 94‐100 80‐99 110‐112	

		 One	MIP	 5 61‐64 n/a 61‐64	 n/a

		 		 75‐118 75‐118	

		 		 77‐80 77‐80	

		 		 92‐102 92‐102	

		 		 100‐103 100‐103	

A.2	PROGRESSIVE	MIP	

In	the	Tables	that	follow,	the	Linear	Obj	Value	is	the	objective	value	of	the	linear	

iterative	solver.		The	NLP	Obj	Value	is	the	value	of	the	nonlinear	solution	when	the	

network	is	fixed	as	the	optimal	network	found	in	the	linear	iterative	solver.		The	MIP	

gap	is	the	gap	between	the	best‐known	linear	solution	and	the	final	MIP	answer.		

New	Opened	Line	designates	which	line	was	opened	in	the	problem.	For	example,	

the	first	problem	only	allows	one	line	open.		Then,	the	next	problem	fixes	that	line	as	

open	and	sees	if	there	is	another	line	that	would	be	beneficial	to	open,	and	so	forth.	

	

14	bus.	

The	linear	and	nonlinear	solutions	in	the	14	bus	case	–	both	for	the	tight	and	loose	

constraints	ሺA.9	and	A.10ሻ	‐	are	within	0.6%	of	each	other	except	for	one	case	where	

the	linear	solver	found	an	infeasible	solution.		Like	the	repeated/one	MIP	

approaches,	the	lines	opened	in	the	tight	current	constraint	case	were	4‐7	and	2‐5.	

The	loose	current	constrained	case	consistently	opens	lines	2‐5,	3‐4,	and	4‐7,	but	

differs	on	the	4th	and	5th	lines	opened	ሺand	a	5th	line	is	only	opened	in	one	caseሻ.		

The	linear	solution	was	very	close	to	the	nonlinear	solution	for	the	corresponding	
Page

50

network.		The	total	CPU	time	for	all	the	linear	cases	up	to	5	lines	open	is	close	to	the	

CPU	time	of	using	one	MIP.		

Table	A.9.		Progressive	MIP:		14	Bus,	Tight	Current	Constraint	

Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16 32	

൏ൌ	Number	of	Lines	

Open	 		

Obj	Value	 ൏ൌ1	 95.05 95.07 95.06	 95.04	

		 ൏ൌ2	 95.04 95.06 95.06	 94.98	

		 ൏ൌ3	 95.33 95.07 95.41	 94.99	

		 ൏ൌ4	 95.14 95.06 95.41	 95.00	

		 ൏ൌ5	 95.19 INF 95.42	 95.04	

NLP	Obj	Value	 ൏ൌ1	 95.06 95.04 95.05	 95.07	

	ሺIPOPTሻ	 ൏ൌ2	 95.06 95.00 95.03	 95.06	

		 ൏ൌ3	 95.41 95.05 95.40	 95.06	

		 ൏ൌ4	 95.41 94.99 95.28	 95.06	

		 ൏ൌ5	 95.43 94.95 95.26	 95.06	

CPU	Time	 ൏ൌ1	 1.3 1.9 1.0	 1.9	

		 ൏ൌ2	 1.1 1.6 0.7	 1.4	

		 ൏ൌ3	 1.7 2.6 1.5	 2.9	

		 ൏ൌ4	 1.4 4.1 1.7	 3.7	

		 ൏ൌ5	 1.7 5.0 1.4	 4.3	

NLP	CPU	Time	 ൏ൌ1	 1.1 1.1 0.9	 3.2	

	ሺIPOPTሻ	 ൏ൌ2	 1.3 1.3 0.9	 1.0	

		 ൏ൌ3	 2.2 1.3 2.4	 1.2	

		 ൏ൌ4	 4.5 1.6 2.1	 2.1	

		 ൏ൌ5	 5.7 2.0 6.7	 1.4	

Linear	

Iterations	 ൏ൌ1	 4 4 4	 4	

		 ൏ൌ2	 4 3 4	 3	

		 ൏ൌ3	 2 4 3	 3	

		 ൏ൌ4	 3 3 4	 2	

		 ൏ൌ5	 2 3 2	 4	

MIP	Gap	 ൏ൌ1	 1.5E‐15 2.7E‐15 1.5E‐15	 2.7E‐15	

		 ൏ൌ2	 2.8E‐04 9.5E‐15 1.0E‐14	 8.4E‐15	

		 ൏ൌ3 4.4E‐15 1.3E‐05 3.9E‐15	 2.8E‐14	

		 ൏ൌ4	 1.3E‐14 9.5E‐04 4.2E‐04	 9.6E‐04	

		 ൏ൌ5	 2.2E‐14 8.5E‐04 1.4E‐14	 3.9E‐14	

New	Opened	

Line		

൏ൌ1 4‐7 4‐7 4‐7	 4‐7	

൏ൌ2	 None None None	 None	

		 ൏ൌ3 2‐5 None 2‐5	 None	

		 ൏ൌ4	 None None None	 None	

		 ൏ൌ5	 None None None	 None	
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51

Table	A.10.		Progressive	MIP:		14	Bus,	Loose	Current	Constraint

	 Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16 32	

൏ൌ	Number	of	Lines	

Open	 		

Obj	Value	 ൏ൌ1	 85.17 85.17 85.28	 85.61	

		 ൏ൌ2	 85.03 84.97 85.11	 85.20	

		 ൏ൌ3 85.04 85.08 85.03	 85.09	

		 ൏ൌ4	 84.98 85.24 85.08	 85.36	

		 ൏ൌ5	 85.01 85.21 85.09	 85.36	

NLP	Obj	Value	 ൏ൌ1	 85.61 85.61 85.61	 85.29	

	ሺIPOPTሻ	 ൏ൌ2	 85.25 85.20 85.20	 85.01	

		 ൏ൌ3 85.10 85.10 85.09	 85.10	

		 ൏ൌ4	 85.09 85.36 85.10	 85.20	

		 ൏ൌ5	 85.02 85.36 85.10	 85.23	

CPU	Time	 ൏ൌ1	 0.9 1.3 0.9	 1.2	

		 ൏ൌ2	 0.8 1.3 2.1	 1.3	

		 ൏ൌ3	 0.8 1.0 2.0	 1.0	

		 ൏ൌ4	 0.6 0.8 2.1	 0.8	

		 ൏ൌ5	 1.2 0.7 2.6	 0.8	

NLP	CPU	Time	 ൏ൌ1	 1.0 1.0 0.8	 1.2	

	ሺIPOPTሻ	 ൏ൌ2	 1.0 1.9 1.2	 1.8	

		 ൏ൌ3	 3.4 4.1 0.7	 1.7	

		 ൏ൌ4	 1.8 3.5 0.7	 2.6	

		 ൏ൌ5	 3.2 4.8 0.6	 1.4	

Linear	

Iterations	 ൏ൌ1	 6 6 4	 4	

		 ൏ൌ2	 4 4 5	 4	

		 ൏ൌ3	 3 3 3	 3	

		 ൏ൌ4	 2 2 4	 3	

		 ൏ൌ5	 3 3 3	 2	

MIP	Gap	 ൏ൌ1	 3.4E‐15 8.5E‐16 3.4E‐15	 8.5E‐16	

		 ൏ൌ2	 3.0E‐14 3.1E‐15 2.8E‐14	 8.6E‐16	

		 ൏ൌ3	 1.4E‐15 1.0E‐15 1.5E‐15	 3.6E‐15	

		 ൏ൌ4	 4.6E‐15 4.8E‐15 2.9E‐15	 1.9E‐14	

		 ൏ൌ5	 1.4E‐15 3.7E‐14 4.5E‐15	 1.1E‐14	

New	Opened	

Line	 ൏ൌ1	 2‐5 2‐5 2‐5	 2‐5	

		 ൏ൌ2	 3‐4 3‐4 3‐4	 3‐4	

		 ൏ൌ3	 4‐7 4‐7 4‐7	 4‐7	

		 ൏ൌ4	 None 9‐14 12‐23	 9‐14	

		 ൏ൌ5	 4‐5,	4‐9 None None	 None	

	
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52

30	bus.	

In	the	30	bus	case	with	tight	current	constraints	ሺA.11ሻ,	the	linear	objective	value	is	

within	3%	of	the	objective	value	except	for	two	cases	–	one	where	they	differ	by	

12%	and	one	where	the	solution	is	infeasible.		With	loose	current	constraints	ሺA.12ሻ,	

the	linear	objective	value	is	within	2%	of	the	nonlinear	objective	value.		The	

repeated	and	one	MIP	approaches	share	some	of	the	same	lines	as	opened	by	the	

progressive	MIP	approach,	but	the	lines	opened	are	not	exactly	the	same	between	

the	approaches	or	within	the	different	step	size	functions	and	number	of	cuts	within	

the	same	approach.		In	the	30	bus	problem,	only	one	solution	is	found	to	be	

infeasible	–	when	lines	6‐28,	10‐21,	and	9‐11	are	opened	in	the	tightly	constrained	

case	with	quadratic	step	size,	32	cuts,	and	up	to	4	lines	open.

	

Table	A.11.		Progressive	MIP:		30	Bus,	Tight	Current	Constraint

Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16 32	

൏ൌ	Number	of	Lines	

Open	 		

Obj	Value	 ൏ൌ1	 5.92 5.93 5.92	 5.93	

		 ൏ൌ2	 5.88 5.89 5.93	 5.89	

൏ൌ3 5.89 5.90 5.94 5.89	

		 ൏ൌ4	 5.90 5.90 5.92	 5.94	

		 ൏ൌ5	 5.89 5.90 5.93	 5.93	

NLP	Obj	Value	 ൏ൌ1	 5.93 6.10 5.97	 6.76	

	ሺIPOPTሻ	 ൏ൌ2	 5.94 5.94 5.97	 5.94	

		 ൏ൌ3 5.94 5.96 5.94	 5.94	

		 ൏ൌ4	 5.95 5.95 5.95	 INF	

		 ൏ൌ5	 5.94 5.98 5.95	 6.07	

CPU	Time	 ൏ൌ1	 6.4 8.8 6.7	 9.1	

		 ൏ൌ2	 5.4 7.7 3.6	 6.5	

൏ൌ3 4.7 6.9 3.8	 5.2	

		 ൏ൌ4	 4.6 5.5 3.3	 19.1	

		 ൏ൌ5	 3.7 5.3 3.2	 1.5	

NLP	CPU	Time	 ൏ൌ1	 3.7 2.2 2.4	 1.9	

	ሺIPOPTሻ	 ൏ൌ2	 11.8 8.2 7.2	 6.0	

		 ൏ൌ3 7.5 5.6 7.3 6.8	

		 ൏ൌ4	 4.8 4.6 6.6	 2.1	

		 ൏ൌ5	 6.6 4.9 7.4	 1.8	

LinearIterations	 ൏ൌ1	 4 3 4	 3	

		 ൏ൌ2	 2 4 2	 4	

		 ൏ൌ3 4 2 4 2	

		 ൏ൌ4	 2 2 3	 9	
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53

		 ൏ൌ5	 2 2 3	 2	

MIP	Gap	 ൏ൌ1	 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

		 ൏ൌ2	 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

		 ൏ൌ3 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

		 ൏ൌ4	 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

		 ൏ൌ5	 1.8E‐07 1.7E‐07 1.7E‐07	 7.6E‐09	

New	Opened	

Line	 ൏ൌ1 6‐28 6‐28 6‐28	 6‐28	

		 ൏ൌ2	 10‐21 10‐21 10‐17	 10‐21	

		 ൏ൌ3 14‐15 15‐18 25‐27	 None	

		 ൏ൌ4	 15‐18 14‐15 18‐19	 9‐11	

		 ൏ൌ5	 12‐15 None 12‐14	 15‐18	

	

Table	A.12.		Progressive	MIP:		30	Bus,	Loose	Current	Constraint

Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16 32	

൏ൌ	Number	of	Lines	

Open	 		

Obj	Value	 ൏ൌ1	 5.92 5.92 5.92	 5.92	

		 ൏ൌ2	 5.92 5.92 5.93	 5.92	

൏ൌ3 5.90 5.89 5.89 5.89	

		 ൏ൌ4	 5.90 5.90 5.90	 5.90	

		 ൏ൌ5	 5.89 5.90 5.90	 5.91	

NLP	Obj	Value ൏ൌ1 5.94 5.92 5.93	 6.04	

	ሺIPOPTሻ	 ൏ൌ2	 5.93 5.92 5.94	 5.92	

		 ൏ൌ3 5.94 5.94 5.96 5.94	

		 ൏ൌ4	 5.96 5.95 5.95	 5.97	

൏ൌ5 5.98 5.95 5.95	 5.98	

CPU	Time	 ൏ൌ1	 4.6 5.5 4.5	 6.2	

		 ൏ൌ2	 5.5 7.7 3.5	 6.4	

൏ൌ3 4.8 6.0 4.5 6.2	

		 ൏ൌ4	 5.1 5.7 3.7	 5.5	

		 ൏ൌ5	 5.9 4.8 3.9	 4.5	

NLP	CPU	Time	 ൏ൌ1	 6.8 6.6 8.5	 2.8	

	ሺIPOPTሻ	 ൏ൌ2	 3.7 3.8 8.9	 4.0	

		 ൏ൌ3 18.5 3.2 8.5 2.8	

		 ൏ൌ4 3.5 6.5 5.3 11.7	

൏ൌ5 5.7 7.6 8.1	 3.6	

LinearIterations	 ൏ൌ1 4 2 4 2	

		 ൏ൌ2	 2 2 2	 2	

		 ൏ൌ3 4 2 4 2	

		 ൏ൌ4	 10 2 4	 2	

		 ൏ൌ5 3 2 4 2	
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54

MIP	Gap	 ൏ൌ1	 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

		 ൏ൌ2	 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

		 ൏ൌ3 1.7E‐07 1.7E‐07 1.7E‐07 1.7E‐07	

൏ൌ4 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

		 ൏ൌ5	 1.7E‐07 1.7E‐07 1.7E‐07	 1.7E‐07	

New	Opened	

Line ൏ൌ1 6‐28 24‐25 6‐28 24‐25	

		 ൏ൌ2	 25‐27 6‐28 19‐20	 6‐28	

		 ൏ൌ3 10‐21 10‐21 10‐21 10‐21	

		 ൏ൌ4	 10‐20 14‐15	 10‐20	

		 ൏ൌ5	

27‐30,			

18‐19 		 12‐14	

	

57	bus.	

The	57	bus	problem	ሺA.13	and	A.14ሻ	exhibits	some	peculiarities.		There	are	many	

cases	where	the	linear	iterative	solver	finds	what	it	believes	to	be	a	feasible	solution,	

but	the	nonlinear	solver	reveals	that	the	configuration	is	not	feasible	ሺusually	due	to	

violating	the	minimum	voltage	constraintሻ.		Out	of	the	20	different	tests	performed	

for	the	tight	current	case,	only	8	find	a	feasible	solution.	However,	this	approach	is	

much	faster	than	even	the	one	MIP	approach	and	does	not	perform	any	worse	than	

the	one	MIP	approach.		The	tightly	constrained	problem	appears	to	be	very	sensitive	

to	what	approach	is	used.		The	first	three	problems,	where	up	to	1,	then	2,	then	3	

lines	are	allowed	open	all	have	the	same	network	configuration;	however,	the	

nonlinear	solver	finds	some	solutions	that	are	feasible	and	some	that	are	infeasible	

for	the	2	and	3	lines	open	cases.		This	is	likely	due	to	different	starting	points	being	

used	in	the	nonlinear	solver.		In	addition,	it	is	surprising	that	some	of	the	

approaches	recommend	opening	additional	lines	even	when	it	appears	to	increase	

the	total	cost	ሺsee	the	linear	step	size	function	with	32	cuts;	each	additional	line	

open	increases	the	cost,	although	the	solver	has	the	option	not	to	open	any	

additional	lines0.	

In	the	loose	current	case,	only	14	of	the	20	tests	find	a	feasible	solution.		The	loosely	

constrained	case	also	appears	to	be	sensitive	to	what	step	size	and	number	of	cuts	

approach	is	used.		It	also	opens	the	same	three	lines	in	all	approaches,	but	for	up	to	

2	or	3	lines	open,	the	nonlinear	solver	can	find	a	feasible	solution	with	the	given	

configuration	in	some	approaches	and	not	in	others,	likely	due	to	the	starting	point	

used.		In	addition,	the	linear	approach	also	opens	lines	in	some	cases	when	it	

increases	the	total	cost	ሺsee	the	quadratic	step	size	function	with	16	cuts,	for	

exampleሻ.	

	
Page

55

Table	A.13.		Progressive	MIP:		57	Bus,	Tight	Current	Constraint

Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16	 32

൏ൌ	Number	of	Lines	

Open	 		

Obj	Value	 ൏ൌ1	 423.10 421.42 423.95	 424.04

		 ൏ൌ2	 423.18 422.85 422.79	 422.84

൏ൌ3 422.14 424.44 423.66	 424.45

		 ൏ൌ4	 427.76 424.84 425.20	 423.44

		 ൏ൌ5	 INF 428.56 426.15	 427.96

NLP	Obj	Value	 ൏ൌ1	 429.56 429.56 429.56	 429.56

	ሺIPOPTሻ	 ൏ൌ2	 INF INF 430.28	 430.30

		 ൏ൌ3 427.43 INF INF INF

		 ൏ൌ4	 INF INF INF	 426.17

൏ൌ5	 INF INF 427.62	 INF

CPU	Time	 ൏ൌ1	 37.1 60.1 35.6	 52.4

		 ൏ൌ2	 20.5 36.5 9.4	 12.7

൏ൌ3 21.7 38.2 14.8	 28.3

		 ൏ൌ4	 32.3 15.8 12.1	 17.5

		 ൏ൌ5	 26.4 31.6 8.4	 28.3

NLP	CPU	Time	 ൏ൌ1	 29.4 28.2 15.9	 27.4

	ሺIPOPTሻ	 ൏ൌ2	 86.9 26.6 33.3	 25.5

		 ൏ൌ3 49.3 248.3 40.2 51.1

		 ൏ൌ4 7.0 32.6 12.7	 48.7

൏ൌ5	 7.3 62.4 32.1	 9.8

Linear	

Iterations	 ൏ൌ1 4 7 4	 4

		 ൏ൌ2	 4 10 3	 3

		 ൏ൌ3 4 20 4 5

		 ൏ൌ4	 20 3 5	 3

		 ൏ൌ5 20 3 3	 20

MIP	Gap	 ൏ൌ1	 2.5E‐15 5.1E‐14 2.5E‐15	 5.1E‐14

		 ൏ൌ2	 1.5E‐14 1.4E‐14 7.3E‐04	 1.0E‐03

		 ൏ൌ3 2.0E‐14 1.9E‐15 9.6E‐16	 1.9E‐15

൏ൌ4	 9.6E‐16 3.3E‐15 3.6E‐15	 4.7E‐04

		 ൏ൌ5	 3.4E‐13 1.9E‐15 7.9E‐04	 6.8E‐16

New	Opened	

Line ൏ൌ1 3‐4 3‐4 3‐4	 3‐4

		 ൏ൌ2	 54‐55 54‐55 53‐54	 53‐54

		 ൏ൌ3 22‐38 22‐38 22‐38	 22‐38

		 ൏ൌ4	 32‐34 1‐15 1‐15	 14‐15

		 ൏ൌ5	 20‐21 3‐15 12‐13	 13‐14
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56

Table	A.14.		Progressive	MIP:		57	Bus,	Loose	Current	Constraint

Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16	 32

൏ൌ	Number	of	Lines	

Open	 		

Obj	Value	 ൏ൌ1	 421.36 422.93 423.09	 423.27

		 ൏ൌ2	 422.19 422.92 423.42	 422.75

൏ൌ3 421.99 421.78 423.61	 423.34

		 ൏ൌ4	 428.17 422.95 424.81	 424.44

		 ൏ൌ5	 INF 426.41 425.79	 424.63

NLP	Obj	Value	 ൏ൌ1	 425.14 425.14 425.14	 425.14

	ሺIPOPTሻ	 ൏ൌ2	 426.43 426.43 426.44	 425.10

		 ൏ൌ3 426.21 INF 426.51 INF

		 ൏ൌ4	 INF 432.88 INF	 426.55

		 ൏ൌ5	 INF INF 428.93	 430.18

CPU	Time	 ൏ൌ1	 33.1 48.0 31.3	 48.7

		 ൏ൌ2	 29.2 26.8 8.6	 14.1

൏ൌ3 18.2 37.1 13.3	 23.5

		 ൏ൌ4	 31.5 32.1 10.8	 17.2

		 ൏ൌ5	 26.1 38.3 7.8	 19.1

NLP	CPU	Time	 ൏ൌ1	 26.8 19.9 21.6	 27.5

	ሺIPOPTሻ	 ൏ൌ2	 53.0 22.2 53.1	 29.3

		 ൏ൌ3 28.8 19.6 39.5 79.5

		 ൏ൌ4	 19.0 47.9 10.6	 27.0

		 ൏ൌ5	 11.8 17.9 70.3	 59.3

Linear	

Iterations	 ൏ൌ1	 4 4 4	 4

		 ൏ൌ2	 10 3 4	 3

		 ൏ൌ3 6 10 4 4

		 ൏ൌ4	 20 11 5	 3

		 ൏ൌ5	 20 20 3	 3

MIP	Gap	 ൏ൌ1	 9.4E‐15 4.9E‐13 9.4E‐15	 4.9E‐13

		 ൏ൌ2	 1.4E‐14 6.0E‐15 7.4E‐04	 7.4E‐04

		 ൏ൌ3 2.7E‐16 5.7E‐15 5.1E‐15	 3.6E‐15

		 ൏ൌ4	 1.8E‐15 1.8E‐15 0.0E൅00	 2.7E‐15

		 ൏ൌ5	 1.4E‐12 9.5E‐04 8.1E‐04	 4.1E‐16

New	Opened	

Line	 ൏ൌ1 3‐4 3‐4 3‐4	 3‐4

		 ൏ൌ2	 54‐55 54‐55 54‐55	 54‐55

		 ൏ൌ3 22‐38 22‐38 22‐38	 22‐38

		 ൏ൌ4	 32‐34 14‐15 1‐15	 1‐15

		 ൏ൌ5	 21‐22 13‐14 12‐13	 9‐11

	

	 	
Page

57

118	bus.	

In	the	tightly	constrained	problem	ሺA.15ሻ,	the	linear	objective	value	is	within	1%	or	

less	of	the	nonlinear	objective	value	except	for	one	case	ሺup	to	2	lines	open,	32	cuts,	

quadratic	step	sizeሻ.		The	first	line	opened	is	the	same	in	all	cases.		The	2nd,	3rd,	and	

4th	lines	opened	are	consistent	within	the	same	step‐size	approach.		However,	the	

nonlinear	program	cannot	find	a	feasible	solution	in	the	2	lines	open,	32	cuts,	

quadratic	step	size	case	while	it	can	find	a	feasible	solution	in	the	2	lines	open,	16	

cuts,	quadratic	step	size	case	with	the	same	network	configuration;	this	difference	is	

likely	due	to	the	difference	in	starting	points.	

In	the	loosely	constrained	problem	ሺA.16ሻ,	the	difference	between	the	objective	

value	the	linear	iterative	program	found	and	the	nonlinear	program	found	is	less	

than	1%,	and	there	were	not	any	infeasible	solutions.		The	first	line	opened	is	

consistent	in	all	approaches,	while	the	other	lines	that	are	opened	differ	between	

approaches.	

Table	A.15.		Progressive	MIP:		57	Bus,	Tight	Current	Constraint

Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16	 32	

൏ൌ	Number	of	Lines	

Open 		

Obj	Value	 ൏ൌ1	 1379.12 1378.16 1385.68	 1385.79

		 ൏ൌ2	 1377.59 1378.33 1385.31	 1384.70

൏ൌ3 1377.37 1376.77 1381.49	 1383.92

		 ൏ൌ4	 1376.41 1376.80 1382.34	 1382.95

		 ൏ൌ5	 1377.49 1377.63 1381.14	 1382.52

NLP	Obj	Value ൏ൌ1	 1385.82 1385.82 1385.82	 1385.97

	ሺIPOPTሻ	 ൏ൌ2	 1384.13 1384.13 1385.63	 INF

		 ൏ൌ3 1383.26 1383.27 1384.40 1383.95

		 ൏ൌ4	 1383.05 1383.05 1383.15	 1383.63

൏ൌ5	 1383.35 1383.37 1382.15	 1382.82

CPU	Time	 ൏ൌ1	 182.7 279.3 186.3	 339.5

		 ൏ൌ2	 108.0 181.2 26.6	 46.0

൏ൌ3 101.3 183.9 82.9	 125.4

		 ൏ൌ4	 106.6 161.6 65.9	 40.8

		 ൏ൌ5	 93.2 130.5 61.0	 100.5

NLP	CPU	Time	 ൏ൌ1	 174.1 8.0 101.4	 54.6

	ሺIPOPTሻ	 ൏ൌ2	 213.6 11.5 148.8	 235.9

		 ൏ൌ3 142.4 9.7 107.0 167.4

		 ൏ൌ4 127.1 10.4 139.6	 119.3

൏ൌ5	 125.5 8.0 178.0	 125.5
Page

58

Linear	Iterations	 ൏ൌ1 3 3 9	 15

		 ൏ൌ2	 3 3 2	 4

		 ൏ൌ3 3 3 3 15

		 ൏ൌ4	 3 3 5	 4

൏ൌ5 3 3 6	 6

MIP	Gap	 ൏ൌ1	 7.2E‐15 3.0E‐14 7.2E‐15	 3.0E‐14

		 ൏ൌ2	 9.9E‐04 7.0E‐04 8.4E‐04	 1.0E‐03

		 ൏ൌ3 7.4E‐04 9.2E‐04 3.4E‐04	 7.5E‐04

൏ൌ4	 5.9E‐04 4.5E‐04 9.3E‐04	 8.2E‐04

		 ൏ൌ5	 8.8E‐04 9.1E‐05 9.8E‐04	 8.8E‐04

New	Opened	Line ൏ൌ1 77‐82 77‐82 77‐82	 77‐82

		 ൏ൌ2	 82‐96 82‐96 60‐62	 60‐62

		 ൏ൌ3 47‐49 47‐49 77‐82	 77‐82

		 ൏ൌ4	 48‐49 48‐49 82‐96	 82‐96

		 ൏ൌ5	 45‐49 45‐49 61‐62	 93‐94

Table	A.16.		Progressive	MIP:		118	Bus,	Loose	Current	Constraint	

Step	Size	Function Linear Quadratic	

Number	of	Cuts 16 32 16 32	

൏ൌ	Number	of	Lines	

Open 		

Obj	Value	 ൏ൌ1	 1310.10 1309.91 1315.61	 1313.40

		 ൏ൌ2	 1309.08 1310.11 1315.14	 1315.23

൏ൌ3 1308.54 1311.02 1316.49 1312.51

		 ൏ൌ4	 1310.71 1308.90 1312.58	 1318.96

		 ൏ൌ5	 1306.34 1307.86 1314.62	 1315.33

NLP	Obj	Value ൏ൌ1 1316.32 1316.32 1316.42	 1316.38

	ሺIPOPTሻ	 ൏ൌ2	 1317.05 1317.11 1321.49	 1317.02

		 ൏ൌ3 1317.17 1317.32 1317.02 1316.68

		 ൏ൌ4	 1317.06 1317.43 1317.26	 1320.89

൏ൌ5 1317.29 1317.29 1325.20	 1322.33

CPU	Time	 ൏ൌ1	 470 1597 437	 1506

		 ൏ൌ2	 192 303 42	 74

൏ൌ3 173 247 54 50

		 ൏ൌ4	 201 250 34	 336

		 ൏ൌ5	 123 245 99	 88

NLP	CPU	Time	 ൏ൌ1	 168 120 91	 136

	ሺIPOPTሻ	 ൏ൌ2	 130 120 179	 155

		 ൏ൌ3 115 119 149 254

		 ൏ൌ4 232 177 123 101

൏ൌ5 200 117 171	 173

Linear	Iterations	 ൏ൌ1 11 11 7 5
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59

		 ൏ൌ2	 11 13 6	 6

		 ൏ൌ3 10 11 8 2

		 ൏ൌ4	 11 10 4	 5

		 ൏ൌ5 2 9 5 2

MIP	Gap	 ൏ൌ1	 3.9E‐14 4.0E‐14 3.9E‐14	 4.0E‐14

		 ൏ൌ2	 9.7E‐04 2.2E‐11 1.0E‐03	 7.3E‐04

		 ൏ൌ3 2.2E‐11 2.2E‐11 9.8E‐04 8.5E‐04

൏ൌ4 4.3E‐09 3.3E‐04 9.6E‐04	 4.4E‐04

		 ൏ൌ5	 2.2E‐11 7.8E‐04 8.4E‐04	 4.0E‐04

New	Opened	Line ൏ൌ1 80‐81 80‐81 80‐81 80‐81

		 ൏ൌ2	 69‐77 69‐77 8‐30	 61‐64

		 ൏ൌ3 61‐64 23‐24 94‐100 23‐24

		 ൏ൌ4	 76‐118 61‐64 47‐49	 59‐63

		 ൏ൌ5	 75‐77 76‐118 65‐68	 94‐100