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December	2012

Optimal	Power	Flow	Paper	2	

Staff	paper	by

Richard	P.	O’Neill

Anya	Castillo

Mary	B.	Cain
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The	IV	Formulation	and	Linear	Approximations	

of	the	AC	Optimal	Power	Flow	Problem		

Optimal	Power	Flow	Paper	2	

Richard	P	O’Neill,	Anya	Castillo	and	Mary	Cain		

richard.oneill@ferc.gov;	anya.castillo@ferc.gov;	mary.cain@ferc.gov	

December		2012			

The	authors	thank	Ross	Baldick	for	his	comments	and	contributions	on	an	earlier	draft

Abstract	and	Executive	Summary	

Understanding	AC	Optimal	Power	Flow	ሺACOPFሻ	is	important	because	a	one	percent	

improvement	in	a	power	system	dispatch	can	save	roughly	tens	of	billion	dollars	

annually.	In	this	paper,	we	formulate	the	ACOPF	in	several	ways,	compare	each	

formulation’s	properties,	and	argue	that	the	current‐voltage	ሺIVሻ	formulation	and	its	

linear	approximations	may	be	easier	to	solve	than	the	traditional	quadratic	power	

flow	formulations.		Unlike	the	DC	model	that	holds	voltage	constant,	ignores	reactive	

power	and	assumes	small	voltage	angle	differences,	the	IV	formulation	solves	a	

linear	system	of	equations	without	decomposition,	unnecessary	constraints	or	

omissions.		The	nonconvex	constraints	need	careful	consideration.	For	problems	

that	are	solved	repetitively	with	minor	variations,	there	is	considerable	potential	for	

individual	parameter	tuning	and	preprocessed	constraints.	For	example,	constraints	

on	angles	and	past	iteratively	added	constraints	may	be	able	to	help.		Initial	results	

indicate	that	the	IV	formulation	and	its	linear	approximations	have	promise	to	meet

practical	computational	requirements.	In	addition,	steady‐state	quadrature	

constraints	on	generators	are	linear	in	the	IV	formulation	and	can	be	included	in	the	

formulation.	Additional	formulation	testing	and	computational	testing	are	needed	to	

determine	commercial	feasibility.	If	a	linear	IV	approximation	to	the	ACOPF	proves	

promising,	it	can	be	embedded	in	the	unit	commitment	models,	optimal	topology	

models	and	other	formulations	that	use	binary	variables.	This	allows	the	use	of	

mixed	integer	linear	programs	ሺMIPሻ	algorithms	that	are	exceptionally	fast	and	

robust	to	better	model	the	power	markets.

	

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	 	 	 	 	 	 	 	 	 	 	4	

3. AC	Flow	Equations	 	 	 	 	 	 	 	 	8	

4. Generator,	Load,	Transmission,	Voltage,	and	Angle	Constraints	 	 	9	

5. ACOPF	Formulation	 	 	 	 	 	 	 	 10	

6. The	Linear	Approximations	to	the	IV	Formulation	 	 	 	 12	

7. Conclusions	 	 	 	 	 	 	 	 	 17	

References		 	 	 	 	 	 	 	 	 	 18	

	 	
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1.		Introduction	

	 The	Alternating	Current	Optimal	Power	Flow	ሺACOPFሻ	problem	is	

important—a	one	percent	improvement	in	dispatch	derived	from	better	solutions	to	

this	problem	can	save	roughly	1	to	5	billion	dollars	per	year	in	the	US,	or	4	to	20	

billion	dollars	per	year	in	the	world	ሺsee	Cain	2012ሻ.	In	this	paper,	we	formulate	the	

ACOPF	in	several	ways,	compare	each	formulation’s	properties,	and	argue	that	the	

current‐voltage	ሺIVሻ	formulation	and	its	linear	approximations	may	be	easier	to	

solve	than	the	traditional	quadratic	power	flow	formulation.		

	 We	focus	on	the	linear	approximation	because	the	linear	program	solvers	are	

robust	and	fast	and	the	power	systems	community	is	already	familiar	with	linear	

programs	for	economic	dispatch.	The	linear	program	approximation	constructs	

linear	relaxations	of	convex	constraints	and	linear	first	order	approximations	to	the	

non‐convex	constraints.	If	a	linear	IV	approximation	to	the	ACOPF	proves	promising,	

it	can	be	embedded	in	the	unit	commitment	models,	optimal	topology	models	and	

other	formulations	that	use	binary	variables.	This	allows	the	use	of	mixed	integer	

program	ሺMIPሻ	algorithms	that	are	exceptionally	fast	and	robust	to	better	model	the	

power	markets.	Since	both	MIP	algorithms	and	computer	hardware	continue	to	

improve,	today’s	proof‐of‐concept	software	may	be	tomorrow’s	commercial	

standard.	Power	system	optimization	has	evolved	with	improvements	in	computer	

hardware	and	optimization	software.		In	companion	papers,	we	present	more	detail	

on	the	formulations	and	on	several	test	problems.	We	also	examine	some	of	the	

arguments	presented	here	with	success.	

	 The	optimal	power	flow	ሺOPFሻ	problem	finds	the	optimal	solution	to	an	

objective	function	subject	to	the	power	flow	constraints.	There	are	a	variety	of	OPF	

formulations	with	different	constraints,	different	objective	functions,	and	different	

solution	methods	that	have	been	labeled	optimal	power	flow.		The	simplest	optimal	

power	flow	model	is	known	as	the	“Direct	Current	ሺDCሻ	OPF”.	It	uses	a	linearized	

approximation	of	the	AC	power	flow	equations	and	linear	constraints.		Formulations	

that	use	the	AC	power	flow	equations	are	known	as	“AC	OPF.”		The	ACOPF	

formulations	are	continuous	nonconvex	optimization	problems	without	binary	

variables.	For	more	detail,	see	Cain	et	al	ሺ2012ሻ.	Most	nonlinear	solvers	only	find	

local	optimal	solutions	for	nonconvex	problems.			

2.		Notation	

	 When	n	and	m	are	subscripts,	they	index	buses;	k	indexes	each	three‐phase	

transmission	element	between	buses	n	and	m.	When	j	is	not	a	superscript,	j	ൌ	ሺ‐

1ሻ1/2;	the	complex	current	is	i.	When	j	is	a	superscript,	it	is	the	‘imaginary’	part	of	a	

complex	number,	while	an	r	superscript	is	the	‘real’	part.	Unless	specified	otherwise,	

scalars	and	complex	numbers	are	lower	case.	Generally,	in	ሺxy zሻw,	x	is	a	variable	or	
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parameter,	y	further	defines	the	variable	or	parameter,	z	is	a	member	of	an	index	

set,	w	is	an	exponent.		

	 Unless	specified	otherwise,	matrices	are	upper	case.	All	vectors	are	column	

vectors	and	T	superscript	represents	a	transpose	operation.		The	element‐by‐

element	or	Hadamard	product	is	•.	If	diagሺAሻ	is	a	square	matrix	with	A	on	the	

diagonal	and	zeroes	elsewhere,	diagሺAሻdiagሺBሻ	ൌ	diagሺA•Bሻ.	The	complex	conjugate	

operator	is	*	ሺsuperscriptሻ	and	*	ሺno	superscriptሻ	is	an	optimal	solution.	

	 We	assume	balanced	three‐phase	steady‐state	conditions.		All	variables	are	

associated	with	a	single‐line	model	of	a	balanced	three‐phase	system.		A	common	

practice	in	power	system	modeling	is	the	per‐unit	ሺp.u.ሻ	representation,	where	base	

quantities	for	voltage,	current,	power,	and	impedance	ሺor	admittanceሻ	may	fully	

define	a	power	network.	Such	normalization	is	a	convenience.	

	 The	topology	of	the	network	consists	of	locations	known	as	buses	or	nodes	

and	transmission	elements	connecting	paired	buses.	The	network	is	an	undirected	

graph	with	weighted	edges	or	lines.		

Indices	and	Sets	

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

k	is	a	three‐phase	transmission	element	index.	Each	transmission	element	k	has	a	

pair	of	terminal	buses	n	and	m.		k	ε	ሼ1,	...,	Kሽ	where	k	is	the	number	of	transmission	

elements		

K’	is	the	number	of	a	connected	bus	pairs	ሺK’	൑	Kሻ.	

Unless	otherwise	stated,	summations	ሺ∑ሻ	are	over	the	full	set	of	indices.		

Variables	

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

qn	is	the	reactive	power	injection	or	withdrawal	at	bus	n	

sn	ൌ	pn	൅	jqn	is	the	net	complex	power	injection	at	bus	n

pnmk	is	the	real	power	flow	at	bus	n	to	bus	m	on	transmission	element	k

qnmk	is	the	reactive	power	flow	at	bus	n	to	bus	m	on	transmission	element	k

θn	is	the	voltage	phase	angle	at	bus	n	

θnm	ൌ	θn	‐	θm	is	the	voltage	phase	angle	differences	from	bus	n	to	bus	m

in	is	the	current	ሺcomplex	phasorሻ	injection	ሺpositiveሻ	or	withdrawal	ሺnegativeሻ	at	

bus	n		 where		in	ൌ	ir n	൅	ji

j

n

inmk	is	the	current	ሺcomplex	phasorሻ	flow	into	transmission	element	k	at	bus	n	ሺto	

bus	mሻ;	inmk	ൌ	ir nmk	൅	ji

j

nmk

snmk	is	the	apparent	complex	power	flow	into	bus	n	on	transmission	element	k	ሺto	

bus	mሻ;	snmk	ൌ	sr nmk	൅	jsj

nmk

vn is	the	complex	voltage	at	bus	n.	vn	ൌ	vr n	൅	jvj

n

ynmk	is	the	complex	admittance	on	transmission	element	k	connecting	bus	n	and	bus

m	ሺIf	buses	n	and	m	are	not	connected	directly,	ynmkൌ	0.ሻ;		

yn	is	the	self‐admittance	ሺto	groundሻ	at	bus	n.		
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ynm	is	the	complex	admittance	connecting	bus	n	and	bus	m	for	all	transmission	

elements	k	between	buses	n	and	m.	

V	ൌ	ሺv1,	...,	vNሻT	is	the	complex	vector	of	bus	voltages;	V	ൌ	Vr 	൅	jVj

I	ൌ	ሺi1,	...,	iNሻT	is	the	complex	vector	of	bus	current	injections;	I	ൌ	Ir 	൅	jI

j

P	ൌ	ሺp1,	...,	pNሻT	is	the	vector	of	real	power	injections	

Q	ൌ	ሺq1,	...,	qNሻT	is	the	vector	of	reactive	power	injections	

G	is	the	N‐by‐N	conductance	matrix	

B	is	the	N‐by‐N	suseptance	matrix	

Note	that	elements	of	G	and	B	will	be	constant	for	passive	transmission	elements	

such	as	transmission	lines,	but	can	be	variable	when	active	transmission	elements	

such	as	phase	shifting	transformers,	switched	capacitors/reactors,	or	FACTs	devices	

are	included.	

Y	ൌ	G	൅	jB	is	the	complex	admittance	N‐by‐N	matrix	

Functions	and	Transformations	

Reሺ	ሻ	is	the	real	part	of	a	complex	number,	for	example,	Reሺi

r n	൅	jij

nሻ	ൌ	i

r n

Imሺ	ሻ	is	the	real	part	of	a	complex	number,	for	example,	Imሺi

r n	൅	jij

nሻ	ൌ	i

j

n

|	|	is	the	magnitude	of	a	complex	number,	for	example,	|vn|	ൌ	ሾሺvr nሻ2൅ሺvj

nሻ2ሿ1/2		

absሺ	ሻ	is	the	absolute	value	function.	

The	transformation	from	rectangular	to	polar	coordinates	for	voltage	is:	

vr n		ൌ	|vn|cosሺθnሻ	 	 	

vj

n		ൌ	|vn|sinሺθnሻ	

ሺvr nሻ2	൅	ሺvj

nሻ2	ൌ	ሾ|vn|sinሺθnሻሿ2	൅	ሾ|vn|cosሺθnሻሿ2	ൌ	|vn|2ሾsinሺθnሻ2	൅	cosሺθnሻ2ሿ	ൌ	|vn|2		

We	drop	the	bus	index	n	and	let	θ	be	the	voltage	angle	and	δ	be	the	current	angle.	

For	real	power,	p	ൌ	vr ir 	൅	vjij	ൌ	|v||i|cosሺθ‐δሻ	and	for	reactive	power,	q	ൌ	vjir 	‐	vr ij	ൌ	

|v||i|sinሺθ	‐	δሻ	where	θ	–	δ	is	the	power	angle.	

Parameters	

rnmk	is	the	resistance	of	transmission	element	k.		

xnmk	is	the	inductance	of	transmission	element	k.		

smax k	is	the	apparent	power	thermal	limit	on	transmission	element	k	at	both	

terminal		 buses.		

i

max k	is	the	current	limit	on	transmission	element	k	at	both	terminal	buses.		

θminnm,	θmax nm	are	the	maximum	and	minimum	voltage	angle	differences	between	n	

and	m	

pminn,	pmax n	are	the	maximum	and	minimum	real	power	for	generator	n	

qminn,	qmax n	are	the	maximum	and	minimum	reactive	power	for	generator	n	

C1	ൌ	ሺc11,	...,	c1NሻT	and	C2	ൌ	ሺc21,	...,	c2NሻT	are	vectors	of	linear	and	quadratic	objective	

function	cost	coefficients	respectively.		

inmk	ൌ	anmk2ynmkvn	‐	anmkynmkvm

imnk	ൌ	‐anmkynmkvn	൅	ynmkvm	 	

For	the	phase	shifting	transformer	ሺPARሻ	with	a	phase	angle	shift	of	φ,	
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vm/vn	ൌ	tnmk	ൌ	anmkejφ

inm/imn	ൌ	tnmk* 	ൌ	‐anmke‐jφ

The	current‐voltage	equations	for	the	phase	shifting	transformer	k	between	buses	n	

and	m	are:		

inmk	ൌ	anmk2ynmkvn	‐	tnmk*ynmkvm

imnk	ൌ	‐tnmkynmkvn	൅	ynmkvm

Admittance	Matrix.	If	there	are	no	transformers	or	flexible	AC		

transmission	system	ሺFACTSሻ	devices,	G	is	positive	semidefinite	and	B	is	negative	

semidefinite.	A	matrix	where	ynn	൒	absሺ∑m ynmሻ	is	called	diagonally	dominant	and	

strictly	diagonally	dominant	if	ynn	൐	absሺ∑m ynmሻ.		

	 If	there	are	no	transformers	and	yn	ൌ	0,	G	and	B	are	weighted	Laplacian	

matrices	of	the	undirected	weighted	graph	that	describes	the	transmission	network.	

Much	is	known	about	the	weighted	Laplacian	matrices.	Y	is	a	complex	weighted	

Laplacian	matrix.	If	absሺynሻ	൏൏	absሺ∑m ynmሻ	close	to	0,	we	will	call	Y	an	approximate	

complex	weighted	Laplacian.		

Y	ൌ	G൅jB.	G	and	B	are	real	symmetric	diagonally	dominant	matrices.	A	

symmetric	diagonally	dominant	matrix	has	a	symmetric	factorization,	for	example,	B	

ൌ	UUT	where	each	column	of	U	has	at	most	two	non‐zeros	and	the	non‐zeroes	have	

the	same	absolute	value.		

	 If	there	are	transformers	and	FACTS	devices,	let	ynmk	be	ynmk	,	anmk2ynmk,	

tnmk*ynmk,	or	‐tnmkynmkvn	as	appropriate	off‐diagonal	element,	then	ynn	ൌ	yn	൅	∑k,m

ynmk,	ynm	ൌ	∑k ynmk,	and	Y	is	the	matrix	ሾynmሿ.	If	there	are	only	ideal	in‐phase	

transformers,	the	Y	matrix	is	symmetric.	If	there	are	phase	shifting	transformers,	the	

symmetry	of	the	Y	matrix	is	lost.		

	 For	large	problems,	the	admittance	matrix,	Yൌ	G൅jB,	is	usually	sparse.	The	

density	of	both	G	and	B	is	ሺN൅2K’ሻ/N2	where	K’	is	the	number	of	off‐diagonal	non‐

zero	entries	ሺthe	aggregate	of	multiple	transmission	elements	between	adjacent	

bussesሻ	and	n	is	the	number	of	buses.	For	example,	in	a	topology	with	1000	buses	

and	1500	transmission	elements,	G	and	B	would	have	a	density	of	

ሺ1000൅3000ሻ/10002	ൌ	.004.	The	lowest	density	for	a	connected	network	is	the	

spanning	tree.	It	has	N‐1	transmission	elements	and	the	density	is	ሺN൅2ሺN‐1ሻሻ/N2.	

For	large	sparse	systems,	ሺN൅2ሺN‐1ሻሻ/N2	ൎ	3/N.		

	

	

	 	
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3.	AC	Flow	Equations		

Kirchhoff’s	current	law.	Kirchhoff’s	current	law	requires	that	the	sum	of	the	currents	

injected	and	withdrawn	at	bus	n	equal	zero:			

	 in	ൌ	∑k inmk	 	 	 	 	 	 	 	 	 ሺ1ሻ	

If	we	define	the	ground	to	be	bus	0,	current	to	ground	to	be	ynሺvn – v0ሻ	and	v0	ൌ	0,	we	

have:		

	 in	ൌ	∑k ynmkሺvn ‐ vmሻ	൅	ynvn		 	 	 	 	 	 	 ሺ2ሻ	

Current	is	a	linear	function	of	voltage.	Rearranging,	 	

	 in	ൌ	vnሺyn	൅	∑k ynmkሻ	‐	∑k ynmkvm	 	 	 	 	 	 ሺ3ሻ	

Expanding,	we	obtain		

	 inmk	ൌ	ynmkሺvn ‐ vmሻ	ൌ	gnmkሺvr n ‐ vr mሻ‐bnmkሺvj

n ‐ vjmሻ	൅	jሺbnmkሺvr n ‐ vr mሻ൅gnmkሺvjn ‐

vj

mሻሻ	

	 where	ir nmk	ൌ	gnmkሺvr n ‐ vr mሻ	‐	bnmkሺvj

n ‐ vjmሻ	and	

	 ij

nmk	ൌ	bnmkሺvr n ‐ vr mሻ	൅	gnmkሺvjn ‐ vjmሻ	

In	matrix	notation,	the	IV	flow	equations	in	terms	of	current	ሺIሻ	and	voltage	ሺVሻ	in	

ሺ3ሻ	are	

I	ൌ	YV	ൌ	ሺG	൅	jBሻሺVr 	൅	jVjሻ	ൌ	GVr 	‐	BVj	൅	jሺBVr 	൅	GVjሻ		 	 	 ሺ4ሻ	

	 where	Ir 	ൌ	GVr 	‐	BVj and	Ij ൌ	BVr 	൅	GVj

In	another	matrix	format,	ሺ4ሻ	is	

I	ൌ	ሺI

r ,	Ij	ሻ	ൌ	YሺVr ,	VjሻT	or	

											I	ൌ	ሺI

r , Ij	ሻ	ൌ		 G		 ‐B	 	 Vr

		where	Y ൌ	

G ‐B 			

B G	 	 Vj B G 			

If	a	and	φ	are	constant,	the	I	ൌYV	equations	are	linear.	If	not,	the	linearity	is	lost	

since	the	some	elements	of	the	Y	matrix	are	variable.	Discrete	setting	on	the	

transmission	assets	like	PARs	and	FACTS	devices	can	be	modeled	with	binary	

variables,	but	otherwise	retain	linearity.		

Power	flow	equations.	The	traditional	rectangular	real	power	ሺPሻ,	reactive	power	

ሺQሻ	and	voltage	ሺVሻ	power	flow	equations	ሺPQVሻ	are	

	 S	ൌ	P	൅	jQ	ൌ	diagሺVሻI

*	ൌ	diagሺVሻሾYVሿ*	ൌ	diagሺVሻY*V*	 	 	 ሺ5ሻ	

The	power	injections	are		

S	ൌ	V•I* 	ൌ	ሺVr 	൅	jVj	ሻ•ሺIr 	‐	jIj	ሻ	ൌ	ሺVr•I

r 	൅	Vj•Ij	ሻ	൅	jሺVj•Ir 	‐	Vr•Ijሻ	 	 ሺ6ሻ	

	 where		

P	ൌ	Vr•I

r 	൅	Vj•Ij		 	 	 	 	 	 	 ሺ7ሻ	

Q	ൌ	Vj•I

r 	‐	Vr•Ij		 	 	 	 	 	 	 ሺ8ሻ	

The	PQV	power	flow	equations,	ሺ5ሻ	and	ሺ6ሻ,	are	quadratic.	The	IV	flow	equations	ሺ4ሻ	

are	linear.	

	 	
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4.	Generator,	Load,	Transmission,	Voltage,	and	Angle	Constraints	

We	present	the	bus	level	constraints	in	terms	of	the	current	and	voltage	at	each	bus.	

Generator	and	Load	Constraints.	The	standard	but	simplified	representation	of	

generator	is	used	in	most	ACOPF	formulation.		The	lower	and	upper	bound	

constraints	for	generation	ሺinjectionሻ	and	load	ሺwithdrawalሻ	at	bus	n	are:	

pminn൑	p	൑	pmax n		 	 	 	 	 	 	 ሺ9ሻ	

qminn	൑	q	൑	qmax n	 	 	 	 	 	 	 ሺ10ሻ	

In	terms	of	v	and	i,	

vr ni

r n	൅	vj

ni

j

n	൑	pmax n			 	 	 	 	 	 ሺ11ሻ	

pmin n	൑	vr ni

r n	൅	vj

ni

j

n					 	 	 	 	 	 ሺ12ሻ	

vj

ni

r n	‐	vr nij

n	൑	qmax n	 	 	 	 	 	 	 ሺ13ሻ	

qminn ൑	vj

ni

r n	‐	vr nij

n	 	 	 	 	 	 	 ሺ14ሻ	

Inequalities	ሺ11ሻ‐ሺ14ሻ	along	with	other	thermal	constraints	on	equipment	enforced	

at	each	generator	bus	constitute	a	four‐dimensional	‘D‐curve’	in	the	IV	space;	ሺ11ሻ‐

ሺ14ሻ	are	non‐convex	constraints.	Additional	D‐curves	defining	the	tradeoff	between	

real	and	reactive	power	are	circles	in	the	p‐q	plane,	are	their	intersection	of	

constitutes	a	convex	set	and	can	be	easily	linearized.	Since	here	we	model	a	single	

period,	ramp	rates	are	unnecessary.	The	model	is	easily	extended	to	multiple	

periods.		

Voltage	constraints.	The	two	constraints	that	limit	the	voltage	magnitude	in	

rectangular	coordinates	at	each	bus	n	are		

	 ሺvr nሻ2	൅ሺvj

nሻ2	൑	ሺvmax nሻ2		 	 	 	 	 	 ሺ15ሻ	

	 ሺvminnሻ2	൑	ሺvr nሻ2	൅ሺvj

nሻ2			 	 	 	 	 	 ሺ16ሻ		

Again,	each	nonlinear	inequality	involves	only	the	voltages	at	bus	n.	The	inequality	

depicted	in	ሺ15ሻ	is	a	convex	constraint	but	the	one	in	ሺ16ሻ	is	not.	

Line	Flow	Constraints.		The	thermal	transmission	limit	on	k,	smax k,	is	a	based	on	the	

asset	materials	ability	to	withstand	temperature	increases.	As	current	increases,	

lines	sag	and	equipment	may	be	damaged	by	overheating.	Transmission	assets	

generally	have	three	progressively	larger	thermal	ratings:	steady‐state,	4	hour	and	

30	minute.	The	apparent	power	at	n	on	k	to	m	is:	

	 	snmk	ൌ	vninmk	ൌ	vnynmkሺvn ‐ vmሻ	ൌ	vnynmkvn ‐ vnynmkvm.	

The	thermal	limit	on	snmk	is	

	 ሺsr nmkሻ2൅	ሺsj

nmkሻ2	ൌ	|snmk|2	൑	ሺsmax kሻ2	 	 	 	 ሺ17ሻ	

These	constraints	are	quadratic	in	sr nmk	and	sj

nmk	and	quartic	in	vr n,	vjn,	vr m,	vjm.			

	 For	the	IV	formulation,	the	constraints	that	limit	the	current	magnitude	in	

rectangular	coordinates	at	each	bus	n	on	k	are		

	 ሺi

r nmkሻ2	൅ሺij

nmkሻ2	൑	ሺi

max nmkሻ2		 	 	 	 	 ሺ18ሻ	

Again,	the	nonlinearities	are	convex	quadratic	and	isolated	to	the	complex	current	at	

the	bus.	The	inequality	in	ሺ18ሻ	is	a	convex	constraint.	Generally,	the	maximum	
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10

currents,	i

max nmk,	are	determined	by	material	science	studies.	Limiting	current	may	

be	a	better	physical	constraint	then	limiting	apparent	power.	

Voltage	Angle	Constraints.	The	power	flowing	over	an	AC	line	is	approximately	

proportional	to	the	sine	of	the	voltage	phase‐angle	difference	at	the	receiving	and	

transmitting	ends.	For	stability	reasons,	the	terminal‐buses	voltage‐angle	

differences	on	transmission	elements	can	be	constrained	as	follows:		

θminnm	൑	θn	‐	θm	൑	θmax nm.	 	 	 	 	 	 ሺ19ሻ	

In	the	rectangular	formulation,	the	arctan	function	appears	in	some	constraints.		

θmin nm	൑	arctanሺvj

n/vr nሻ	‐	arctanሺvjm/vr mሻ	൑	θmax nm.		 	 ሺ20ሻ	

5.	ACOPF	Formulation	

Objective	function.		For	a	generator,	the	cost	of	generation	is	a	function	of	the	

apparent	power	generated,	cሺSሻ	ൌ	cPሺPሻ	൅	cQሺQሻ,	where	S	ൌ	ሺP2൅Q2ሻ1/2.		Most	of	the	

literature	assumes	cQሺQሻ	ൌ	0	and	cPሺPሻ	is	quadratic	in	P.	There	is	little	empirical	

evidence	for	these	assumptions	and	they	may	have	been	made	to	fit	the	nonlinear	

solver.		There	is	evidence	that	the	cost	functions	are	better	approximated	by	

piecewise	linear	functions.		With	binary	variables	and	linear	functions,	there	is	no	

need	to	assume	that	the	generator	cost	function	is	monotonic	non‐decreasing.		All	

ISOs	use	piecewise	linear	functions.	If	we	assume	that	the	cost	of	reactive	power	is	

small	compared	to	the	cost	of	real	power	and	if	the	cost	function,	cሺSሻ,	is	linear	in	S,	

an	approximation	of	cሺSሻ	is		

cሺSሻ	ൎ	cPሺPሻ	൅	cQሺ|Q|ሻ.	

For	most	generators,	the	normal	operating	range	is	|Q|	൏	0.3P.	The	absolute	value	

function,	|Q|,	can	be	made	a	linear	function	with	the	transformation	|Q|	ൌ	Q൅൅Q‐

	and	

Q	ൌ	Q൅‐Q‐	where	Q൅,	Q‐	൒	0.	By	virtue	of	the	minimization	and	the	constraint	

formulation,	Q൅•Q‐

	ൌ	0.		

If	there	is	a	value	function	for	load	or	demand,	dሺSሻ,	the	objective	function	is	to	

maximize	the	market	benefits	from	trade,	dሺSሻ	–	cሺSሻ.	Similar	arguments	hold	to	

simplify	dሺSሻ	but	are	beyond	the	scope	of	this	paper.		

Rectangular	Power	Voltage	Formulation	The	rectangular	power	voltage	ሺrectangular	

PQVሻ	ACOPF	ሺrectangular	ACOPF‐PQVሻ	formulation	is:	

Network‐wide	objective	function:	Min	cሺSሻ	 	 	 ሺ21ሻ	

	 Network‐wide	constraint:	P	൅	jQ	ൌ	S	ൌ	V•I* 	ൌ	V•Y*V* 	 	 ሺ22ሻ	

	 Bus‐specific	constraints	

Pmin	൑	P൑	Pmax 	 	 	 	 	 	 	 ሺ23ሻ	

Qmin	൑	Q		൑	Qmax 	 	 	 	 	 	 	 ሺ24ሻ	

Vr•Vr 	൅	Vj•Vj	൑	ሺVmax ሻ2		 	 	 	 	 	 ሺ25ሻ	

	 ሺVminሻ2	൑	Vr •Vr 	൅	Vj•Vj	 	 	 	 	 	 ሺ26ሻ	

	 ሺsnmkሻ2	൑	ሺsmax kሻ2	 	for	all	k	 	 	 	 	 ሺ27ሻ	
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11

θmin nm	൑	arctanሺvj

n/vr nሻ	‐	arctanሺvjm/vr mሻ	൑	θmax nm	 	 ሺ28ሻ	

In	this	formulation,	network‐wide	constraints	in	ሺ22ሻ	are	2N	quadratic	equalities	

that	apply	throughout	the	network;	the	bus‐specific	constraints	in	ሺ23ሻ‐ሺ24ሻ	are	

simple	variable	bounds	at	each	bus;	the	constraints	in	ሺ25ሻ	are	convex	quadratic	

inequalities	at	each	bus;	the	constraints	in	ሺ26ሻ	are	nonconvex	quadratic	

inequalities	at	each	bus;	the	constraints	in	ሺ27ሻ	are	quartic		inequalities	in	v	an	i	at	

each	bus;	and	the	constraints	in	ሺ28ሻ	are	nonconvex	inequalities.		

Polar	Power	Voltage	Formulation	The	polar	power	voltage	ሺpolar	PQVሻ	ACOPF	

ሺpolar	ACOPF‐PQVሻ	replaces	quadratic	equality	constraints	in	ሺ22ሻ	with	the	polar	

formulation	of	ሺ22ሻ:	

Network‐wide	constraints:	

pn	ൌ	∑mk vnvmሺgnmkcosθnm	൅	bnmksinθnmሻ		 for	all	n	 	 ሺ29ሻ	

qn	ൌ	∑mk vnvmሺgnmksinθnm	‐	bnmkcosθnmሻ	 for	all	n	 	 ሺ30ሻ

ሺ25ሻ	‐	ሺ26ሻ	is	replaced	by							Vmin	൑	V	൑	Vmax 		 	 	 		ሺ31ሻ	‐ሺ32ሻ	

ሺ28ሻ	is	replaced	by															θmin nm	൑	θn	‐	θm	൑	θmax nm.		 for	all	n,	m ሺ33ሻ	

In	this	formulation,	the	network‐wide	ሺ29ሻ‐ሺ30ሻ	are	2N	nonlinear	equality	

constraints	with	quadratic	terms	and	sine	and	cosine	functions	that	apply	

throughout	the	network.	The	arctan	functions	disappear	in	the	angle	difference	

constraints.	

Rectangular	ACOPF‐IV		formulation.	The	rectangular	ACOPF‐IV	formulation	is:	

Network‐wide	objective	function:	Min	cሺSሻ		 		 	 	 ሺ34ሻ	

Network‐wide	constraint:	I	ൌ	YV		 	 	 	 ሺ35ሻ	

Bus‐specific	constraints:	

P	ൌ	Vr•Ir 	൅	Vj•Ij	൑	Pmax 		 	 	 	 	 	 ሺ36ሻ	

Pmin	൑	P	ൌ	Vr•Ir 	൅	Vj•Ij	 	 	 	 	 	 ሺ37ሻ	

Q	ൌ	Vj•Ir 	‐	Vr•Ij	൑	Qmax 	 	 	 	 	 	 ሺ38ሻ	

Qmin	൑	Q	ൌ	Vj•Ir 	‐	Vr•Ij 	 	 	 	 	 ሺ39ሻ	

Vr•Vr 	൅	Vj•Vj	൑	ሺVmax ሻ2		 	 	 	 	 	 ሺ40ሻ	

	 ሺVminሻ2	൑	Vr •Vr 	൅	Vj•Vj 	 	 	 	 	 ሺ41ሻ	

	 ሺi

r nmkሻ2	൅	ሺij

nmkሻ2		൑	ሺi

max kሻ2	 	for	all	k	 	 	 	 ሺ42ሻ	

θmin nm	൑	arctanሺvj

n/vr nሻ	‐	arctanሺvjm/vr mሻ	൑	θmax nm	 	 ሺ43ሻ	

In	this	formulation,	the	constraints	in	ሺ35ሻ	are	2N	linear	equality	constraints	that	

apply	throughout	the	network.	This	is	in	contrast	to	the	PQV	formulations	where	

quadratic	and	trigonometric	constraints	apply	throughout	the	network	and	linear	

constraints	are	isolated	at	each	bus.		The	constraints	in	ሺ36ሻ	–	ሺ39ሻ	are	locally	

quadratic	and	non‐convex.	The	constraints	in	ሺ40ሻ	and	ሺ42ሻ	are	convex	locally	

quadratic	inequality	constraints,	but	the	ones	in	ሺ41ሻ	are	non‐convex	locally	

quadratic	inequality	constraints.	Overall,	the	constraint	set	is	still	non‐convex,	but	it	

would	appear	that	this	formulation	may	be	easier	to	solve	since	the	nonlinearities	

are	isolated	to	each	bus	and	each	transmission	element,	while	the	constraint	that	
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12

applies	throughout	the	network	is	linear.	The	constraint	in	ሺ43ሻ	could	be	eliminated	

by	using	the	constraints	in	ሺ42ሻ	as	a	surrogate	and	the	problem	becomes	locally	

quadratic	with	linear	network	equations.			

The	IV	formulation	has	6N	variables	ሺP,	Q,	Vr ,	Vj,	I

r ,	Ijሻ	and	the	polar	PQV	has	4N	

variables	ሺP,	Q,	|V|,	Θሻ.	

6.	The	Linear	Approximations	to	the	IV	Formulation	

A	judicious	choice	of	constraint	formulations	may	produce	better	approximations.	

We	take	two	approaches	to	constraint	formulation.	If	the	constraint	is	convex,	we	

add	linear	cutting	planes	to	remove	from	the	linear	feasible	region	points	that	are	

infeasible	in	the	nonlinear	formulation.	If	the	constraint	is	non‐convex,	we	use	the	

first	order	Taylor	series	approximation	to	the	constraint	and	update	it	after	optimal	

iteration	of	the	linear	program.				

Taylor’s	series.	The	Taylor	series	approximation	for	a	function	of	n	variables,		

fሺXሻ	where	X	ൌ	ሺx1,	...,	xNሻ	is		

In	matrix	notation,	

	 fሺXሻ	ൌ	fሺXሻ	൅	f’ሺXሻሺX‐Xሻ	൅	ሺX‐XሻTf’’ሺXሻሺX‐Xሻ	൅	oሾሺX‐Xሻ•ሺX‐Xሻ•ሺX‐Xሻሿ	

where	f’ሺXሻ	ൌ	ሾሺ∂fሺXሻ/∂x1ሻ,	...	∂fሺXሻ/∂xNሻሿT	is	the	Jacobian	vector	and	

										f’’ሺXሻ	ൌ	ሾ∂2fሺXሻ/∂xn∂xmሿ	is	the	Hessian	matrix.	

Convexity.	If	fሺXሻ	is	convex	and	differentiable,	fሺXሻ	൒	fሺXሻ	൅	f’ሺXሻTሺX‐Xሻ	or		

	 fሺXሻ	‐	f’ሺXሻTX	൒	fሺXሻ	‐	f’ሺXሻTX

and	f’’ሺXሻ	is	positive	semidefinite.	

If	fሺXሻ	൑	fup	and	fሺXሻ	൒	fሺXሻ	‐	f’ሺXሻTX		൅	f’ሺXሻTX,	

		 fሺXሻ	‐	f’ሺXሻTX		൅	f’ሺXሻTX	൑	fup

is	a	relaxation	of	the	convex	constraint	that	includes	the	feasible	set,	but	excludes	X.		

If	flow	൑	fሺXሻ,	and	fሺXሻ	൒	fሺXሻ	‐	f’ሺXሻTX		൅	f’ሺXሻTX,	

		 flow	൑	fሺXሻ	‐	f’ሺXሻTX		൅	f’ሺXሻTX		

is	not	a	conservative	linear	approximation	and	not	a	convex	constraint	and	we	may	

need	to	limit	the	linear	approximation	to	small	excursions	from	fሺXሻ,	for	example,		

	 flow	ൌ	maxሼflow,	rfሺXሻሽ	where	r	is	the	acceptable	range	of	the	approximation.	

Linear	Voltage	Approximations.	We	can	linearize	the	voltage	constraint	with	a	first	

order	Taylor’s	series	approximation	about	ሺvr ,	vjሻ.	at	each	bus	n,	the	voltage	

magnitude	approximation	is		

vr nvr n 	൅	vj

nvjn		ൎ	vr nvr n	൅	vjnvjn	൅	2vr nሺvr n‐	vr nሻ൅	2vjnሺvjn	–	vjnሻ	 	 ሺ44ሻ	

Collecting	terms,	we	obtain	

vr nvr n	൅	vj

nvjn		ൎ	2vr nvr n൅	2vjnvjn	‐vr nvr n	‐	vjnvjn	 	 	 	 ሺ45ሻ		

The	Hessian	is	

	 	
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13

vr n vj

n 	 	

∂2ሺ|vn|2ሻ/∂vr n∂vj

n ൌ	∂2ሺvr nvr n	൅	vjnvjnሻ/∂vr n∂vjn ൌ 2 0 vr n ሺ46ሻ	 0 2 vj

n

Since	the	Hessian	is	positive	definite,	the	voltage	magnitude	function	is	convex,	ሺ40ሻ	

is	a	convex	constraint	and	ሺ41ሻ	is	a	non‐convex	constraint.			

Iterative	Voltage	Constraints.	Let	vr n,	vj

n be	the	optimal	solution	to	the	LP	

approximation	that	may	violate	the	nonlinear	constraint	ሺ40ሻ.	We	can	add	a	linear	

constraint	to	create	an	outer	bound	on	the	constraint	set.		

Let	a	ൌ	vj

n/vr n,		

vr nvr n	൅	vj

nvjn	ൌ	vr nvr n	൅	a2vr nvr nൌ	ሺ1൅a2ሻvr nvr n.		

If	vr nvr n	൅	vj

nvjn	ൌ	ሺ1൅	a2ሻvr nvr n	൐	ሺvmax nሻ2,		

	 reset	vr n	ൌ	signሺaሻሾሺvmax nሻ2/ሺ1൅a2ሻሿ1/2	and	vj

n	ൌ	avr n

Now	vj

n,	vr n	is	a	point	on	the	maximum	voltage	constraint	ሺ40ሻ.	

	 add	the	linear	constraint:		vr nvr n൅	vj

nvjn	൑	ሺvmax nሻ2	 	 ሺ47ሻ	

ሺ47ሻ	cuts	off	the	linear	program	solution,	and	is	tangent	to	and	contains	ሺ40ሻ.		

Figure	1	illustrates	the	new	constraint	where	the	shaded	area	is	the	non‐convex	

feasible	region	of	the	maximum	and	minimum	voltage	at	a	bus.	Since	the	maximum	

voltage	constraint	is	convex,	the	outer	approximation	linear	constraints	on	

maximum	voltage	contain	the	feasible	region.		

Figure	1.	Adding	a	maximum‐voltage	linear	constraint.		

The	outer	approximation	can	accumulate	iteratively	and/or	can	be	generated	as	a	

part	of	preprocessing	without	eliminating	any	part	of	the	feasible	solution.	We	can	

create	a	linear	approximation	by	the	following	process.	We	can	choose	several	

points	on	the	boundary	of	the	maximum	voltage	constraint	and	add	the	outer	

linearization	to	the	formulation	in	the	preprocessing.			

Preprocessed	Voltage	Constraints.	We	can	start	with	simple	linear	bounds	on	the	

maximum	voltage,	if	vj

n	ൌ	0,	the	outer	linearization	constraint	at	the	voltage	angles	0	

and	π,	

	 ‐vmax n		൑	vr n൑	vmax n	 	 	 	 	 	 	 ሺ48ሻ	

if	vr n	ൌ	0,	the	outer	linearizations	at	the	voltage	angles	π/2	and	3π/2	are:	

vj

vr

(vr , vj)
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14

	 ‐vmax n		൑	vj

n൑	vmax n	 	 	 	 	 	 ሺ49ሻ	

We	also	bound	at	π/4	and	5π/4,	vr nൌ	vj

n,	at	the	boundary:		

vr nvr n൅	vj

nvjn	ൌ	2vr nvr n	ൌ	ሺvmax nሻ2		

the	tangent	points	are	vr n	ൌ	േvmax n/21/2and	vj

n	ൌ	േ2‐1/2vmax n

The	constraints	at	π/4	and	5π/4	are:	

	 ‐21/2vmax n	൑	vr n൅	vj

n	൑	21/2vmax n	 	 	 	 ሺ50ሻ	

In	a	similar	manner,	at	3π/4	and	7π/4,	with	vr nൌ	‐vj

n,	the	constraints	are:	

	 ‐21/2vmax n	൑	vr n ‐	vj

n	൑	21/2vmax n	 	 	 	 ሺ51ሻ

Inequalities	ሺ48ሻ,	ሺ49ሻ,	ሺ50ሻ	and	ሺ51ሻ	create	polygon	constraints	as	shown	in	Figure	

2.	

Figure	2.	Regular	8‐polygon	bounds	on	voltage	variables	

Non‐Convex	Minimum	Voltage	Constraints.	For	a	minimum	voltage	constraint,	since	

it	is	non‐convex,	the	linear	approximation	is	more	problematic.	This	may	be	a	

serious	problem.	Since	higher	losses	occur	at	lower	voltages,	the	natural	tendency	of	

the	optimization	will	be	toward	higher	voltages.	It	is	an	inner	approximation	and	

eliminates	parts	of	the	feasible	region	ሺsee	figure	4ሻ.	Although	this	may	be	

reasonable,	we	cannot	accumulate	these	constraints	and	should	relinearize	after	

each	linear	program	pass.		

Let	vr n,	vj

n be	the	optimal	solution	to	the	LP	approximation	and	assume	it	violates	

the	nonlinear	nonconvex	constraint	ሺ41ሻ.	Let	 vj

n	ൌ	avr n	.	If	vr nvr n	൅	vjnvjn	൏	ሺvminnሻ2,		

	 reset	vr n	ൌ	signሺaሻሾሺvminnሻ2/ሺ1൅a2ሻሿ1/2	and	vj

n	ൌ	avr n

Add	the	linear	constraint:		

vr nvr n൅	vj

nvjn	൒	ሺvminnሻ2	 	 	 	 	 ሺ52ሻ	

Figure	4.	Adding	a	minimum‐voltage	linear	constraint.		

vj

vr

vj

vr

(vr , vj)
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15

If	we	have	voltage	angle	constraints,	there	may	be	a	better	idea.	If	there	are	voltage	

angle	constraints,	a	linear	representation	of	the	voltage	constraints	may	be	at	hand.	

If	we	add	a	constraint,	vminn	൑	vr n.	Let	vj

n	–	a1vr n	൑	0	and	vjn	൅	a2vr n	൑	0	represent	

voltage	angle	constraints.	The	resulting	constraint	set	is	convex,	see	Figure	5	and	

contains	most	of	the	nonconvex	constraint	set.	The	stability	of	the	power	system	is	a	

function	of	the	angle	differences.		

Figure	5.	Minimum‐voltage	with	voltage	angle	constraints	

.	

When	combined	with	a	maximum	voltage	linearization,	the	approximation	

formulation	could	be	preprocessed	to	obtain	a	good	fit	to	the	nonlinear	constraints.	

Since	the	natural	process	of	optimization	pushes	the	voltage	higher	to	avoid	losses,	

the	minimum	voltage	constraint	is	only	approximated	when	it	is	violated.		

Linear	Approximation	of	Thermal	Transmission	Constraints	

The	constraints	that	limit	MVA	flow,	ሺ27ሻ,	and	the	constraint	on	current	flow	

perform	similar	functions.	There	are	no	minimum	current	constraints.		To	eliminate	

subscript	clutter,	we	will	drop	the	subscripts	n	and	m,	that	is,	i

r nmk,	ij

nmk	becomes	i

r k,	

i

j

k	.		Similar	to	the	maximum	voltage	constraints,	we	can	linearize	the	current	

constraint	with	a	first	order	Taylor’s	series	approximation	about	ሺi

r nmk,	ij

nmkሻ		

i

r kir k൅	ij

ki

j

k	ൎ	i

r kir k൅	ij

ki

j

k൅	2i

r kሺir k ‐ir kሻ	൅	2ij

kሺi

j

k‐i

j

kሻ	 	 	 ሺ53ሻ	

Collecting	terms,	we	obtain	the	linear	approximation	is:	

i

r kir k൅	ij

ki

j

k	ൎ	2i

r kir k൅	2ij

ki

j

k	‐	i

r kir k	‐	ij

ki

j

k	 	 	 	 ሺ54ሻ		

The	Hessian	is	

i

r k ij

k 		 	

					Hሺikሻ	ൌ	∂2ሺ|ik|2ሻ/∂i

r k∂ij

k	ൌ	∂2ሺi

r kir k൅	ij

ki

j

kሻ/∂i

r k∂ij

kൌ 20	 i

r k ሺ55ሻ	 02	 i

j

k

Since	the	Hessian	is	positive	definite,	the	current	magnitude	function	is	convex	in	

ሺi

r k,	ij

kሻ.	In	the	IV	formulation,	the	current	magnitude	is	functionally	the	same	as	the	

voltage	magnitude.	

Preprocessed	Current	Constraints.	We	can	start	with	simple	linear	bounds	on	the	

maximum	current,	if	i

j

k	ൌ	0,	the	outer	linearization	constraint	the	current	angles	0	

and	π,	

	 ‐i

max k		൑	ir k൑	imax k	 	 	 	 	 	 	 ሺ56ሻ	

Vr

Vi
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16

If	i

r k	ൌ	0,	the	outer	linearizations	at	the	current	angles	π/2	and	3π/2	are:	

	 ‐i

max k		൑	ij

k൑	i

max k	 	 	 	 	 	 	 ሺ57ሻ	

Inequalities	ሺ56ሻ	and	ሺ57ሻ	create	box	constraints	as	shown	in	Figure	2.	

We	also	bound	at	π/4	and	5π/4,	i

r kൌ	ij

k,	at	the	boundary:		

i

r kir k൅	ij

ki

j

k	ൌ	i

r kir k	ൌ	ሺimax kሻ2/2.	

The	constraints	at	π/4	and	5π/4	are:	

	 ‐21/2i

max k	൑	ir k൅	ij

k	൑	21/2i

max k	 	 	 	 	 ሺ58ሻ	

At	3π/4	and	7π/4,	with	i

r kൌ	‐ij

k	the	constraints	are:	

	 ‐21/2i

max k	൑	ir k ‐	ij

k	൑	21/2i

max k	 	 	 	 	 ሺ59ሻ

Inequalities	ሺ56ሻ,	ሺ57ሻ,	ሺ58ሻ	and	ሺ59ሻ	create	the	polygon	constraints	shown	in	

Figure	2.	

Iterative	Current	Constraints.	Let	i

r k,	ij

kbe	the	optimal	solution	to	the	LP	

approximation	that	may	violate	the	nonlinear	constraint.	We	develop	the	maximum	

current	constraints	in	a	similar	manner	to	the	maximum	voltage	constraints.	We	can	

add	a	linear	constraint	to	create	an	outer	bound	on	the	constraint	set.		

Let	a	ൌ	i

j

k/i

r k.	If	ir kir k	൅	ij

ki

j

k	൐	ሺi

max kሻ2,		

	 reset	i

r k	ൌ	signሺaሻሾሺimax kሻ2/ሺ1൅a2ሻሿ1/2	and	ij

k	ൌ	ai

r k

	 add	the	linear	constraint:		i

r kir k൅	ij

ki

j

k	൑	ሺi

max kሻ2	 	 	 ሺ60ሻ	

If	we	change	the	voltage	variables	to	current	variables,	Figure	1	illustrates	the	new	

constraint	where	the	outer	circle	defines	the	boundary	of	the	convex	feasible	region	

ሺfor	current	there	is	no	inner	circle	minimum	constraintሻ.	Since	the	max	current	

constraint	is	convex,	the	outer	approximation	linear	constraints	on	maximum	

current	contain	the	nonlinear	feasible	region.		

The	outer	approximation	can	accumulate	iteratively	and/or	can	be	generated	as	a	

part	of	prepossessing	without	eliminating	any	part	of	the	feasible	solution.	We	can	

create	a	linear	approximation	by	the	following	process.	We	can	choose	several	

points	on	the	boundary	of	the	maximum	current	constraint	and	add	the	outer	

linearization	to	the	formulation	in	the	prepossessing.		As	experience	with	this	model	

increases,	the	constraints	can	be	chosen	based	on	experience	and	specific	system	

behavior.	

Voltage‐angle	difference	constraints	serve	the	purpose	of	limiting	the	line	flows,	but	

in	rectangular	coordinates	are	not	convex.	Limiting	current	flow	has	a	similar	effect,	

but	the	current	flow	limits	are	convex.	Therefore,	substituting	current	for	angle	

constraints	may	be	a	better	computational	formulation.	

Real	Power	Constraints.	Constraints,	ሺ36ሻ	‐	ሺ37ሻ,	are	real	power	generator	

constraints.	At	each	bus	n,	pn	is	the	sum	of	two	hyperbolas	in	the	real	and	imaginary	

planes.	We	can	linearize	them	with	a	first	order	approximation	as	follows:		

	 pn	ൌ	vr ni

r n	൅	vj

ni

j

n		

	 ൎ	vr•i

r 	൅	vj•ij	൅	vr•ir 	൅	vj•ij	൅	vr•ir 	൅	vj•ij ‐	ሺvr•ir 	൅	vj•ij	൅	vr•ir 	൅	vj•ijሻ	 ሺ61ሻ	

collecting	terms,	we	obtain	
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17

	 pൎ	ൌ	vr ni

r n	൅	vj

ni

j

n	൅	vr ni

r n	൅	vj

ni

j

n ‐	ሺvr ni

r n	൅	vj

ni

j

nሻ	 	 	 ሺ62ሻ	

The	Hessian	is	

	 	 vr n vj

n i

r n ij

n

hሺpnሻൌ∂2pn/∂vn∂in	ൌ		

	 0	 0 1 0 vr n

ሺ63ሻ	

	 0	 0 0 1 vj

n

	 1	 0 0 0 i

r n

	 0	 1 0 0 i

j

n

The	Hessian	is	an	indefinite	symmetric	matrix	with	2	eigenvalues	equal	to	1	and	2	

eigenvalues	equal	to	‐1.	Since	the	Hessian	is	indefinite,	the	real	power	function	is	

non‐convex.		We	add	constraints:	

	 Pmin	‐	Prmin	൑	Pൎ	൑	Pmax	൅	Prmax 	 	 	 	 	 ሺ64ሻ	

	 where	Prmin	and	Prmax 	൒	0	with	high	objective	function	coefficients.	

Reactive	Power	Constraints.	Similarly,	for	reactive	power,	

qൎ	ൌ	vj

ni

r n	‐	vr nij

n	‐	vr ni

j

n	൅	vjni

r n		‐	ሺvj

ni

r n	‐	vr nij

nሻ		 	 	 ሺ65ሻ	

The	Hessian	is	

vr n vj

n i

r n ij

n

hሺqሻ	ൌ	∂2qn/∂vn∂in	ൌ		

0 0 0 ‐1 vr n

ሺ66ሻ	 0 01 0 vj

n

0100 i

r n

‐1 0 0 0 i

j

n

The	Hessian	is	an	indefinite	symmetric	matrix	consisting	with	2	eigenvalues	equal	to	

1	and	2	eigenvalues	equal	to	‐1.	Since	the	Hessian	is	indefinite,	the	reactive	power	

function	is	non‐convex.	We	added	the	constraints:		

Qmin	‐	Qrmin	൑	Qൎ	൑	Qmax 	൅	Qrmax 	 	 	 	 	 ሺ67ሻ	

	 where	Qrmin	൒	0	and	Qrmax 	൒	0	with	high	objective	function	coefficients	

7.	Conclusions	

Unlike	the	DC	model	that	holds	V	constant,	ignores	reactive	power	and	assumes	

small	angle	differences,	the	IV	formulation	solves	a	linear	system	of	equations	

without	decomposition,	unnecessary	constraints	or	omissions.		The	nonconvex	

constraints	need	careful	consideration.	It	appears	that	the	IV	formulation	and	its	

linear	approximations	have	promise	to	meet	practical	computational	requirements.	

For	problems	that	are	solved	repetitively	with	minor	variations,	there	is	

considerable	potential	for	individual	parameter	tuning	and	preprocessed	

constraints.	For	example,	constraints	on	angles	and	past	iteratively	added	

constraints	may	be	able	to	help.		In	addition,	steady‐state	quadrature	constraints	on	

generators	are	linear	in	the	IV	formulation	and	can	be	included	in	the	formulation.	
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18

Additional	formulation	testing	and	computational	testing	are	needed	to	determine	

commercial	feasibility.	If	a	linear	IV	approximation	to	the	ACOPF	proves	promising,	

it	can	be	embedded	in	the	unit	commitment	models,	optimal	topology	models	and	

other	formulations	that	use	binary	variables.	This	allows	the	use	of	MIP	algorithms	

that	are	exceptionally	fast	and	robust	to	better	model	the	power	markets.

	

	

References	

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

Formulations,”	FERC	staff	technical	paper,	December,	2012