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Enhancing state estimation in distribution and transmission systems using advanced metering infrastructure Alimardani, Arash 2015

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 ENHANCING	STATE	ESTIMATION	IN	DISTRIBUTION	AND	TRANSMISSION	SYSTEMS	USING	ADVANCED	METERING	INFRASTRUCTURE	by	Arash	Alimardani		B.Sc.	Isfahan	University	of	Technology,	2008	M.Sc.	Amirkabir	University	of	Technology	(Tehran	Polytechnic),	2011		A	THESIS	SUBMITTED	IN	PARTIAL	FULFILLMENT	OF	THE	REQUIREMENTS	FOR	THE	DEGREE	OF	DOCTOR	OF	PHILOSOPHY	in	The	Faculty	of	Graduate	and	Postdoctoral	Studies	(Electrical	and	Computer	Engineering)		THE	UNIVERSITY	OF	BRITISH	COLUMBIA		(Vancouver)	August	2015		©	Arash	Alimardani,	2015	ii Abstract	State	 estimation	 is	 the	 heart	 of	 many	 tools	 used	 in	 operations	 of	 distribution	 and	transmission	power	 systems.	The	quality	of	distribution	systems	state	estimation	 (DSSE)	typically	 suffers	 from	 a	 lack	 of	 adequate/accurate	measurements	 and	 has	 not	 been	 fully	implemented	by	many	utilities.	Recently,	as	part	of	many	smart	grid	related	 initiatives	to	modernize	 power	 systems,	 electric	 utilities	 started	 to	 invest	 in	 advanced	 metering	infrastructure	(AMI)	throughout	their	distribution	systems.	The	main	challenge	in	this	area	is	that	AMI	measurements	are	generally	not	synchronized,	and	the	difference	between	the	measuring	 times	 of	 smart	 meters	 can	 be	 significant.	 In	 generation	 and	 transmission	systems,	 the	 transmission	 system	 state	 estimation	 (TSSE)	 is	 already	 prevalent	 in	 many	utilities.	However,	TSSE	typically	suffers	from	four	major	problems:	partial	unobservability,	numerical	 ill‐conditioning,	 bad	 data,	 and	 low	 accuracy.	 This	 thesis	 is	 based	 on	 three	contributions.	 Firstly,	 an	 innovative	 method	 is	 developed	 to	 incorporate	 the	 non‐synchronized	measurements	 coming	 from	AMI	based	 on	 the	 credibility	 of	 each	 available	measurement	 and	 appropriately	 adjusting	 the	 statistical	 property	 of	 the	 measurement	signals.	 To	 illustrate	 the	 effectiveness	 of	 the	 proposed	 method,	 it	 is	 compared	 with	traditional	approach	used	in	DSSE	and	the	results	show	the	improvements	in	the	accuracy	of	DSSE.	Next,	 based	on	 the	 interconnection	 of	 the	 transmission	 system	and	distribution	systems	at	PQ	buses	(feeder	heads),	a	novel	approach	in	TSSE	method	is	presented	which	uses	 the	 DSSE	 results	 to	 provide	 additional	 measurements	 at	 the	 PQ	 buses	 of	 the	transmission	 system.	 Comparisons	 between	 the	 traditional	 TSSE	 and	 the	 proposed	 TSSE	show	 that	 significant	 improvements	 are	 achieved.	 The	 third	 contribution	 is	 the	iii methodology	 for	 identification	 of	 electricity	 theft	 points	 in	 distribution	 systems	without	violating	 privacy	 of	 consumers.	 The	 proposed	 approach	 models	 theft	 as	 bad	 data	 and	consists	of	two	stages.	Firstly,	the	multiple	bad	data	identification	problem	is	solved	using	a	heuristic	 optimization	 method	 to	 locate	 the	 points	 of	 theft	 which	 have	 redundant	measurements.	 In	 the	 second	 stage,	 regarding	 identification	of	 theft	 points	which	do	not	have	redundant	measurements,	a	method	is	proposed	based	on	the	discrepancies	between	the	 measured	 and	 estimated	 voltage	 magnitudes.	 Simulations	 results	 demonstrate	 the	effectiveness	of	the	proposed	approach.	iv Preface	Based	 on	 the	 research	 presented	 in	 this	 thesis,	 several	 papers	 have	 been	 published	 in	conference	 proceedings,	 and	 published	 and/or	 submitted	 as	 journal	 articles.	 In	 all	publications,	 I	 have	 developed	 mathematical	 formulations,	 implemented	 models,	conducted	 simulations,	 analyzed	 results,	 and	 prepared	 the	 initial	 manuscripts.	 My	supervisors,	Dr.	Jatskevich	and	Dr.	Vaahedi,	have	provided	me	with	instructive	comments	and	 corrections	 throughout	 the	 process	 of	 conducting	 research	 studies,	 preparing	 and	editing	manuscripts.	 The	 following	 describes	 published	 and	 submitted	 papers	 as	well	 as	contributions	of	other	co‐authors.	Part	of	Chapter	2	was	presented	at	a	conference.	A.	Alimardani,	S.	Zadkhast,	J.	Jatskevich,	E.	Vaahedi,	 "Using	 smart	meters	 in	 state	 estimation	 of	 distribution	 networks,"	PES	General	Meeting	 |	 Conference	 &	 Exposition,	 2014	 IEEE	 ,	 vol.,	 no.,	 pp.1,5,	 27‐31	 July	 2014.	 As	 a	colleague	 graduate	 student	 in	 our	 group,	 S.	 Zadkhast	 was	 working	 with	 me	 on	 this	manuscript	 and	 provided	 comments	 and	 suggestions	 on	 the	 studies	 and	 content	 of	 this	paper.	Chapter	 2	 has	 been	 published	 as	 a	 journal	 article.	 A.	 Alimardani,	 F.	 Therrien,	 D.	Atanackovic,	J.	Jatskevich,	E.	Vaahedi,	“Distribution	System	State	Estimation	Based	on	Non‐Synchronized	 Smart	 Meters,”	 IEEE	 Transaction	 on	 Smart	 Grids,	 DIO:	10.1109/TSG.2015.2429640.	I	took	advice	from	F.	Therrien	based	on	his	experience	from	working	 for	CYME	International	T&D	on	development	of	distribution	system	simulations.	Also,	 Dr.	 Atanackovic,	 as	 the	 current	 manager	 of	 Real‐Time	 System	 department,	 Grid	v Operations	of	BC	Hydro,	BC.,	Canada,	helped	with	explaining	the	asynchronicity	problem	of	smart	meters	from	industrial	perspective.		Part	of	Chapter	3	was	presented	at	a	conference.	A.	Alimardani,	S.	Zadkhast,	F.	Therrien,	J.	Jatskevich,	E.	Vaahedi,	 "Impact	of	employing	state	estimation	of	distribution	networks	on	state	estimation	of	transmission	networks,"	PES	General	Meeting	|	Conference	&	Exposition,	2014	IEEE	,	vol.,	no.,	pp.1,5,	27‐31	July	2014.	As	colleague	graduate	students	in	our	group,	S.	Zadkhast	and	F.	Therrian	worked	with	me	on	ideas	on	presentation	of	this	manuscript	and	provided	comments	and	suggestions	on	the	studies	and	content	of	this	paper.		Chapter	 3	 is	 submitted	 for	 peer	 review.	 A.	 Alimardani,	 F.	 Therrien,	 D.	 Atanackovic,	 J.	Jatskevich,	 E.	 Vaahedi,	 “Incorporating	 Distribution	 System	 State	 Estimation	 Results	 at	Transmission	 Level,”	 F.	 Therrien	 advised	 on	 the	 simulation	 process,	 and	 D.	 Atanackovic	explained	the	industrial	implementation	of	state	estimation	in	DMS	and	EMS	in	BC	Hydro,	BC.,	Canada.		Chapter	4	is	under	preparation	as	A.	Alimardani,	J.	Jatskevich,	E.	Vaahedi,	“Identification	of	Electricity	Theft	in	Distribution	Systems”,	to	be	submitted.		vi Table	of	Contents	ABSTRACT	................................................................................................................................................	ii PREFACE	..................................................................................................................................................	iv TABLE	OF	CONTENTS	..........................................................................................................................	vi LIST	OF	TABLES	.......................................................................................................................................	x LIST	OF	FIGURES	..................................................................................................................................	xii LIST	OF	ABBREVIATIONS	.................................................................................................................	xvi ACKNOWLEDGEMENTS	....................................................................................................................	xvii DEDICATION	.........................................................................................................................................	xix CHAPTER	1:  INTRODUCTION	.............................................................................................	1 1.1  Motivation	......................................................................................................................................	1 1.2  Background	of	State	Estimation	...........................................................................................	4 1.2.1  Transmission	system	state	estimation	.............................................................	4 1.2.2  Distribution	system	state	estimation	................................................................	8 1.2.3  Electricity	theft	detection	.....................................................................................	11 1.3  State‐of‐the‐Art	Research	......................................................................................................	12 1.3.1  Distribution	system	state	estimation	..............................................................	12 vii 1.3.1.1  DSSE	based	on	AMI	measurements	..............................................	13 1.3.2  Transmission	system	state	estimation	...........................................................	14 1.3.2.1  Enhancing	TSSE	using	the	results	of	DSSE	................................	15 1.3.3  Electricity	theft	detection	.....................................................................................	16 1.4  Research	Objectives	and	Anticipated	Impacts	.............................................................	17 CHAPTER	2:  DSSE	 BASED	 ON	 NON‐SYNCHRONIZED	 AMI	MEASUREMENTS	.....................................................................................................................	20 2.1  Weighted	Least	Squares	Approach	....................................................................................	20 2.2  State	Estimation	Based	on	Non‐synchronized	Smart	Meters	................................	24 2.2.1  Load	variation	modeling.......................................................................................	25 2.2.2  Implementation	of	load	variation	modeling	technique	..........................	30 2.3  Simulation	of	Case	Studies	....................................................................................................	32 2.3.1  IEEE	13‐bus	distribution	system	......................................................................	37 2.3.1.1  Impact	of	random	noise	.....................................................................	40 2.3.1.2  Impact	of	load	uncertainty	...............................................................	42 2.3.2  IEEE	123‐bus	distribution	system	...................................................................	43 CHAPTER	3:  INCORPORATION	OF	DSSE	INTO	TRANSMISSION	LEVEL	.............	46 3.1  Transmission	System	State	Estimation	...........................................................................	46 3.2  Incorporating	DSSE	Results	into	TSSE	.............................................................................	47 3.3  Areas	of	Impact	in	TSSE	..........................................................................................................	50 viii 3.3.1  Observability	.............................................................................................................	50 3.3.2  Numerical	conditioning	........................................................................................	51 3.3.3  Bad	data	identification	..........................................................................................	52 3.3.4  Accuracy	......................................................................................................................	54 3.4  Case	Studies	.................................................................................................................................	55 3.4.1  Restoring	observability	.........................................................................................	59 3.4.2  Improving	numerical	conditioning	..................................................................	62 3.4.3  Impact	on	accuracy	of	TSSE	................................................................................	66 3.4.4  Facilitating	 bad	 data	 identification	 and	 accuracy	improvement	.............................................................................................................	69 CHAPTER	4:  BAD	DATA	IDENTIFICATION	AND	ELECTRICITY	THEFT	..............	72 4.1  Modeling	Theft	as	Bad	Data	..................................................................................................	72 4.2  Problem	Formulation	of	Bad	Data	Identification	........................................................	74 4.3  Solution	Method	.........................................................................................................................	79 4.3.1  Shuffled	frog	leaping	algorithm	.........................................................................	80 4.3.2  Satisfying	the	constraints	.....................................................................................	81 4.3.3  Implementation	of	SFLA	as	the	proposed	solution	...................................	82 4.3.4  Case	study	...................................................................................................................	84 4.4  Discrepancies	between	Estimated	and	Measured	Voltages	...................................	87 4.4.1  Case	study	...................................................................................................................	90 ix CHAPTER	5:  SUMMARY	OF	CONTRIBUTIONS	AND	FUTURE	WORKS	................	94 5.1  Conclusions	and	Contributions	...........................................................................................	94 5.2  Potential	Impacts	of	Contributions	...................................................................................	98 5.3  Future	Work	.............................................................................................................................	100 REFERENCES	.......................................................................................................................................	102 		x List	of	Tables	Table	2―1	Normality	Evaluation	of	Figure	2‐2	Data	.............................................................................	28 Table	2―2	Statistical	Performance	of	DSSE	Error	for	a	Typical	Load	Proϐile	in	the	IEEE	13‐bus	Distribution	System	.....................................................................................................................................	41 Table	2―3	Statistical	Performance	of	DSSE	Error	 for	a	Distorted	Load	Proϐile	 in	 the	 IEEE	13‐bus	Distribution	System	..............................................................................................................................	43 Table	2―4	Statistical	Performance	of	DSSE	Error	for	a	Typical	Load	Proϐile	in	the	IEEE	123‐bus	Distribution	System	.....................................................................................................................................	45 Table	2―	5	Statistical	Performance	of	DSSE	Error	for	a	Distorted	Load	Profile	in	the	IEEE	123‐bus	Distribution	System	...........................................................................................................................	45 	Table	3―	1	Statistical	Results	of	Calculating	the	Condition	Number	of	TSSE	for	100	Runs	.	63 Table	 3―2	 Number	 of	 Iterations	 Taken	 by	 the	 Traditional	 and	 Proposed	 TSSE	 Methods	Over	2	Hour	Interval.	...........................................................................................................................................	66 Table	3―3	Normalized	Residuals	of	TSSE	of	the	IEEE	14‐bus	Transmission	System	for	Bus	10	before	Bad	Data	Detection	..........................................................................................................................	70 Table	3―4	Total	Error	of	TSSE	for	the	IEEE	14‐bus	Transmission	System	with	Bad	data	at	Bus	10	and	Its	Elimination	................................................................................................................................	71 xi 	Table	4―1	Percentage	of	the	Unmeasured	Consumption	at	Each	Theft	Point	...........................	86 Table	 4―2	 Results	 of	 Multiple	 Bad	 Data	 Identiϐication	 Solution	 Achieved	 by	 Different	Methods	.....................................................................................................................................................................	87 Table	 4―3	 Comparison	 between	 Measured	 and	 Estimated	 Voltage	 Magnitudes	Discrepancies	with	3σ	.........................................................................................................................................	91 	xii List	of	Figures	Figure	1―1	Typical	architecture	of	EMS.	Measurements	from	RTUs	are	send	to	the	SCADA	system.	TSSE	uses	these	data	and	provides	other	applications	with	required	information.	..	5 Figure	 1―2	 Typical	 architecture	 of	 DMS.	 Measurements	 from	 the	 distribution	 power	system	are	transferred	based	on	which	DSSE	is	executed	providing	necessary	information	for	other	applications.	.........................................................................................................................................	10 	Figure	2―1	Historical	active	power	consumption	proϐile	of	a	building	at	The	University	of	British	Columbia.	...................................................................................................................................................	26 Figure	2―2	Power	consumption	variation	from	2:00PM	to	2:15PM	based	on	the	historical	data	depicted	in	Figure	2‐1.	..............................................................................................................................	28 Figure	2―3	DSSE	process	addressing	asynchronicity	of	measurements.	.....................................	31 Figure	2―4	A	typical	load	proϐile	for	24	hours	in	pu	with	respect	to	the	peak	value	based	on	the	historical	data	recorded	at	The	University	of	British	Columbia.	..............................................	32 Figure	2―5	Pattern	of	 the	updating	 time	of	each	smart	meter	measurements	 installed	on	the	loads	of	the	IEEE	13‐bus	distribution	system.	..................................................................................	34 Figure	2―6	Simulation	process	for	the	case	studies.	.............................................................................	36 xiii Figure	2―7	Norm	error	of	 real	power	measurements:	 (a)	 ideal	 case	 ‐	 error	due	 to	 smart	meters	imprecision	only;	(b)	error	only	due	to	out‐of‐date	measurements,	and	(c)	error	due	to	smart	meters	imprecision	combined	with	error	due	to	out‐of‐date	measurements.	........	38 Figure	2―8	Accuracy	comparison	of	the	three	considered	methods:	(a)	DSSE	error	during	one	day;	and	(b)	accumulated	error	of	DSSE	during	one	day	for	each	method.	........................	39 Figure	2―9	Total	error	of	DSSE	in	24	hours	for	10000	runs.	............................................................	41 Figure	2―10	Typical	and	distorted	load	proϐiles	for	24	hours	in	pu.	.............................................	42 Figure	2―11	Considered	IEEE	123	distribution	systems.	...................................................................	44 	Figure	3―1	Equivalent	circuit	for	a	transmission	line.	.........................................................................	46 Figure	3―2	Simpliϐied	diagram	depicting	a	transmission	system	connected	to	distribution	systems	via	PQ	buses	(feeder	heads).	...........................................................................................................	48 Figure	 3―3	 Conϐiguration	 of	 the	 ϐirst	 case	 study:	 (a)	Modiϐied	 IEEE	 14‐bus	 transmission	system	with	allocated	loads,	generators,	and	measurement	devices;	(b)	The	feeder	head	of	the	IEEE	13‐bus	distribution	system	is	connected	to	bus	9	of	the	IEEE	14‐bus	transmission	system.........................................................................................................................................................................	57 Figure	3―4	Typical	24	hour	load	profile	normalized	with	respect	to	the	peak	value	based	on	historical	data	recorded	at	The	University	of	British	Columbia	[91];	(b)	Recorded	2‐hour	fragment	of	load	profile	from	2pm	to	4pm.	...............................................................................................	58 xiv Figure	3―5	Procedure	of	calculating	the	results	of	Proposed	and	Traditional	TSSE	at	each	step	of	the	simulation.	.........................................................................................................................................	60 Figure	 3―6	 IEEE	 14‐bus	 transmission	 system	with	 a	 faulty	 critical	measurement	 device.	The	 subsystem	 which	 has	 a	 faulty	 measurement	 device	 is	 depicted.	 Red	 measurement	devices	show	the	injection	power	from	the	rest	of	the	system	to	the	subsystem.	....................	62 Figure	3―7	Number	of	 iterations	 taken	 to	achieve	a	converged	TSSE	solution	 for:	 (a)	 the	IEEE	14‐bus	transmission	system;	and	(b)	the	IEEE‐57	bus	transmission	system.	.................	65 Figure	3―8	Total	error	of	TSSE	evaluated	over	all	buses	at	each	time	step	and	its	average	for	the	2‐hour	study	interval	with	smart	meters	measurements	updating	every	15	minutes	for:	(a)	the	IEEE	14‐bus	transmission	system;	and	(b)	the	IEEE	57‐bus	transmission	system.	.......................................................................................................................................................................................	68 Figure	3―9	Total	error	of	TSSE	evaluated	over	all	buses	at	each	time	step	and	its	average	for	the	2‐hour	study	interval	when	the	smart	meters	measurements	are	updated	every	30	minutes	for	the	IEEE	57‐bus	transmission	system.	................................................................................	69 	Figure	 4―1	 A	 simpliϐied	 diagram	 depicting	 connection	 of	 smart	 meters	 for	 modeling	electricity	 theft	 as	 bad	data:	 	 (a)	 load	2	 is	 partially	 bypassing	 its	 smart	meter	 to	 commit	electricity	theft;	and	(b)	smart	meter	of	load	2	is	malfunctioning.	..................................................	73 Figure	 4―2	Probability	 density	 function	 of	 the	 chi‐square	 distribution	with	 8	 degrees	 of	freedom.	.....................................................................................................................................................................	76 xv Figure	4―3	The	optimization	process	using	SFLA.	.................................................................................	84 Figure	4―4	Considered	IEEE	123	distribution	systems	depicting	eight	loads	with	electricity	theft.	............................................................................................................................................................................	85 xvi List	of	Abbreviations	Abbreviation	 Meaning	AMI	 Advanced	Metering	Infrastructure	DMS	 Distribution	Management	System	DSSE	 Distribution	System	State	Estimation	TSSE	 Transmission	System	State	Estimation	EA	 Evolutionary	Algorithms	EMS	 Energy	Management	Systems	GA	 Genetic	Algorithm	IPM	 Interior	Point	Method	MA	 Memetic	Algorithms	MDMS	 Meter	Data	Management	Systems	NLP	 Nonlinear	Programming	NN	 Neural	Network	OD	 Out‐of‐date	PDF	 Probability	Density	Function	PSO	 Particle	Swarm	Optimization	QP	 Quadratic	Programming	RTU	 Remote	Terminal	Units	SA	 Simulated	Annealing	xvii Acknowledgements	Foremost,	 I	 would	 like	 to	 express	 my	 sincere	 gratitude	 to	 my	 supervisors	 Prof.	 Juri	Jatskevich	 and	 Dr.	 Ebrahim	 Vaahedi	 for	 their	 continuous	 support,	 patience,	 great	inspiration,	 immense	 knowledge	 and	 insights,	 and	 enthusiasm	 that	 they	 provided	 me	throughout	my	 studies.	 I	 appreciate	 all	 their	 contribution	 of	 time	 and	 ideas	 to	make	my	doctoral	experience	productive	and	stimulating.	The	financial	support	for	this	research	was	made	possible	by	the	Natural	Science	and	Engineering	Research	Council	(NSERC)	of	Canada	under	the	Strategic	Project	Grant	“Enabling	Solutions	for	the	Future	Canadian	Smart	Grid”	led	by	D.	Jatskevich	as	a	principal	investigator.	I	am	also	grateful	to	Dr.	Atanackovic,	the	manager	of	the	department	of	Real‐Time	Systems,	BC	 Hydro,	 BC,	 Canada	 and	 a	 member	 of	 IEEE	 Working	 Group	 on	 State	 Estimation	Algorithms,	 for	 his	 advice	 on	 a	 variety	 of	 issues	 regarding	 real‐time	 power	 systems	operations	 and	 practical	 concerns	 from	 the	 industry	 point	 of	 view	 as	 well	 as	 verifying	simulations.	My	 sincere	 thanks	 also	 go	 to	 Prof.	 Uri	 Ascher.	 His	 lectures	 on	 numerical	 conditioning	 of	systems	were	very	helpful	in	my	research.	Also,	I	would	like	to	thank	Dr.	Ebrahim	Vaahedi	and	Mr.	Nathan	Ozog	for	the	wonderful	teaching	experience	I	gained	while	assisting	them.	In	 addition,	 I	would	 like	 to	 thank	my	 fellow	graduate	 students	 and	 colleagues	 in	Electric	Power	and	Energy	Systems	research	group	at	 the	University	of	British	Columbia:	Francis	Therrien,	 Pouya	 Zadkhast,	 Arash	 Taavighi,	 Mehrdad	 Chapariha,	 and	 Yingwei	 Huang	 for	xviii many	stimulating	discussions	and	all	the	memorable	moments	we	have	had	in	the	past	four	years	at	UBC.	Also,	thanks	to	Eva	Yu	for	proofreading	parts	of	the	text.	Lastly,	I’d	like	to	thank	my	parents.	They	supported	me	all	along	the	way	and	I	am	grateful	for	everything	they	did	for	me.	xix Dedication		To	My	Wonderful	Parents	and	Beloved	Friends	1 CHAPTER	1: INTRODUCTION			1.1 Motivation	Modern	electrical	power	and	energy	systems	are	fast	evolving	and	undergoing	significant	changes	 to	 accommodate	 the	 new	 and	 renewable	 generation	 and	 satisfy	 the	 increasing	power	 demand,	 while	 making	 the	 system	 more	 reliable,	 more	 resilient,	 and	 “smarter”.	Significant	 resources	 are	 being	 deployed	 to	 enable	 the	 future	 smart	 grids	 [1],	 [2].	 An	integral	part	of	the	philosophy	of	smart	grids	is	to	use	modern	communication	and	advance	metering	 technologies	 in	 power	 systems.	 As	 an	 investment	 towards	 Advanced	Metering	Infrastructure	 (AMI)	 development,	 the	 number	 of	 installed	 measurement	 devices	 in	distribution	 systems	 has	 increased	 significantly	 in	 recent	 years	 [3].	 Smart	 meters	 have	been	 installed	 in	 very	 large	 numbers	 in	many	 regions.	 For	 instance,	 in	 British	 Columbia	(BC),	 Canada,	 BC	 Hydro	 is	 installing	 smart	 meters	 for	 every	 consumer	 throughout	 the	province,	which	will	 provide	 1.8	million	 accurate	 and	 reliable	measurements	 [4].	 Future	distribution	management	systems	(DMSs)	are	expected	to	be	capable	of	handling	a	growing	number	 of	 applications,	 including	 real‐time	 control	 and	 monitoring,	 distribution	transformer	usage	optimization,	feeder	reconfiguration	and	restoration,	control	of	switches	and	reclosers,	demand	side	management,	and	capacitor	switching	[5]‐[10].	The	deployment	of	distributed	generation	and	renewable	energy	sources,	which	is	crucial	 for	enabling	the		2 future	smart	grid,	also	poses	new	challenges,	such	as	the	occurrence	of	overvoltages	at	the	distribution	level.	Therefore,	smart	distribution	systems	are	also	expected	to	have	volt/VAr	control	capabilities.	Since	distribution	system	state	estimation	(DSSE)	provides	the	 initial	condition	of	all	of	the	mentioned	DMS	applications,	its	accuracy	and	reliability	will	have	a	significant	impact	on	the	operation	of	the	future	grid.	Another	recent	trend	in	distribution	systems	is	the	introduction	of	meshed	topologies,	which	are	incompatible	with	traditional	DSSE	 algorithms	 (or	 makes	 them	 less	 efficient)	 [1].	 New	 DSSE	 techniques	 should	 be	capable	of	dealing	efficiently	with	highly	meshed	and	radial	topologies.	A	 major	 challenge	 when	 using	 data	 from	 smart	 meters	 is	 that	 they	 do	 not	 provide	synchronized	measurements	 (i.e.	 the	measurement	 signals	 are	 not	 updated	 at	 the	 same	time)	[11],	[12].	Additionally,	their	measuring	resolution	is	lower	than	measurements	from	RTUs	of	reclosers.	This	problem	has	made	utilities	reluctant	to	use	smart	meters	for	some	real‐time	applications	including	state	estimation.	For	instance,	in	many	utilities	such	as	BC	Hydro,	smart	meters	are	only	considered	as	a	tool	for	hourly	billing	and	dynamic	pricing.	Solving	the	problem	with	non‐synchronized	smart	meter	measurements	is	a	necessity	and	would	 significantly	 improve	 the	 capabilities	 of	 DMS	with	 respect	 to	 distribution	 system	applications	which	are	based	on	DSSE	results.	This	objective	 is	one	of	 the	motivations	 in	defining	this	thesis.	In	 energy	 management	 system	 (EMS),	 referring	 to	 transmission	 and	 generation	systems	 management,	 state	 estimation	 is	 in	 the	 heart	 of	 all	 major	 applications	 such	 as	transient	 stability	 analysis,	 voltage	 stability	 analysis,	 optimal	 dispatch,	 optimal	 reactive		3 power	flow,	preventing	overloading,	etc.	[5]‐[10].	One	of	the	major	problems	with	respect	to	TSSE	is	lack	of	sufficient	measurement	devices	in	lower	voltage	level	parts	of	the	system.	For	instance,	the	25kV	and	12kV	substations	in	BC,	Canada	have	this	problem	of	inadequate	measurement	devices.	As	a	result,	the	transmission	system	may	be	partially	unobservable.	Utilities	typically	overcome	this	problem	by	defining	pseudo	measurements	to	render	the	system	 observable.	 However,	 since	 pseudo	 measurements	 are	 assumptions	 based	 on	historic	data	and	tend	to	be	not	very	accurate,	their	use	also	reduces	the	accuracy	of	state	estimation.	Another	problem	in	TSSE	is	that	the	system	is	often	numerically	ill‐conditioned.	Numerous	studies	have	been	conducted	to	investigate	numerical	robustness	of	TSSE	[13].	It	is	shown	that	increasing	the	redundancy	of	voltage	magnitude	measurements	generally	improves	 the	 numerical	 condition	 of	 the	 problem	 by	 reducing	 the	 condition	 number	 of	system	 matrices	 [14].	 However,	 installing	 Remote	 Terminal	 Units	 (RTUs)	 with	measurement	devices	has	very	significant	costs	in	transmission	systems.	Alternatively,	if	an	AMI‐based	DSSE	method	with	acceptable	accuracy	is	developed,	the	results	at	feeder	heads	of	 distribution	 systems	 could	 be	 used	 to	 provide	measurements	 for	 the	 TSSE	 PQ	 buses.	These	 additional	 measurements	 could	 render	 the	 unobservable	 parts	 of	 transmission	system	 to	 become	 observable,	 and	 at	 the	 same	 time	 improve	 numerical	 condition	 and	increase	 the	 accuracy	 of	 TSSE	 without	 more	 investments	 on	 new	 measuring	 devices.	Improvement	 in	 numerical	 condition	 of	 the	 system	would	 further	 improve	 convergence	and	accuracy	of	TSSE.	Bad	data	as	a	result	of	telemetry	problems,	measuring	malfunction,	or	malicious	attacks	are	a	concern	in	TSSE.	Since	DSSE	results	at	feeder	heads	provide	more	redundancy	 for	 measurements,	 bad	 data	 detection	 and	 identification	 in	 TSSE	 could	 be	improved	consequently.		4 The	 problem	 with	 AMI	 is	 that	 rich	 information	 exchange	 and	 hierarchical	 network	structure	increases	the	vulnerability	for	cyber‐attacks	[15],	[16].	In	particular,	energy	theft	is	a	growing	concern	in	both	developing	and	developed	countries	[17].	Many	studies	have	been	 conducted	 to	 address	 this	 problem	 based	 on	 load	 profile	 analysis.	 Analyzing	 the	behavior	of	a	load	requires	access	to	its	consumption	periodically,	which	is	also	recognized	as	 private	 information.	 Disclosing	 such	 information	 could	 be	 a	 violation	 of	 privacy	 of	customers	 [16].	 This	 means	 that	 the	 electricity	 theft	 and	 privacy	 have	 become	 two	contradictory	objectives	 in	distribution	system	management.	Developing	a	method	which	can	mitigate	the	electricity	theft	without	disclosing	load	profile	could	incentivize	utilities	to	continue	investing	in	AMI	and	prevent	a	loss	of	billions	of	dollars	in	their	revenue.	1.2 Background	of	State	Estimation	The	first	step	in	power	system	operation	is	to	monitor	the	current	state	of	the	system.	This	application	is	referred	to	as	the	state	estimation	(SE)	[6],	[13].	If	the	state	of	the	system	can	be	 calculated	based	 on	 the	 given	measurements,	 the	 system	 is	 referred	 to	 as	 observable	[18].	SE	of	 transmission	and	distribution	systems	 is	 impacted	by	 the	architecture	of	data	acquisition	from	measurements	devices	in	EMS	and	DMS.	1.2.1 Transmission	system	state	estimation	Based	on	the	estimated	system	state,	other	real‐time	system	applications	such	as	security	analysis,	 automatic	 generation	 control,	 and	 optimal	 power	 flow	 with	 contingencies	constraints,	etc.	can	be	executed.	Figure	1‐1	schematically	depicts	how	the	TSSE	program	is	related	to	other	real‐time	applications	in	a	power	systems	control	centre.	As	shown	in			5 Observability AnalysisEstimation of System StateBad Data ProcessingTopology ProcessingNetwork ModelContingency AnalysisOptimal Power FlowTransient Stability AnalysisTSSE ProgramPower System Security(Real-Time Applications)Control ActionRTURTURTURTUSCADA processorsRTUsMeasuring DeviceMeasuring DeviceMeasuring DeviceMeasuring DeviceOptimal Reactive power DispatchPower System Operation(Real-Time Applications)Performance Evaluation Capital InvestmentsPower System Planning(Off-Line Applications)GeneratorGeneratorData From RTUs in Transmission System Transferred via Optical Fiber, Microwave or Satellite Communication SystemsAVRGovernerGovernerAVRTransmission Power System	Figure	1―1	Typical	architecture	of	EMS.	Measurements	from	RTUs	are	send	to	the	SCADA	system.	TSSE	uses	these	data	and	provides	other	applications	with	required	information.	Figure	1‐1,	in	transmission	systems,	RTUs	collect	various	types	of	measurements	from	the	transmission	power	system	and	transfer	them	via	supervisory	control	and	data	acquisition		6 (SCADA)	communication	system.	The	SCADA	system	provides	the	TSSE	program	with	the	measurement	data.	The	TSSE	program	consists	of	three	tasks	including	1) Observability	analysis:	Determining	whether	the	system	is	observable	based	on	the	type	and	placement	of	measurements	and	the	given	system	topology.	2) State	 estimation:	 Generally,	measurements	 include	 line	 active	 and	 reactive	 power	flows,	 line	 current	 magnitudes,	 bus	 voltage	 magnitude,	 generator	 outputs,	 loads,	status	information	of	circuit	breakers	and	switches,	transformer	tap	positions,	and	information	of	connected	switchable	capacitor	banks.	These	measurements	may	not	always	 be	 reliable	 due	 to	 the	 error	 of	 measurements,	 telemetry	 failures,	 and	communication	noise.	These	raw	data	are	processed	by	the	state	estimator	in	order	to	filter	the	measurement	noise	and	detect	gross	errors.	3) Bad	 data	 detection	 and	 identification:	 It	 may	 occur	 that	 the	 error	 of	 some	measurements	is	significantly	more	than	typical	noise	error.	It	is	important	to	find	a	plausible	explanation	for	the	inconsistency	in	the	data.	Bad	data	may	be	caused	by	measuring	 devices’	 malfunctions	 and	 telemetry	 problems	 in	 the	 communication	network.	Several	studies	have	been	done	in	this	field	to	detect	and	identify	bad	data	[19]‐[24].	 This	 task	 identifies	 bad	 data	 and	 elimination	 of	 them	 from	 the	 state	estimation	procedure.	The	results	of	TSSE	program	are	used	to	build	the	network	model.	This	model	includes	the	system	 topology	 taking	 into	 account	 the	 status	 of	 switching	 devices	 and	 out	 of	 service		7 equipment,	 transmission	 line	 power	 flows,	 bus	 voltage	 magnitudes	 and	 angles,	 and	injection	powers.	Based	on	 this	model,	 different	 system	applications	 are	 executed.	These	applications	can	be	categorized	as	real‐time	and	offline	applications.	Real‐time	applications	contribute	to	real‐time	operation	of	the	system.	As	shown	in	Figure	1‐1,	these	applications	generate	 control	 actions	 and	 send	 them	 to	 the	 transmission	 system	 [25].	 Off‐line	applications	analyze	and	evaluate	the	performance	of	the	system	for	planning	and	design	purposes.	They	also	predict	long	term	demand	growth	for	capital	investment	decisions.	Since	 the	 introduction	 of	 state	 estimation	 in	 transmission	networks	 by	 Schweppe	 in	 late	1960s	[26]‐[28],	the	TSSE	has	benefited	from	a	broad	range	of	advances	and	developments.	In	early	stages,	as	there	were	not	enough	measurement	devices	installed	in	the	system	and	it	 was	 partially	 unobservable,	 the	 concept	 of	 observable	 islands	 was	 developed.	 An	observable	 island	 is	 an	 island	 for	 which	 all	 branch	 flows	 can	 be	 calculated	 from	 the	available	set	of	measurements	independent	of	the	values	adopted	for	reference	values.	As	more	 measuring	 devices	 have	 been	 installed	 in	 the	 last	 4	 decades,	 a	 larger	 portion	 of	transmission	 systems	 have	 become	 observable.	 Presently,	 most	 of	 the	 transmission	systems	are	observable	in	higher	voltage	levels―	e.g.,	60kV	and	higher.	The	observability	of	transmission	 systems	 in	 lower	 voltage	 levels	 is	 usually	 obtained	 using	 pseudo	measurements.	Another	 common	 problem	 in	 TSSE	 is	 ill‐conditioning	 which	 leads	 to	 inaccuracy	 and	divergence.	 In	 ill‐condition	problems,	 small	 errors	 in	 the	entries	 could	 lead	 to	 significant	error	 in	 the	 solution	 [29].	 In	 state	 estimation	 methods,	 a	 weight	 is	 assigned	 to	 each		8 measurement	 to	 represent	 its	 credibility	 [13].	Widely	different	weighting	 factors	and	 the	representation	 of	 low	 impedance	 branches	 could	 result	 in	 Ill‐conditioning	 [30].	 Adding	more	measurement	devices	could	improve	the	numerical	stability	[14].	However,	since	it	is	a	costly	solution,	utilities	prefer	various	mathematical	methods	that	have	been	developed	to	 improve	 numerical	 conditioning	 as	 a	 widely	 accepted	 practice.	 These	 methods	reformulate	TSSE	in	order	to	increase	robustness	and	achieve	convergence	in	trade	off	with	more	computation.	 In	addition,	numerical	conditioning	methods	may	 include	partitioning	the	matrices	 into	 smaller	 ones	 and	 solving	 them	 in	 an	 interconnected	 fashion	 [30],	 [31].	Recently,	 researches	 have	 been	 done	 on	 deployment	 of	 phasor	 measurement	 units	 for	monitoring	and	outage	response	purposes	[32]‐[35].	1.2.2 Distribution	system	state	estimation	Similar	to	EMS,	the	DMS	applications	depend	on	the	results	of	the	state	estimation	program	of	the	distribution	system.	Traditional	DSSE	methods	have	been	developed	with	emphasize	on	three	phase	radial	or	weakly	meshed	topologies.	One	major	problem	in	industrial	DSSE	implementation	 is	 that	 distribution	 systems	 are	 not	 observable.	 Therefore,	 DSSE	developments	 gave	 rise	 to	 more	 studies	 on	 methods	 to	 create	 pseudo	 measurements.	Pseudo	 measurements	 are	 constructed	 from	 the	 historical,	 sample	 load	 profile,	 and	customer	 billing	 information	 [36].	 Although	 this	 method	 is	 inexpensive	 and	 therefore	desirable,	it	is	typically	very	inaccurate	and	unreliable.		AMI	is	a	recent	development	in	distribution	systems.	The	AMI	measurements	complement	the	 existing	 RTUs	 measurements.	 This	 infrastructure	 includes	 smart	 meters,	 reclosers’		9 measurements,	 and	 their	 communication	 system,	 meter	 data	 management	 systems	(MDMS),	and	means	to	integrate	the	collected	data	into	application	programs	[37].	Another	recent	 development	 in	 distribution	 networks	 is	 the	 introduction	 of	 meshed	 topologies	which	impacts	some	of	the	methods	used	in	DSSE	[1].	In	radial	networks,	it	was	possible	to	trace	the	power	flow	from	loads	to	the	feeder	and	estimate	the	voltages	merely	based	on	pseudo	measurements	at	the	user	end	side.	However,	as	non‐radial	distribution	networks	are	being	designed,	DSSE	techniques	which	are	only	applicable	for	these	specific	topologies	should	be	revised	or	dismissed	[38].	Figure	1‐2	depicts	typical	architecture	of	DMS	with	recently	developed	AMI.	In	this	figure,	measuring	devices	of	reclosers	and	substations	and	smart	meters	are	the	main	sources	of	data,	and	smart	meters	are	the	dominant	one.	The	number	of	smart	meters	 installed	 in	a	typical	 distribution	 system	 is	 in	 the	 order	 of	 tens	 of	 thousands.	 Since	 certain	 utilities	operate	several	distribution	systems,	it	is	not	uncommon	for	large	utilities	to	have	a	great	number	 of	 smart	 meters.	 For	 example,	 BC	 Hydro	 has	 installed	 about	 1.8	 million	 smart	meters	 throughout	 BC,	 Canada.	 As	 shown	 in	 Figure	 1‐2,	 due	 to	 the	 limited	 available	bandwidth,	 smart	meters	 do	 not	 incessantly	 transmit	 data	 to	 the	 aggregators	 (and	 then	from	the	aggregator	to	the	control	center).	Since	synchronizing	all	smart	meters	with	each	other	 is	 considered	 too	 costly,	 smart	 meters	 practically	 do	 not	 transfer	 measurement	signals	simultaneously	[12].	For	example,	each	available	measurement	in	the	control	center	is	a	telemetered	signal	that	has	been	measured	somewhere	between	0	to	15	minutes	ago.	Figure	1‐2	shows	that	the	reclosers	use	RTUs	to	transfer	their	measurement	signals	every	4	seconds.			10  Figure	1―2	Typical	architecture	of	DMS.	Measurements	from	the	distribution	power	system	are	transferred	based	on	which	DSSE	is	executed	providing	necessary	information	for	other	applications.	During	normal	operation,	 the	DSSE	 is	 executed	e.g.,	 every	5	minutes.	However,	 there	are	several	irregular	event	triggers	in	power	control	systems,	such	as	drastic	variation	of	RTU		11 signals	and	switching,	which	modify	the	topology	of	the	system.	Whenever	an	event	trigger	occurs,	 the	DSSE	must	be	executed	again.	Therefore,	DSSE	will	not	always	be	executed	at	regular	intervals.	The	 smart	 meters	 measuring	 resolution	 is	 lower	 than	 the	 rate	 of	 DSSE	 execution.	Therefore,	 for	an	available	signal	which	 is,	 for	example,	measured	10	minutes	before,	 the	error	could	be	substantial	as	the	load	may	have	considerably	varied	since	it	was	measured.	This	 error,	 if	 ignored,	 could	 decrease	 the	 accuracy	 of	 state	 DSSE	 and	 negatively	 impact	other	applications	which	are	based	on	the	estimated	system	state.		Figure	 1‐2	 shows	 real‐time	 and	 offline	 applications	 using	 the	 results	 of	 DSSE.	 Similar	 to	EMS,	 real‐time	 applications	 issue	 control	 actions	 and	 send	 them	 to	 the	 system.	 Offline	applications	are	exemplified	by	capital	investment	in	Figure	1‐2.	1.2.3 Electricity	theft	detection	With	the	AMI,	not	only	can	people	read	the	meter	remotely,	but	also	implement	customized	control	 and	 demand	 response	 methods	 [39].	 However,	 rich	 information	 exchange	 and	hierarchical	 network	 structure	 in	AMI	 expands	 the	 attack	 surface	 for	metering	 to	 public	networks	and	promotes	vulnerability	for	cyber‐attacks	[40],	[41].	There	are	different	kinds	of	attackers	to	violate	the	AMI.	Attackers	 in	AMI	can	be	classified	as	four	categories	[16];	Curious	 attackers	 that	 are	 only	 interested	 in	 the	 activity	 of	 their	 neighbors,	 greedy	attackers	that	want	to	crack	the	AMI	in	order	to	steal	electricity,	malicious	eavesdroppers		attackers	that	collect	data	for	vicious	purposes	such	as	house	breaking,	and	active	attackers	that	aim	to	compromise	power	systems	to	launch	large‐scale	terrorist	attacks.		12 Among	 various	 violation	 of	 AMI,	 energy	 theft	 in	 particular	 is	 a	 growing	 field	 in	 both	developing	and	developed	countries	 [16].	A	World	Bank	report	 shows	 that	up	 to	50%	of	electricity	in	developing	countries	is	supplied	through	theft	[42].	Each	year,	over	6	billion	dollars	 are	 lost	 in	 the	 United	 States	 as	 a	 result	 of	 energy	 theft	 [43].	 In	 2009,	 the	 FBI	reported	an	organized	energy	 theft	attempt	 that	may	have	cost	up	 to	400	million	dollars	annually	for	a	utility	after	installing	AMI	[16].	In	Canada,	BC	Hydro	reported	a	100	million	dollars	of	losses	every	year	[44].	In	India	and	Brazil,	utility	companies	incur	losses	around	$4.5	billion	and	$5	billion,	respectively	[16].	In	Netherlands,	non‐technical	loss	represents	about	23%	of	total	loss	with	an	additional	loss	of	about	1200	GWh/year	for	illegal	use	[45].	After	installing	new	meters	at	5%	of	the	points	of	delivery	in	Italy	to	check	power	usage	of	downstream	customer	meters,	the	theft	detection	increased	up	to	50%.	Since	electricity	 theft	has	greatly	 impacted	the	revenue	of	utilities,	 some	companies	have	started	to	invest	on	implementation	of	methods	to	counter	this	problem.	1.3 State‐of‐the‐Art	Research	1.3.1 Distribution	system	state	estimation	Researchers	have	been	working	on	enhancing	the	DSSE	as	many	advance	technologies	are	being	 deployed	 in	 distribution	 systems.	 Baran	 and	 Kelly	 proposed	 a	 DSSE	 technique	 for	real‐time	 monitoring	 of	 the	 distribution	 system	 based	 on	 the	 Weighted	 Least	 Squares	(WLS)	 approach	 and	 uses	 a	 three‐phase	 node	 voltage	 formulation	 [46].	 Branch‐current	based	DSSE	was	developed	in	[47]	and	[48].	Similarly,	different	methods	were	proposed	to		13 solve	 the	 DSSE	 for	 unbalanced	 and/or	 asymmetric	 systems	 [47],	 [49],	 [50]	 and	 radial	networks	 [51]‐[53].	 However,	 introduction	 of	 meshed	 topologies	 in	 future	 smart	distribution	networks	renders	some	of	these	methods	inapplicable	[1].	Admittance	matrix	based	methods	is	introduced	in	[54]	and	improved	in	[55].	Therrion	et	al.	proposed	a	DSSE	which	unifies	power	flow	and	short‐circuit	calculation	algorithms	to	achieve	a	unified	DSSE	with	desirable	numerical	characteristics	using	modified	augmented	matrices	[56].	 1.3.1.1 DSSE based on AMI measurements At	this	point,	few	studies	focused	on	the	DSSE	based	on	smart	meters	can	be	found	in	the	literature.	 In	 [57],	 a	 DSSE	 method	 based	 on	 synchronized	 AMI	 measurements	 is	established.	 Studies	 in	 [58]	 show	 that	 smart	 meters	 can	 improve	 the	 accuracy	 of	 DSSE	assuming	that,	similarly	to	transmission	networks,	the	measurements	of	smart	meters	are	reasonably	 synchronized.	 However,	 treating	 the	 signals	 as	 if	 they	 are	 all	 synchronized	could	reduce	the	quality	of	DSSE	since	the	time	difference	between	the	measurements	may	be	 significant	 in	 practice.	 To	 take	 the	 asynchronicity	 of	 smart	meters	 into	 account,	 [59]	assumes	available	smart	meter	measurements	have	two	levels	of	error	(2%	or	10%),	while	[11]	considers	all	smart	meters	to	have	the	same	error	(10%).	However,	the	approaches	in	[59]	and	[11]	do	not	take	into	account	the	historical	short‐term	load	variation	patterns	nor	the	length	of	time	that	has	passed	between	updating	of	the	measurement	and	the	execution	of	 DSSE.	 Alternative	 approach	 includes	 modeling	 pseudo	 loads	 using	 a	 neural	 network	(NN)	 in	 for	 DSSE	 applications	 [60].	 Despite	 the	 contributions	 to	 modeling	 pseudo	measurements,	 the	 problem	 with	 this	 method	 is	 that	 the	 high	 uncertainty	 and	 poor	accuracy	of	the	generated	pseudo	measurements	for	real‐time	applications.			14 1.3.2 Transmission	system	state	estimation	There	have	been	numerous	studies	on	TSSE	over	the	last	four	decades.	The	computational	burden	of	traditional	TSSE	is	significantly	reduced	by	fast	decoupled	TSSE	in	[31].	In	order	to	analyze	the	observability	of	the	system,	topological	and	numerical	analysis	methods	are	developed	[61]‐[64].	Based	on	these	methods,	in	order	to	ensure	full	network	observability,	an	optimal	measurement	placement	method	is	proposed	in	[18].	Various	studies	have	tried	to	 deal	 with	 the	 numerical	 ill‐conditioning	 of	 TSSE.	 Measurements	 which	 are	 based	 on	topological	 information,	 such	 as	 zero	 injection	 when	 there	 is	 no	 load	 at	 a	 bus	 or	 zero	current	 for	 open	 circiuts,	 are	 referred	 to	 as	 virtual	 measurements.	 Authors	 of	 [65]	proposed	a	method	that	improved	the	numerical	conditioning	by	formulating	normal	TSSE	equations	 in	 combination	 with	 the	 virtual	 measurements	 that	 are	 considered	 as	constraints.	 By	 separating	 the	 residuals	 as	 independent	 variables,	 [66]	 proposed	 a	numerically	 stable	 method	 for	 TSSE,	 which	 is	 still	 practiced	 in	 state	 estimation	 of	tranmission	 and	 distributions	 systems.	 In	 [67],	 orthogonal	 transformation	 method	 is	applied	in	order	to	improve	the	numerical	condition	of	TSSE,	which	is	widely	accepted	in	industry.	Later,	it	is	shown	that	orthogonal	method	in	hybrid	with	normal	equations	would	significantly	 improve	 the	 numerical	 stability	 and	 solution	 accuracy	 compared	 against	alternatives,	such	as	the	Peters‐Wilkinson	and	Hachtel	methods	[68].	However,	this	method	is	computationally	expensive.	Normalized	residuals	were	used	in	order	to	develop	a	basic	bad	 data	 identification	 method	 in	 TSSE	 [24].	 To	 evaluate	 the	 quality	 of	 TSSE	 results,	hypothesis	test	is	developed	to	check	the	solution	and	detect	the	existence	of	bad	data	[69].		15 Later,	 in	 order	 to	 identify	multiple	bad	data	 in	measurements,	 [70]	proposed	a	bad	data	identification	test	as	an	optimization	problem.	1.3.2.1 Enhancing TSSE using the results of DSSE The	possibility	of	using	DSSE	for	enhancing	TSSE	has	yet	been	fully	exploited.	Historically,	utilities	 developed	 energy	 management	 systems	 and	 distribution	 management	 systems	separately,	 and	 TSSE	 and	 DSSE	 have	 often	 been	 pursued	 by	 independent	 departments.	Moreover,	until	recently,	the	results	of	DSSE	were	unavailable	(due	to	unobservability)	or	too	inaccurate	to	be	considered	useful.	Consequently,	incorporating	these	results	into	TSSE	was	 unappealing.	 To	 the	 best	 of	 the	 authors’	 knowledge,	 only	 a	 few	 studies	 on	relating/combining	TSSE	and	DSSE	are	available	in	the	literature.	In	[71],	a	SE	algorithm	is	proposed	for	observable	transmission	systems	and	unobservable	distribution	systems.	The	method	is	implemented	in	two	stages.	Typical	SE	is	first	executed	at	the	transmission	level,	followed	 by	 an	 external	 estimation	 of	 the	 unobservable	 distribution	 systems.	 Therefore,	TSSE	is	not	impacted	by	the	DSSE	results.	Reference	[72]	proposes	a	global	SE	framework	to	 simultaneously	 perform	 SE	 for	 transmission	 and	 distribution	 systems.	 Despite	 the	attempt	of	proposing	a	comprehensive	method,	the	primary	problem	with	this	approach	is	that,	 realistically,	 each	 of	 the	 distribution	 systems	 connected	 to	 the	 transmission	 system	has	 thousands	 of	 buses,	 resulting	 in	 a	 very	 large	 overall	 system	 of	 equations.	 More	importantly,	 the	 commonly	used	 industrial	 SE	programs	would	need	 to	be	modified	 to	 a	great	 extent	 in	 order	 to	 allow	 iterations	 between	 DSSE	 and	 TSSE,	 which	 is	 not	 easily	accomplished	in	practice.		16 1.3.3 Electricity	theft	detection	Several	methods	to	mitigate	energy	theft	have	been	proposed	recently.	The	first	category	of	these	methods	 is	based	on	classification	 techniques.	The	 idea	 is	 to	 study	 the	 load	profile	over	 a	 period	 of	 time	 and	 extract	 patterns	 to	 distinguish	 abnormal	 energy	 usage	 form	normal	energy	usage	[73]‐[83].	Machine	learning	and	data	mining	technologies	are	used	to	generate	 a	 good	 classifier	 based	 on	 sample	 data.	 The	 problem	 is	 that	 in	many	 practical	situations,	 an	 example	 of	 the	 attack	 class	 is	 not	 obtainable	 [16].	 Moreover,	 most	 of	 the	methods	in	this	category	are	based	on	the	assumption	that	the	attackers	are	not	adaptive	and	 do	 not	 try	 to	 evade	 the	 detection	mechanisms,	 leading	 to	missing	 intelligent	 energy	theft.	As	a	result,	detection	rate	can	be	affected	significantly	[16].	The	second	category	for	electricity	 theft	 is	 state‐based	 methods.	 In	 these	 methods,	 the	 goal	 is	 to	 detect	 theft	 by	monitoring	 the	state	of	 the	system.	This	monitoring	could	be	provided	by	AMIs,	wireless	networks,	 mutual	 inspection,	 etc.	 in	 combination	 with	modeling	 and	 statistical	 methods	[15],	[17].	The	main	problem	with	these	methods	is	the	cost	of	extra	investment	by	utility	companies,	 which	 includes	 measurement	 device	 costs,	 software	 costs,	 system	implementation	costs	and	training	costs.	Third	category	of	methods	for	electricity	theft	is	game	 theory	 based.	 This	 is	 a	 relatively	 recent	 technique	 for	 electricity	 theft.	 In	 the	developed	methods,	a	tariff	and	investment	strategy	is	proposed	from	a	game	theory	point	of	view;	assuming	that	involving	actors	are	rational.	These	methods	are	not	mature	yet,	but	they	provide	a	new	perspective	to	recognize	electricity	theft	[84],	[85].	Privacy	 preservation	 is	 one	 of	 the	 main	 challenges	 in	 practical	 implementation	 of	 the	developed	 theft	 identification	 methods.	 Using	 information	 about	 the	 behavior	 of	 a	 load,		17 including	 load	profiles,	 is	 recognized	 as	 a	 violation	 of	 privacy,	 and	would	 raise	 concerns	[16].	 In	particular,	 the	private	 information	of	 customers	may	be	 sold	 to	 third	parties	 for	marketing	or	to	insurance	companies.	It	could	also	be	used	for	criminal	activities;	robbers	could	 analyze	 the	 pattern	 of	 consumption	 by	 a	 potential	 victim	 to	 deduce	 their	 daily	activities	 and	 behavior.	 The	 consumption	 pattern	 could	 also	 be	 used	 to	 determine	household	 routines	 and	 vacancy	 times.	Many	 researchers	 have	 called	 for	 legislators	 and	authorities	 to	address	 this	 threat	 [16].	This	means	 that	electricity	 theft	and	privacy	have	become	two	contradictory	notions	in	distribution	system	management.	1.4 Research	Objectives	and	Anticipated	Impacts	Objective	1:	DSSE	based	on	AMI	measurements	This	 first	 objective	 of	 this	 thesis	 addresses	 the	 asynchronicity	 problem	 of	 AMI	measurements	 in	 DSSE.	 The	 proposed	 method	 should	 improve	 the	 accuracy	 of	 DSSE	 in	comparison	to	the	traditional	method	which	treats	the	non‐synchronized	measurements	as	synchronized.	Comparison	of	traditional	and	the	proposed	DSSE	on	different	case	studies	should	 be	 delivered.	 Since	 the	 measurement	 noise	 is	 a	 random	 phenomenon,	 the	performance	of	the	proposed	method	should	be	tested	different	measurement	noise,	and	it	should	consistently	outperform	the	traditional	DSSE.	The	observability	analysis	should	be	presented	 in	 order	 to	 explain	 whether	 or	 not	 the	 AMI	 measurements	 will	 result	 in	 an	observable	system.			18 The	 proposed	 algorithm	 should	 require	 minimum	 modifications	 to	 the	 existing	 DSSE	algorithms	 in	 order	 to	be	 appealing	 for	practical	 implementation	and	usage	 in	 industrial	applications	and	commercial	packages.	Moreover,	the	proposed	method	should	not	depend	on	any	specific	distribution	system	topology,	and	it	should	be	effective	for	both	radial	and	meshed	distribution	systems.		Objective	2:	Enhancing	TSSE	using	the	results	of	DSSE		As	 the	 second	 objective,	 this	 thesis	 presents	 an	 innovative	 and	practical	 framework	 that	utilizes	the	estimated	feeder	heads	power	flows	and	bus	voltage	magnitudes	as	equivalent	power	 injection	 and	 voltage	 magnitude	 measurements	 in	 transmission	 systems.	Specifically,	the	feeder	heads	power	flows	and	voltage	magnitudes	calculated	by	the	DSSE	algorithm	(and	their	variances)	need	to	be	transmitted	to	the	TSSE	program.	Moreover,	the	estimated	 voltage	 magnitude	 at	 distribution	 substation	 is	 used	 as	 the	 PQ	 bus	 voltage	magnitude	measurement.	In	the	proposed	approach,	the	DSSE	and	TSSE	programs	should	may	 be	 executed	 independently	 using	 different	 commercial	 packages,	 making	 this	approach	practical	for	industry.	It	 is	 important	 to	 investigate	 and	 demonstrate	 how	 the	 DSSE	 results	 could	 restore	 the	observability	 of	 the	 unobservable	 parts	 of	 transmission	 network	 and	 thus,	 enable	 the	execution	of	TSSE.	The	proposed	approach	should	also	be	examined	regarding	 its	 impact	on	 the	 numerical	 conditioning	 of	 TSSE	 and	 the	 number	 of	 iterations	 that	 TSSE	 takes	 to	converge.	Moreover,	the	impact	of	the	new	approach	in	facilitating	bad	data	identification	in	 transmission	 systems	 should	 be	 shown.	 Finally,	 the	 impact	 of	 AMI‐based	 DSSE		19 measurements,	 better	 numerical	 conditioning,	 and	 higher	 rate	 of	 success	 in	 bad	 data	identification	on	accuracy	of	TSSE	should	be	presented.	Objective	3:	Electricity	theft	detection	The	goal	of	this	objective	is	to	enhance	the	DMS	by	identifying	electricity	theft	points	in	the	distribution	system	using	the	DSSE	method	developed	in	Objective	1.	Electricity	theft	at	a	load	is	modeled	as	(hypothetical)	malfunctioning	of	the	measurement	device	with	no	theft	at	that	load.	Therefore,	bad	data	identification	methods	will	find	the	measurements	which	are	either	the	results	of	truly	malfunctioning	measuring	devices	or	the	results	of	electricity	theft.	A	criterion	should	be	presented	to	distinguish	as	many	cases	of	theft	as	possible	from	truly	 malfunctioning	 devices.	 Moreover,	 the	 problem	 with	 preserving	 the	 privacy	 of	consumers	 should	 be	 considered,	 which	 implies	 that	 using	 the	 customer’s	 consumption	profile	is	not	desirable	and	should	be	avoided.			20 CHAPTER	2: DSSE	BASED	ON	NON‐SYNCHRONIZED	AMI	MEASUREMENTS		2.1 Weighted	Least	Squares	Approach	Different	state	estimation	techniques	have	been	developed	and	used	for	transmission	and	distribution	systems.	The	most	established	approach	is	the	weighted	least	squares	(WLS)	method	[13],	[86],	[87].	The	formulation	of	WLS	DSSE	is	based	on	the	linearization	of	the	relationship	 between	 measurements	 and	 state	 variables.	 The	 nonlinear	 relationship	between	the	state	vector	and	the	measured	electric	variables	can	be	stated	as	vxhz  )(  (2‐1)where z denotes	the	vector	containing	the	measurements,	 x represents	the	state	variables,	vector	function,	 )(xh relates	the	measurements	to	the	state	variables,	and v represents	the	noise	in	the	measurements.	Several	combinations	of	variables	can	be	used	to	formx .	In	this	section,	the	state	variables	are	defined	as	the	angle	and	magnitude	of	voltages	at	all	buses,	for	a	total	of	six	state	variables	per	three‐phase	bus	[81].	As	an	example,	in	a	system	with	3‐phase	buses,	the	active	and	reactive	powers	injected	at	bus	 i 	on	phase	 p 	can	be	written	as	a	function	of	state	variables	as:		  nk llpkilpkilpkilpkilkpipi BGVVP131,,,,,,,, )sin()cos(   (2‐2)	21   nk llpkilpkilpkilpkilkpipi BGVVQ131,,,,,,,, )cos()sin(   (2‐3)wheren is	the	number	of	3‐phase	buses,	and	 BGY j 	 is	the	system	admittance	matrix.	Equations	(2‐2)	and	(2‐3)	are	used	to	construct )(xh .	A	 power	 system	 measurement	 can	 be	 considered	 as	 a	 random	 variable	 with	 Gaussian	(normal)	probability	density	function	(PDF).	The	mean	of	this	variable	is	the	actual	value	of	the	 variable	 which	 is	 measured,	 and	 the	 standard	 deviation	 of	 the	 random	 variable	represents	the	error	of	the	measurement.	The	Gaussian	PDF	of	a	measurement	is	defined	as	22121)(   zezf  (2‐4)where	 	is	the	standard	deviation	and	  	is	the	mean.	This	function	is	also	referred	to	as	the	likelihood	function	[13].	For	a	system,	the	joint	PDF	(which	represents	the	probability	function	 of	 m 	 independent	measurements	 with	 Gaussian	 PDF)	 can	 be	 expressed	 as	 the	product	of	individual	PDFs	(assuming	that	each	measurement	is	independent	of	the	rest)	as	[13]	)()...()()( 21 mzfzfzff z  (2‐5)where	 iz 	is	the	 i th	measurement	and	 ],...,,[ 21 mT zzzz .	For	a	given	set	of	measurements	and	their	corresponding	standard	deviation,	the	likelihood	function	(joint	PDF)	will	attain	its	peak	when	the	unknown	mean	values	are	chosen	closest	to	their	true	values.	Thus,	the	goal	 is	to	maximize	the	likelihood	function	by	assigning	mean	values	which	are	closest	to		22 the	measured	values.	The	mean	value	for	a	measured	electric	variable	is	a	function	of	state	variables,	which	can	be	expressed	as		)(),...,,( 21 xinii hxxxh  . (2‐6)Based	on	 the	assumption	 that	 the	mean	of	 the	measurement	noise	 is	 zero,	 the	explained	process	is	formulated	as		 miiiii mzff1m1i2log2log2 21-))(log(max)(maxzz. (2‐7)In	(2‐7),	without	loss	of	generality,	the	log	function	is	used	to	simplify	the	equations.	Since	the	last	two	terms	in	(2‐7)	are	constant,	maximizing )(zf is	equivalent	to	  m1i2)(miniiizJ x  (2‐8)Accordingly,	the	objective	function	of	the	minimization	problem	is	rewritten	as	)]([)]([))(()(11212xhzRxhzRxx Tmi iiiimi iii hzzJ  (2‐9)where	  22221 ,...,, mdiagR  .	This	means	 that	 the	 inverse	of	 the	variance	weigh	each	measurement,	 i.e.	measurements	with	larger	variance	have	a	lesser	impact	on	the	optimization	process.			23 The	optimality	condition	is	satisfied	at	the	minimum	point	of	 )(xJ .	Therefore,	0)]([)()()( 1   xhzRxHxxx TJg  (2‐10)where	  xxhxH )()( 	is	the	measurement	Jacobian.	To	 solve	 (2‐10),	 the	 Gauss‐Newton	 method	 is	 employed	 which	 iteratively	 calculates	 x 	using	)()]([ 11 kkkk xgxGxx    (2‐11)where	 )()()()( 1 xHRxHxxxG  Tg 	 is	referred	to	as	the	gain	matrix,	and	the	superscript	k 	 is	 the	 iteration	 number.	 Various	 considerations	 such	 as	 equality	 constraints	 for	 zero	injection	points	could	be	added	to	the	equations	above	as	explained	in	[81].	Generally,	 the	 vector	 z 	 includes	 the	 measurements	 at	 different	 substations,	 buses,	 and	branches,	 i.e.,	 bus	 voltage	magnitudes,	 active	 and	 reactive	 power	 injections,	 transmitted	active	and	reactive	powers,	and	line	currents.	These	measurements	should	be	sufficient	to	ensure	that	there	is	a	unique	solution	of x .	If	there	are	enough	measurements	to	uniquely	determinex ,	the	network	is	referred	to	as	observable.		In	 traditional	 distribution	 systems,	 the	 network	 is	 typically	 unobservable	 due	 to	 an	insufficient	 number	 of	 measurements.	 Therefore,	 inaccurate	 and/or	 unreliable	 pseudo	measurements	 are	 often	 employed	 in	 DSSE	 making	 its	 solution	 also	 inaccurate	 and/or		24 unreliable.	In	smart	distribution	systems,	active	and	reactive	powers	of	all	the	PQ	buses	are	measured	 by	 smart	meters.	 In	 addition,	 voltage	 angle	 is	 typically	 assigned	 at	 the	 feeder	head,	which	is	analogous	to	the	slack	bus	in	transmission	networks.	Hence,	similarly	to	the	case	of	power	flow	problem	in	transmission	systems	with	known	injection	powers,	smart	distribution	 networks	 can	 be	 considered	 as	 observable.	 Distribution	 systems	 commonly	contain	 voltage	 magnitude	 and	 power	 measurements	 in	 substations	 and	 a	 number	 of	reclosers	along	the	feeders.	Those	measurements	provide	some	redundancy	to	the	system	measurements.	2.2 State	Estimation	Based	on	Non‐synchronized	Smart	Meters	Typically,	 smart	 meter	 are	 assumed	 to	 update	 their	 measured	 electric	 variables	 signal	every	 15	 minutes	 [88],	 [89].	 These	 measurements,	 which	 are	 not	 updated	 during	 this	interval,	are	reffered	to	as	out‐of‐date	(OD)	signals.	Referring	to	 0(2‐9),	the	credibility	of	a	measurement	 in	 WLS‐based	 DSSE	 is	 represented	 by	 the	 inverse	 of	 its	 variance.	 Since	consumer	 loads	 change	 constantly,	 OD	 signals	 are	 expected	 to	 have	 additional	 error.	Specifically,	if	a	smart	meter	has	measured	the	power	and	voltage	of	a	load	some	minutes	ago,	 the	 total	error	of	 the	available	measurement	signal	 can	be	considered	 to	come	 from	two	sources:	i)	the	smart	meter’s	own	intrinsic	imprecision;	and	ii)	the	result	of	variation	of	load	consumption	since	it	was	measured.	As	the	measurement	sample	is	updated	every	15	minutes,	until	 the	new	sample	signal	 is	 transferred,	 the	available	OD	signal	will	not	 track	the	changes	of	 the	 load	within	the	15	minute	 interval.	 In	 the	proposed	method,	 the	error	caused	by	the	short‐term	load	variation	is	modeled	as	a	random	variable.	In	the	following,	the	properties	of	this	random	variable	are	studied.		25 2.2.1 Load	variation	modeling	The	error	of	an	OD	signal	at	any	point	is	thus	assumed	to	consist	of	two	components:	the	error	 of	 the	 measurement	 device;	 and	 the	 error	 resulting	 from	 the	 short‐term	 load	variation.	Therefore,	the	total	error	itotale 	of	the	 i th	meter	could	be	represented	as	iii ODmtotal eee   (2‐12)where	 ime 	 is	 the	 device’s	 measurement	 error,	 and	 iODe 	 is	 the	 load	 variation	 error.	 The	objective	of	 this	section	 is	 to	model	 the	total	error	 in	a	manner	which	 is	compatible	with	the	WLS‐based	DSSE	process	presented	in	Section	2.1.		The	first	component	of	the	total	error	has	been	well	studied.	It	was	concluded	to	roughly	follow	a	Gaussian	PDF	with	zero	mean	and	a	specific	standard	deviation	[13].	Generally,	the	precision	 of	 a	 measuring	 device	 is	 given	 by	 the	 manufacturer	 as	 determined	 through	several	tests.	The	 second	component	of	 the	 total	 error	 is	 the	variation	of	 the	 load	 from	 the	moment	 it	was	 sampled	 to	 the	 time	 when	 DSSE	 is	 executed.	 In	 some	 studies,	 the	 consumption	 of	residual	loads	has	been	studied,	and	it	is	suggested	that	the	load	can	be	modeled	as	normal,	log‐normal,	and	gamma	distributions	[90].	However,	to	the	best	of	our	knowledge,	no	study	has	 been	 conducted	 to	 assess	 the	 statistical	 distribution	 of	 short‐term	 distribution‐level	variation	of	load	consumption	over	different	points	of	a	day.			26 To	 do	 so,	 the	 historical	 data	 from	 a	 sampled	 building	 consumption	 at	 The	 University	 of	British	Columbia	 campus	has	been	analyzed	 [91].	 Its	active	power	 consumption	 sampled	every	15	minutes	over	more	than	1	month	is	represented	in	Figure	0 200 400 600 800 1000405060708090100110120130Time (hour)Demand (kW)	Figure	2―1	Historical	active	power	consumption	profile	of	a	building	at	The	University	of	British	Columbia.		2‐1.	The	objective	is	to	verify	whether	the	variation	of	the	load	from	any	time	of	every	day	to	 the	next	sampling	point	 follows	normal	distribution	characteristic.	 If	 so,	 then	 the	 total	error	could	be	modeled	as	the	combination	of	two	normally	distributed	random	variables	which	is	also	a	normal	variable	[92]	and	could	be	incorporated	in	the	DSSE	process	as	iiii ODODmtotal    (2‐13)  	27 222iii ODmtotal    (2‐14)  where	itotal ,	 im ,	and	 iOD are	the	mean	values	of	the	total,	measurement	device,	and	OD	signal	 errors;	 and	 itotal ,	 im ,	 and	 iOD 	 are	 the	 standard	 deviations	 of	 the	 total,	measurement	device,	and	OD	signal	errors,	respectively.	Therefore,	some	tests	are	required	to	evaluate	the	normality	of	the	load	variation.	There	 are	 a	 handful	 of	 fitness	 tests	 to	 check	 whether	 or	 not	 a	 set	 of	 data	 is	 normally	distributed	 [93]‐[96].	Two	of	 the	most	 reliable	 tests	are	 the	Shapiro‐Wilk	and	Anderson‐Darling	tests	[95],	[97],	[98],	which	are	implemented	in	this	study.	Presuming	that	a	signal	is	 normally	 distributed	 (this	 assumption	 is	 referred	 as	 the	 ‘null	 hypothesis’),	 each	 test	attempts	to	calculate	an	indicator	which	is	called	the	p‐value	to	evaluate	the	normality	of	a	signal.	 If	 the	p‐value	 is	 less	 than	 a	 threshold	 (0.05	based	on	 [99]),	 the	null	 hypothesis	 is	refuted	meaning	 that	 the	 signal	 is	 not	 normally	 distributed.	 To	 apply	 these	 tests	 in	 our	study,	we	 first	 calculate	 the	 load	 variation	 for	 any	 time	 of	 the	 day	 for	 several	 days.	 For	instance,	Figure	2‐2	shows	the	signal	of	load	variation	(active	power)	from	2pm	to	2:15pm	for	38	days.	Once	these	signals	are	prepared,	they	are	fed	to	the	tests	to	see	whether	they	follow	 a	 normally	 distributed	 pattern.	 The	 results	 of	 the	 normality	 tests	 for	 the	 data	 in	Figure	2‐2	are	presented	in	Table	2‐1.	As	one	can	see,	the	p‐value	for	both	tests	is	clearly	above	0.05.	This	means	that	the	null	hypothesis	is	not	refuted	and	that	it	is	reasonable	to	assume	 that	 the	 signal	 has	 a	 normal	 distribution	 characteristic.	 This	 table	 includes	 the	same	analysis	for	reactive	power.	The	p‐value	for	other	points	of	the	day	are	also	found	to	be	well	above	the	0.05	threshold,	which	means	that	the	second	component	of	the	error	in			28 0 5 10 15 20 25 30 35 40-4-3-2-1012345DayLoad variation (kW)	Figure	2―2	Power	consumption	variation	from	2:00PM	to	2:15PM	based	on	the	historical	data	depicted	in	Figure	2‐1.	Table	2―1	Normality	Evaluation	of	Figure	2‐2	Data	Normality	Evaluation	Test	 p‐value	Statistical	Analysis	on	Active	Power	Anderson‐Darling	Test	 0.2282	Shapiro‐Wilk	Test	 0.3367	Statistical	Analysis	on	Reactive	Power	Anderson‐Darling	Test	 0.2495	Shapiro‐Wilk	Test	 0.3218		(2‐12)	(i.e.	resulting	from	the	signal	being	OD)	can	be	modeled	as	a	random	variable	with	normal	distribution.	Thereafter,	 the	mean	and	standard	deviation	at	any	point	of	 the	day		29 and	for	any	time	distance	(between	0	and15	minutes	for	this	case	study)	can	be	calculated	accordingly.	 It	 is	worth	mentioning	 that	 the	mean	 value	 of	 load	 variation	 as	 depicted	 in	Figure	 2‐2	 as	 an	 instance	 is	 very	 close	 to	 zero.	 This	 shows	 that	 in	 fact,	 it	 is	 mainly	 the	adjustment	of	variance	which	is	changing	the	results.	The	load	historical	data	of	vast	majority	of	utilities	in	distribution	system	is	not	available	at	any	sampling	rate	higher	than	the	resolution	of	smart	meters.	Without	loss	of	generality,	it	can	be	assumed	that	the	mean	and	standard	deviation	vary	linearly	in	the	interval	between	two	consecutive	events	of	updating	measurements.	For	an	OD	signal,	the	time	passed	since	the	 last	 update	 is	 represented	 by	 the	 variable	 LUt .	 If	 smart	meters	 update	 their	 signals	every	15	minutes,	 LUt 	can	be	anywhere	between	0	to	900	seconds.	The	variable	 LUt 	is	reset	to	 0	 every	 time	 the	 signal	 is	 updated.	 As	 a	 result,	 the	 mean	 and	 standard	 deviation	employed	in	the	DSSE	process	are	calculated	as	)0())0()(()( maxmaxiiitODODLUODLULULUOD tttt. (2‐15)  )0())0()(()(22max2max2iiiiODODLUODLULULUOD tttt. (2‐16)  where	 )0(iOD ,	 )( maxLUOD ti , )0(iOD ,	 and	 )( maxLUOD ti 	 	 are	 the	 calculated	 mean	 and	standard	 deviation	 of	 load	 variation	 of	 the	 first	 and	 the	 last	 points	 of	 the	 interval,	 and	maxLUt 	is	the	interval	between	two	consecutive	updating	events.	Therefore,	by	providing	the		30 mean	 and	 standard	 deviation	 at	 updating	 points,	 DSSE	 could	 be	 performed	 at	 any	 point	within	the	interval.	2.2.2 Implementation	of	load	variation	modeling	technique	To	implement	the	proposed	technique	in	existing	DSSE	algorithms,	the	mean	and	standard	deviation	 of	 the	 load	 variation	 at	 each	measurement	 updating	 point	 based	 on	 historical	data	are	first	calculated.	Every	time	that	the	DSSE	process	is	being	executed,	the	variance	of	each	 measurement	 is	 updated	 by	 evaluating	 (2‐14)	 and	 updating	 2i s	 in	 the	 Rmatrix	accordingly.	 As	 mentioned,	 in	 (2‐7),	 it	 is	 assumed	 that	 the	 mean	 value	 of	 the	 noise	measurement	error	is	zero.	However,	in	the	proposed	method,	the	mean	value	of	the	total	error	is	calculated	based	on	(2‐13)	and	(2‐15).	Given	that	the	measurement	is	modeled	as	ΔLvxhz  )(  (2‐17)where	 ΔL 	 is	 the	 load	 variation	 vector	 as	 a	 result	 of	 OD	 measurements,	 the	 objective	function	in	(2‐7)	is	modified	to	 miiiODii mzffi1m1i2log2log2 21-))(log(max)(maxzz. (2‐18)where	 iOD 	 is	 the	mean	of	 load	variation	calculated	in	(2‐15).	Therefore,	 the	mean	of	the	error	 must	 be	 subtracted	 from	 the	 measured	 value	 for	 each	 measurement,	 and	 (2‐10)	would	be	modified	to		31 0)]([)()()( 1   xhμzRxHxxx TJg  (2‐19)whereμ 	is	a	vector	containing	the	means	of	all	the	load	variation	errors.		A	flowchart	summarizing	the	proposed	approach	is	shown	in	Figure	2‐3.	It	should	be	noted	that	 the	proposed	method	 is	 general	 in	 the	 sense	 that	 it	 can	be	 incorporated	 in	all	WLS‐based	 DSSE	 algorithms,	 independently	 of	 their	 sets	 of	 state	 variables	 or	 their	 ability	 to	model	radial	and/or	meshed	topologies.		Figure	2―3	DSSE	process	addressing	asynchronicity	of	measurements.		32 It	is	important	to	mention	that	the	DSSE	is	an	iterative	method	which	requires	calculation	of	 active	and	 reactive	powers	 from	magnitude	and	angles	of	 the	voltages,	 calculating	 the	Jacobian	 for	 the	next	 iteration	 to	 solve	 the	WLS	optimization	problem,	 and	matrix	based	calculations	 (including	 factorization	 of	 the	 gain	 matrix)	 to	 obtain	 the	 solution.	 The	proposed	method	adds	a	very	simple	set	of	operations	(2‐13)‐(2‐16),	calculated	only	at	the	beginning	of	the	DSSE	process	(not	in	each	iteration),	which	is	numerically	insignificant	in	comparison	to	the	overall	DSSE	iterative	solution	of	the	WLS	problem	as	explained	above.	2.3 Simulation	of	Case	Studies	In	this	section,	the	effectiveness	of	the	proposed	method	is	examined	using	the	IEEE	13‐bus	and	 123‐bus	 distribution	 test	 systems	 [100].	 Each	 case	 study	 includes	 the	 execution	 of	DSSE	 every	 200	 seconds	 over	 the	 course	 of	 24	 hours,	 during	 which	 the	 loads	 change	following	a	pattern	similar	to	Figure	2‐4.	Every	200	seconds,	initially,	a	3‐phase	unbalanced	0 5 10 15 200.50.550.60.650.70.750.80.850.90.951Time (hour)Load (p.u.)	Figure	2―4	A	typical	load	profile	for	24	hours	in	pu	with	respect	to	the	peak	value	based	on	the	historical	data	recorded	at	The	University	of	British	Columbia.		33 	power	 flow	 is	 performed	 to	 obtain	 a	 reference	 solution	 in	 accordance	 to	 the	new	 loads.	Afterwards,	 Gaussian	 noise	 is	 added	 to	 the	 actual	 value	 of	 electric	 variables	 in	 order	 to	emulate	 the	measurements’	 error.	 As	 mentioned,	 it	 is	 assumed	 that	 each	 smart	 meter’s	measurement	 signals	 are	 updates	 every	 900	 seconds.	 However,	 smart	 meters	 are	 not	synchronized.	To	simulate	asynchronous	measurements,	a	random	integer	between	0	and	900	is	assigned	to	each	smart	meter.	This	number	represents	the	time	of	first	measurement	update	for	each	smart	meter	from	the	starting	point	of	the	simulation.	At	the	starting	point	of	 the	 simulation,	 it	 is	 assumed	 that	 all	 the	 smart	 meter	 measurements	 are	 available.	Afterwards,	each	smart	meter’s	measurement	signal	 is	updated	at	the	assigned	time	to	 it.	For	 example,	 the	 assigned	 times	 for	 the	 IEEE	 13‐bus	 system	 are	 depicted	 in	 Figure	 2‐5.	Following	the		34 0 5 10 15 20 250200400600800Smart Meters' NumberTime of sampling of smart meters (sec)	Figure	2―5	Pattern	of	the	updating	time	of	each	smart	meter	measurements	installed	on	the	loads	of	the	IEEE	13‐bus	distribution	system.		first	measurement,	each	smart	meter’s	measurement	signal	is	updated	every	900	seconds.	In	this	 interval,	DSSE	uses	the	 last	updated	signal	of	 the	smart	meter	(OD	measurement).	The	 precision	 of	 power	 measurements	 is	 assumed	 to	 be	 1%	 [101].	 The	 uncertainty	 of	network	 parameters	 is	 considered	 using	 a	 uniform	 distribution	 [102].	 The	 considered	bound	of	uncertainty	is	5%.	For	comparison	purposes,	three	approaches	are	implemented.	Figure	2‐6	summarizes	the	simulation	 process.	 In	 the	 first	 approach,	 it	 is	 assumed	 that	 all	 the	 smart	 meters	transferring	data	to	the	control	center	incessantly.	This	is	an	ideal	assumption	(which	is	not	realistic),	 and	 it	 is	 only	 considered	 for	 comparison	 and	 providing	 a	 better	 insight.	 This		35 approach	with	ideal	assumption	will	be	referred	to	as	“Ideal	DSSE”.	In	the	second	and	third	approaches,	the	smart	meters	measuements	are	updated	every	900	seconds	following	the	approach	 explained	 in	 the	 previous	 paragraph.	 In	 the	 second	 approach,	 the	 traditional	DSSE	is	performed	and	the	asynchronicity	of	measurements	is	not	considered,	whereas	in	the	 third	 approach,	 the	 proposed	 method	 of	 adjusting	 the	 variance	 is	 employed.	 These	methods	 are	 referred	 to	 as	 “Traditional	 DSSE”	 and	 “Proposed	 DSSE”,	 respectively.	 After	having	computed	the	state	vector	using	each	method,	the	error	of	DSSE	can	be	calculated.	In	this	study,	the	2‐norm	error	of	DSSE	at	each	execution	is	defined	over	all	bus	voltages	as	  busnoi actiestierror _1 2)( vv  (2‐20)where	 busno _ 	 is	 the	 number	 of	 buses	 (including	 all	 single	 phase	 buses),	 estiv is	 the	estimated	voltage	of	the	 i th	bus,	and activ is	the	actual	(errorless)	value	of	 i th	bus	voltage.	Each	voltage	variable	 in	this	equation	is	a	complex	number	 in	per	unit.	The	accumulative	error	in	(2‐20)	over	the	24	hour	simulation	interval	is	referred	to	as	the	total	error	of	the	estimation.		36 Update loadsExecute power flow for the system and obtain voltages and powersAdd noise to electric variables to form the measurements of each smart meterRun traditional DSSE and calculate the errorUpdate the OD measurements of smart meters according to the updating patternUpdate the OD measurements of smart meters according to the updating patternCalculate µ and σ vectors based on Section IIIRun proposed DSSE and calculate the errorUpdate the OD measurements for ALL of the smart metersRun ideal DSSE and calculate the errorCompare the accuracy of DSSE of three methods	Figure	2―6	Simulation	process	for	the	case	studies.	As	 Figure	 2‐6	 shows,	 the	 proposed	 method	 has	 an	 additional	 block	 compared	 to	 the	traditional	 DSSE	which	 is	 explained	 in	 Section	 2.2.	 The	measurements	 of	 the	 traditional		37 DSSE	and	the	proposed	DSSE	are	identical	whereas	the	measurements	of	the	ideal	DSSE	are	always	up	to	date	as	explained.	2.3.1 IEEE	13‐bus	distribution	system	The	IEEE	13‐bus	distribution	test	system	[100]	is	used	for	the	first	case	study.	This	3‐phase	system	is	unbalanced	with	32	single‐phase	buses.	For	the	considered	system,	the	2‐norm	of	all	 errors	 in	 measurements	 of	 active	 power	 [calculated	 similarly	 to	 (2‐20)]	 at	 each	execution	of	DSSE	for	three	cases	are	shown	in	Figure	2‐7.	In	case	(a),	it	is	assumed	that	the	smart	 meters	 measurement	 signals	 are	 always	 up‐to‐date	 and	 the	 measurement	 errors	result	only	from	the	imprecision	(i.e.	noise)	of	the	measuring	devices.	In	case	(b),	the	error	due	 to	 OD	 measurements,	 which	 depends	 on	 the	 variation	 of	 the	 load	 since	 the	 last	measurement,	 is	shown.	This	portion	of	 the	error	becomes	especially	pronounced	during	increases	and	decreases	of	the	power	demand,	as	can	be	observed	in	Figure	2‐7.	Case	(c)	is	the	combination	of	cases	(a)	and	(b),	i.e.	the	summation	of	the	device	imprecision	and	OD	measurement	errors.	To	demonstrate	the	performance	of	the	proposed	method,	the	DSSE	error	of	each	method	has	been	calculated	using	(2‐20)	and	the	results	are	shown	in	Figure	2‐8.	Specifically,	the	DSSE	 error	 achieved	 by	 each	 method	 during	 the	 day	 is	 depicted	 in	 Figure	 2‐8(a).	 To	demonstrate	the	cumulative	measure	of	 these	errors,	 the	DSSE	errors	are	also	 integrated	and	the	results	are	shown	in	Figure	2‐8(b).	As	expected,	the	best	state	estimation	results			38 0 5 10 15 20 2500.020.040.060.080.10.120.140.160.180.2Time (hour)2-Norm of Total Measurement Error  	Figure	2―7	Norm	error	of	real	power	measurements:	(a)	ideal	case	‐	error	due	to	smart	meters	imprecision	only;	(b)	error	only	due	to	out‐of‐date	measurements,	and	(c)	error	due	to	smart	meters	imprecision	combined	with	error	due	to	out‐of‐date	measurements.	are	obtained	under	the	ideal	case	when	all	smart	meters	are	assumed	to	be	always	up‐to‐date	(see	Figure	2‐8,	“Ideal	DSSE”).	Additionally,	it	is	seen	in	Figure	2‐8	that	the	proposed	DSSE	method	results	in	a	noticeable	reduction	of	the	total	accumulated	error	over	the	day	to	12.89	compared	with	to	the	traditional	DSSE	method	(15.08).	It	is	also	observed	that	as	the	 load	 changes	during	 the	day	 (see	Figure	2‐4),	 the	DSSE	error	 is	 increasing	when	 the	load	 varies	 greatly,	 i.e.	 during	 the	 morning	 load	 pickup	 and	 evening	 load	 decrease.	However,	when	the	rate	of	load	variation	is	low,	the	estimation	error	is	small,	and	the			39 0 5 10 15 2000.020.040.060.080.10.120.14Time (hour)2-Norm of the error of DSSE  Ideal DSSETraditional DSSEProposed DSSE	(a)	0 5 10 15 200246810121416Time (hour)Accumulated Error of DSSE  Ideal DSSETraditional DSSEProposed DSSE	(b)	Figure	2―8	Accuracy	comparison	of	the	three	considered	methods:	(a)	DSSE	error	during	one	day;	and	(b)	accumulated	error	of	DSSE	during	one	day	for	each	method.		40 results	of	the	traditional	DSSE	and	the	proposed	DSSE	are	almost	the	same,	which	can	also	be	observed	in	Figure	2‐8(a).	As	expected,	the	DSSE	error	reproduced	in	Figure	2‐8	are	in	good	agreement	with	the	measurement	errors	depicted	in	Figure	2‐7.	While	the	traditional	DSSE	 and	 proposed	 DSSE	 have	 the	 same	 error	 in	 their	 precision,	 the	 proposed	method	treats	 the	 measurements	 differently	 by	 appropriately	 adjusting	 their	 variances,	 and	therefore	achieves	higher	accuracy.	2.3.1.1 Impact of random noise Since	the	measurement	noise	is	assumed	to	be	random,	it	can	be	expected	that	the	results	will	change	every	time	the	simulation	is	run.	To	consider	this	issue,	the	simulation	has	been	run	 10,000	 times,	 and	 the	 total	 accumulated	 over	 24	 hours	 DSSE	 error	 for	 each	 of	 the	methods	is	depicted	in	Figure	2‐9.	Therein,	it	is	observed	that	the	random	nature	of	noise	has	a	negligible	impact	on	the	total	error	of	each	method.	Additionally,	the	best,	worst,	and	mean	 performances	 for	 each	method	 are	 summarized	 in	 Table	 2‐2.	 As	 also	 observed	 in	Figure	2‐9	and	Table	2‐2,	the	impact	of	measurements	noise	is	minimal,	e.g.	the	total	DSSE	error	 of	 the	 traditional	 DSSE	 varies	 from	 14.83	 to	 15.4;	 whereas,	 the	 total	 error	 of	 the	proposed	DSSE	varies	from	12.7	to	13.16.	To	give	a	better	insight,	Table	2‐2	also	includes	the	 percentage	 of	 error	 reduction	 achieved	 by	 the	 proposed	 DSSE	 method	 over	 the	traditional	DSSE	method.	As	one	can	see,	the	proposed	approach	improves	the	accuracy	of	DSSE	by	an	average	of	13.89%.	This	suggests	that	the	approach	introduced	in	this	chapter	has	a	significant	potential	to	improve	the	quality	of	DSSE	in	distribution	systems.			41 0 2000 4000 6000 8000 10000246810121416Run NumberDSSE error accumulated in 24 hours  Ideal DSSETraditional DSSEProposed DSSE	Figure	2―9	Total	error	of	DSSE	in	24	hours	for	10000	runs.	Table	2―2	Statistical	Performance	of	DSSE	Error	for	a	Typical	Load	Profile	in	the	IEEE	13‐bus	Distribution	System		Best	DSSE	(Least	total	error)	Mean	DSSE	(Average	total	error)	Worst	DSSE	(Most	total	error)Ideal	DSSE	 2.6412	 2.7714	 2.9239	Traditional	DSSE	 14.8335	 15.0972	 15.4202	Proposed	DSSE	 12.7026	 12.9867	 13.1678	Improvement	of	DSSE	accuracy	by	the	Proposed	DSSE	compared	to	the	Traditional	DSSE	 14.37%	 13.89%	 14.61%			42 2.3.1.2 Impact of load uncertainty  The	performance	of	the	proposed	method	is	investigated	further	for	a	scenario	where	the	load	profile	has	significant	unexpected	fluctuation.	The	typical	(Scenario	I)	and	unexpected	(Scenario	II)	load	profiles	are	shown	in	Figure	2‐10.	The	historical	data	used	to	update	the	mean	and	standard	deviation	in	the	following	simulation	is	based	on	Scenario	I.	As	depicted	in	Figure	2‐10,	the	load	variation	is	significant	and	represents	a	case	where	the	day’s	load	profile	does	not	follow	the	pattern	of	historical	data.		0 5 10 15 200.40.50.60.70.80.911.1Time (hour)Load variation (p.u.)  Scenario IScenario II	Figure	2―10	Typical	and	distorted	load	profiles	for	24	hours	in	pu.	Now,	 considering	 Scenario	 II,	 the	 simulation	 is	 performed	 10000	 times,	 but	 the	 same	historical	data	as	in	the	previous	section	(Scenario	I)	is	used.	The	results	are	summarized	in		43 Table	 2‐3.	 Although	 the	 total	 error	 has	 increased	 compared	 to	 the	 scenario	 with	 a	similar/consistent	load	profile	(e.g.	the	mean	of	the	total	error	22.31	compared	to	18.95	for	the	proposed	DSSE),	the	proposed	approach	still	yields	good	improvement	of	the	accuracy	over	the	traditional	DSSE.	These	results	show	the	robustness	of	the	proposed	method.	As	can	be	seen	in	Table	2‐3,	in	scenario	II,	the	average	improvement	percentage	is	increased	compared	 to	scenario	 I,	because	 the	OD	error,	which	 is	not	considered	by	 the	 traditional	DSSE,	is	increased	as	a	result	of	more	drastic	variation	of	loads.	Table	2―3	Statistical	Performance	of	DSSE	Error	for	a	Distorted	Load	Profile	in	the	IEEE	13‐bus	Distribution	System		Best	DSSE		(Least	total	error)	Mean	DSSE	(Average	total	error)	Worst	DSSE	(Most	total	error)	Ideal	DSSE	 2.6312	 2.7653	 2.9137	Traditional	DSSE	 22.0601	 22.3125	 22.6274	Proposed	DSSE	 18.6772	 18.9518	 19.3143	Improvement	of	DSSE	accuracy	by	the	Proposed	DSSE	compared	to	the	Traditional	DSSE	 15.33%	 15.06%	 14.64%		2.3.2 IEEE	123‐bus	distribution	system	To	evaluate	 the	performance	of	 the	proposed	method	on	a	 larger	system,	 the	considered	IEEE	123‐bus	 distribution	 system	 [100]	 is	 implemented	 in	 Figure	2‐11.	 In	 this	 figure,	 to	evaluate	the	performance	of	the	proposed	method	for	non‐radial	systems,	the	lines	in	blue	are	added	to	create	a	meshed	system.	Similarly	to	the	previous	system,	DSSE	is	performed	10,000	times	for	the	load	profiles	given	in	Figure	2‐4	and	Figure	2‐10.	Table	2‐4	and	Table		44 Feeder Head	Figure	2―11	Considered	IEEE	123	distribution	systems.	2‐5	 summarize	 the	 results	 of	 DSSE	 for	 this	 system	 when	 using	 typical	 load	 profiles	(Scenario	I)	and	highly	distorted	load	profiles	(Scenario	II),	respectively.	It	can	be	seen	in	Tables	 2‐4	 and	 2‐5	 that	 in	 both	 cases,	 the	 proposed	 approach	 reduces	 the	 total	 error	compared	 to	 the	 traditional	DSSE	by	15.09%	and	15.68%,	respectively.	These	results	are	similar	 to	 those	 obtained	 for	 the	 IEEE	 13‐bus	 test	 system.	 Thus,	 the	 improvement	 of	accuracy	achieved	by	the	proposed	method	is	consistent	and	promising.			45 Table	2―4	Statistical	Performance	of	DSSE	Error	for	a	Typical	Load	Profile	in	the	IEEE	123‐bus	Distribution	System		Best	DSSE	(Least	total	error)	Mean	DSSE	(Average	total	error)	Worst	DSSE		(Most	total	error)	Ideal	DSSE	 12.3209	 12.4765	 12.9398	Traditional	DSSE	 25.5312	 25.5590	 25.6807	Proposed	DSSE	 21.5711	 21.7034	 22.0211	Improvement	of	DSSE	accuracy	by	the	Proposed	DSSE	compared	to	the	Traditional	DSSE	 15.51%	 15.09%	 14.25%		Table	2―	5	Statistical	Performance	of	DSSE	Error	for	a	Distorted	Load	Profile	in	the	IEEE	123‐bus	Distribution	System		Best	DSSE	(Least	total	error)	Mean	DSSE	(Average	total	error)	Worst	DSSE	(Most	total	error)	Ideal	SE	 12.3309	 12.4680	 12.9362	Traditional	SE	 39.1400	 39.9794	 40.6145	Proposed	SE	 33.5231	 33.7123	 33.8581	Improvement	of	DSSE	accuracy	by	the	Proposed	DSSE	compared	to	the	Traditional	DSSE	 14.33%	 15.68%	 16.64%				46 CHAPTER	3: INCORPORATION	OF	DSSE	INTO	TRANSMISSION	LEVEL	3.1 Transmission	System	State	Estimation	As	 explained	 in	 chapter	 2,	 the	 most	 established	 approach	 for	 state	 estimation	 is	 the	weighted	 least	squares	(WLS)	method.	Unlike	distribution	systems,	 transmission	systems	are	assumed	to	be	balanced.	Therefore,	only	the	positive	sequence	system	is	analyzed.	For	example,	transmission	lines	are	modeled	as	a	two	part	π	model	 in	Figure	3‐1	whereas,	 in	the	distribution	system,	they	were	modeled	with	the	imbalanced	3	phase	matrix	of	the	line	[100].	Therefore,	it	is	assumed	that,	in	transmission	systems,	phases	are	not	coupled	with	each	other	as	only	the	positive	sequence	is	considered.		  Figure	3―1	Equivalent	circuit	for	a	transmission	line.	Denoting	the	total	number	of	buses	as	 n ,	in	an	observable	distribution	system,	the	rank	of	the	 Jacobian	 matrix	 is	 32  n 	 since	 the	 feeder	 head	 bus	 has	 three	 buses	 with	 known		47 angles.	However,	 in	 transmission	distribution	 systems,	 the	 rank	of	 the	 Jacobian	matrix	 is	12  n 	since	the	transition	bus	has	one	bus	with	known	angle.	3.2 Incorporating	DSSE	Results	into	TSSE	A	 schematic	 diagram	 of	 a	 transmission	 system	 interconnected	 with	 several	 distribution	systems	 is	 displayed	 in	 Figure	 3‐2.	 From	 the	 transmission	 system	 point	 of	 view,	 the	common	 bus	 connecting	 the	 two	 systems	 is	 referred	 to	 as	 a	 PQ	 (load)	 bus.	 From	 the	distribution	system	viewpoint,	this	interconnecting	point	is	referred	to	as	the	feeder	head.	In	many	 systems,	 the	 power	 transferred	 from	 the	 transmission	 to	 the	 distribution	 level	through	 a	 given	 PQ	 bus	 is	 measured	 frequently	 (e.g.,	 every	 4	 seconds)	 using	 the	 RTUs	installed	at	 substations.	This	power	 is	used	as	an	 injection	measurement	 in	TSSE,	whose	solution	will	also	include	the	voltage	of	all	the	PQ	buses	(or	equivalently,	as	seen	from	the	distribution	side,	feeder	head	estimated	voltages).		48  Figure	3―2	Simplified	diagram	depicting	a	transmission	system	connected	to	distribution	systems	via	PQ	buses	(feeder	heads).	In	 this	 thesis,	 it	 is	 proposed	 to	 use	 the	 power	 flow	 and	 voltage	magnitude	 at	 the	 feeder	heads	 estimated	 by	 DSSE	 as	 additional	 power	 injection	 and	 voltage	 magnitude	measurements,	 respectively,	 to	 improve	 TSSE.	 As	 (2‐7)	 indicates,	 the	 noise	 of	 these	additional	 measurements	 (estimations)	 should	 be	 represented	 by	 their	 variances.	 To	calculate	these	variances,	(2‐11)	is	rewritten	as	follows:	        xhzRxHxGΔx   1T1  (3‐1)where the iteration superscript k  has been omitted for clarity. Given that the covariance matrix of the measurements is given by R , the covariance matrix xR  of the state vector is 	49              xGRxHxGRRxHxGRx 1T1T11T1   . (3‐2)The diagonal elements of  xG 1  are the variances of the estimated bus voltage magnitudes and angles of the distribution system. Hereinafter, the variance matrix of estimated voltage magnitudes of feeder head is referred to as vmfhR . Similarly to (2-1), the vector of estimated feeder head transferred active and reactive powers, denoted by PQfhz ,  is related to the state vector through   xyz PQfh . (3‐3)Next, defining     xxyxY , (3-3) can be linearized as   xxYz  PQfh . (3‐4)The covariance matrix of the estimated powers at the feeder head becomes          xYxGxYxYRxYR TTx 1PQfh . (3‐5)Thereafter, in the proposed TSSE, the diagonal elements of PQfhR  will be used as the variances of the feeder head active and reactive powers calculated based on DSSE. Let fhz  denote a vector consisting of estimated feeder head voltage magnitudes and calculated feeder head powers ( PQfhz ), and  PQfhvmfvfh RRR ,diag . According to the proposed approach in 	50 this chapter, the objective function in (2-9) becomes           xhzRxhzxhxhzzRRxhxhzztotaltotaltotalTtotaltotalfhmfhmfhmTfhmfhmJ1100 (3‐6)where	 the	 subscript	 “m”	 refers	 to	 the	 transmission	 system	measurements,	 the	 subscript	“fh”	refers	to	the	feeder	head	voltage	magnitudes	and	powers	which	are	calculated	based	on	the	DSSE	results,	and	the	subscript	“total”	refers	to	the	combination	of	the	transmission	system	 measurements	 and	 the	 additional	 measurements	 stemming	 from	 DSSE.	 The	solution	to	the	proposed	TSSE	objective	function	in	(3‐6)	is	similar	to	the	method	used	in	(2‐10)	 and	 (2‐11)	 for	 DSSE.	 The	 potential	 improvements	 from	 solving	 the	 objective	function	(3‐6)	are	discussed	below.	3.3 Areas	of	Impact	in	TSSE	3.3.1 Observability	In	 many	 practical	 power	 systems,	 the	 transmission	 grid	 may	 be	 partially	 unobservable,	particularly	 on	 the	 lower	 voltages	due	 to	 insufficient	measurements	 at	 those	 levels.	 This	problem	 is	 often	mitigated	 by	 electrical	 companies	 by	 utilizing	 pseudo	measurements	 at	the	expense	of	a	decrement	in	the	accuracy	and	reliability	of	the	results.	Generally,	pseudo	measurements	consist	of	forecasted	loads	based	on	historical	data.	The	advantage	of	using	pseudo	measurements	is	that	they	are	inexpensive	to	obtain	without	installing	new	costly	measurement	devices	in	the	system.	In	addition	to	a	lack	of	installed	measurement	devices,		51 telecommunication	 or	 measuring	 problems	 may	 lead	 to	 the	 unavailability	 of	 certain	measurements	 which	 could	 also	 render	 the	 system	 partially	 unobservable	 [103],	 in	particular	if	any	of	these	unavailable	measurements	are	critical	[13].		If	the	measurements	of	a	PQ	bus	(feeder	head)	are	unavailable	due	to	technical	problems	or	lack	of	 installed	measuring	devices,	 the	corresponding	DSSE	could	still	be	executed	given	that	 smart	meters	are	 installed	on	all	 the	 loads	and	reclosers	are	 installed	along	 feeders.	Once	 DSSE	 has	 been	 executed	 and	 the	 state	 of	 the	 system	 is	 known,	 a	 power	 flow	calculation	 can	 provide	 the	 complex	 power	 transferred	 to	 the	 distribution	 system.	 This	estimated	power	along	with	the	estimated	feeder	head	voltage	magnitude	can	thus	be	used	as	 equivalent	measurements	 at	 the	 transmission	 level	 to	 restore	 the	 observability	 of	 the	network	and	achieve	a	TSSE	solution.	3.3.2 Numerical	conditioning	Numerical instability (divergence) is another major problem in TSSE [29], which could be a result of lack of redundancy and/or bad initial conditions. A good indicator of numerical stability is the condition number of the TSSE gain matrix –gain matrix is defined in (2-11)– which is calculated as     212 xGxG Gk  (3‐7)where	 2. 	is	the	2‐norm	of	a	matrix	[103].	Higher	condition	numbers	indicate	that	the	TSSE	process	will	be	more	sensitive	to	noise,	and	therefore,	such	systems	are	referred	to	as	ill‐	52 conditioned	 as	 they	 will	 be	 more	 prone	 to	 numerical	 instability.	 Condition	 number	 is	function	 of	 types	 and	 locations	 of	 the	measurements	 in	 a	 system	 [29].	 As	 the	 proposed	TSSE	 provides	 additional	 measurements	 with	 different	 types	 at	 different	 locations,	 the	impact	of	the	proposed	approach	on	the	condition	number	should	be	examined.	3.3.3 Bad	data	identification	Undetected	bad	data	caused	by	telecommunication	issues,	measuring	malfunctions	such	as	biases	or	drifts,	or	malicious	attacks	may	also	contribute	to	the	inaccuracy	of	TSSE.	Thus,	one	 of	 the	 integral	 parts	 of	 TSSE	 is	 to	 detect	 bad	 data,	 and	 if	 possible	 to	 exclude	 them.	Therefore,	there	is	a	procedure	for	bad	data	identification	in	TSSE	program.	Some	bad	data	are	 just	 eliminated	 by	 a	 simple	 verification	 process,	 e.g.	 negative	 voltages	 and	 large	differences	between	incoming	and	leaving	currents.	Some	bad	data	need	to	be	treated	more	carefully	 through	 a	 rather	 complicated	 process.	 If	 there	 is	 sufficient	 redundancy	 in	 a	system,	 such	 bad	 data	 are	 expected	 to	 be	 filtered.	 For	 this	 purpose,	 both	 pseudo	measurements	and	DSSE	results	could	provide	the	required	redundancy.	However,	pseudo	measurements	tend	to	be	very	inaccurate	to	the	extent	that	they	could	themselves	become	bad	data	and	complicate	the	bad	data	identification	procedure.	Since	measurements	based	on	 DSSE	 results	 as	 presented	 in	 chapter	 2	 are	more	 reliable,	 they	 have	 the	 potential	 to	improve	the	ability	of	existing	algorithms	to	identify	and	remove	the	bad	data.		It	is	important	to	mention	that	bad	data	detection	methods	are	only	effective	against	non‐critical	measurements	[13].	Therefore,	if	measurements	of	a	PQ	bus	are	critical	and	include	bad	data,	 they	are	not	detectable	regardless	of	 the	magnitude	of	the	error	[13].	Since	the		53 DSSE	 results	 could	 render	 those	 bad	measurements	 non‐critical,	 the	 proposed	 approach	may	also	enable	identifying	and	eliminating	them.		There	have	been	many	studies	on	bad	data	identification	in	power	systems,	e.g.,	[13],	[19],	[20],	[23],	[24].	In	this	section,	the	normalized	residual	method	is	employed	[13],	[23].	This	method	 is	 based	 on	 the	 properties	 of	 measurement	 residuals.	 Consider	 the	 linearized	measurement	equation	(2‐1)	in	the	following	form	[25]	  vxxHz   (3‐8)The	WLS	estimation	will	give	the	state	vector	as	           ΔzRxHxGΔzRxHxHRxHxΔ 1T11T11Tˆ    (3‐9)where	 xΔˆ is	the	estimate	of	the	state	vector.	Then,	the	estimated	vector	of	measurements	is	  zKxΔxHzΔ  ˆˆ  (3‐10)where	 zΔˆ 	 is	 the	 calculated	measurements	based	on	 the	estimate	of	 the	 state	vector	and	      1T1  RxHxGxHK .	The	measurement	residuals	can	be	expressed	as	follows:		  zSzKIzΔΔzr  ˆ  (3‐11)	54 where	 I 	 is	the	identity	matrix,	and	 KIS  .	To	obtain	the	vector	of	normalized	residuals	Nr ,	the	absolute	value	of	each	residual	is	divided	by	the	square	root	of	the	corresponding	diagonal	element	of	S 	as	in	iiiNi Srr   (3‐12)where	 the	 subscript	 i 	 refers	 to	 the	 i th	 measurement.	 When	 a	 normalized	 residual	 is	greater	 than	 a	 specific	 threshold,	 its	 corresponding	 measurement	 is	 assumed	 to	 be	incorrect.	Some	utilities	eliminate	the	measurement	with	 the	biggest	normalized	residual	(above	 a	 certain	 threshold)	 and	 perform	 the	 TSSE	 again	 recursively	 until	 all	 normalized	residuals	are	below	the	assumed	threshold.	3.3.4 Accuracy	Due	to	the	use	of	finite	precision	arithmetic	in	computers,	the	ill‐conditioned	system	may	converge	 to	 a	 point	 away	 from	 the	 optimal	 solution.	Moreover,	 the	 normalized	 error	 of	TSSE	is	bounded	by	the	condition	number	as	[103],	[104]	      2222222222 1ˆ xxHzxHxxHzzvxxxGG kk . (3‐13)where	 v 	is	the	noise	vector	introduced	in	(2‐1),  xH 	is	the	Jacobian	matrix	defined	in	(2‐10),	 z 	 is	 the	measurement	 vector,	 x 	 is	 the	 actual/ideal	 state	 of	 the	 system	without	 any	errors,	 xˆ 	is	the	estimate	of	the	state,	and	 Gk 	is	the	condition	number	of	the	gain	matrix.	It		55 has	been	shown	 that	when	 Gk 	 rises,	 the	normalized	error	also	 rises	 [103].	Therefore,	by	improving	 the	 numerical	 conditioning	 of	 TSSE	 using	 DSSE	 results,	 more	 accurate	estimation	at	the	transmission	level	can	be	expected.	Generally,	 the	 accuracy	 of	 measuring	 devices	 improves	 as	 technology	 progresses.	 For	example,	 the	 recently	 installed	measuring	 devices,	 e.g.	 smart	meters,	 are	 generally	more	accurate	 than	RTUs	which	were	 installed	decades	 ago.	Consequently,	 as	 utilities	 keep	on	improving	 their	 distribution	 system	 measuring	 infrastructure	 by	 installing	 more	 smart	meters	while	also	refining	their	modeling	database,	it	is	expected	that	DSSE	will	be	able	to	provide	 accurate	 estimates	 of	 the	 state	 of	 the	 distribution	 system	 (including	 voltage	magnitudes	 of	 feeder	 heads)	 and	 the	 power	 transferred	 through	 the	 PQ	 buses.	 These	estimates	 will	 then	 replace	 the	 less	 reliable	 pseudo	 measurements,	 thus	 increasing	 the	TSSE	accuracy.	3.4 Case	Studies	To	demonstrate	the	proposed	approach,	this	section	presents	studies	in	which	the	results	of	DSSE	are	 incorporated	into	TSSE	using	two	transmission	benchmark	systems.	The	two	transmission	 systems	 are	 the	 IEEE	14‐bus	 system	 and	 the	 IEEE	57‐bus	 system	 [105].	 In	each	case	study,	the	transmission	system	is	connected	to	a	distribution	system	at	every	PQ	bus.	The	 considered	distribution	 system	 is	 the	 IEEE	13‐bus	 system	 [100].	Given	 that	 the	distribution	 systems	 are	 typically	 unbalanced,	 three‐phase	 component	 models	 are	considered	 in	 each	 distribution	 system.	 Network	 parameters	 and	 load	 base	 values	 are		56 modified	to	ensure	a	feasible	solution	with	voltages	in	acceptable	range	exists	for	different	load	levels.	In	 the	 first	 case	 study,	with	 the	 IEEE	 14‐bus	 transmission	 system,	 the	 overall	 combined	system	 contains	 330	 single‐phase	 buses.	 Figure	 3‐3	 demonstrates	 the	 topology	 of	 this	system.	In	Figure	3‐3(a),	the	IEEE	14‐bus	transmission	system	is	shown.	In	Figure	3‐3(b),	the	connection	of	one	typical	PQ	bus	of	 the	transmission	system	(bus	9)	 to	a	distribution	system	 is	 shown.	 The	 transition	 buses	 in	 Figure	 3‐3(b)	 are	 considered	 the	 border	 line	between	modeling	transmission	and	distribution	systems.	If	a	PQ	bus	in	Figure	3‐3(a)	has	measurement	 devices,	 it	 means	 the	 active	 and	 reactive	 powers	 are	 measured	 on	 the	secondary	side	of	the	corresponding	transformers	connected	to	it.	In	the	second	case	study,	with	the	IEEE	57‐bus	transmission	system,	the	overall	system	has	1771	single‐phase	buses.		To	create	more	realistic	case	studies,	it	is	assumed	that	the	network	parameters	have	some	degree	 of	 uncertainty.	 The	 uncertainty	 of	 network	 parameters	 is	 considered	 using	 a	uniform	 distributed	 random	 variable	 [102].	 Bounds	 of	 uncertainty	 of	 2%	 [106]	 for	transmission	systems	and	5%	for	distribution	systems	are	considered.	The	 measurements	 noise	 is	 modeled	 as	 a	 normally	 distributed	 random	 variable	 with	 a	specific	 standard	 deviation.	 The	 standard	 deviation	 for	 each	 measurement	 is	 assigned	based	on	the	precision	of	the	measuring	device	as		57 12 13 146 11 1098 75 4321GGG92 3 5 64137 8 9 1011 12RegulatorTransformer #1 Transformer #2Feeder headTransition BusesTo bus 14To bus 10Measurement DeviceLoadG GeneratorTo bus 4To bus 79(a) (b)RecloserG Figure	3―3	Configuration	of	the	first	case	study:	(a)	Modified	IEEE	14‐bus	transmission	system	with	allocated	loads,	generators,	and	measurement	devices;	(b)	The	feeder	head	of	the	IEEE	13‐bus	distribution	system	is	connected	to	bus	9	of	the	IEEE	14‐bus	transmission	system	3zpr   (3where	 pr is	the	device	precision,	and	 z is	the	absolute	of	the	true	value	of	the	measured	electric	variable.	Typical	power	and	voltage	magnitude	measurement	precisions	of	3%	and	1.5%,	respectively,	are	assumed	for	transmission	systems	[7].		58 A	daily	load	profile	measured	for	a	building	at	The	University	of	British	Columbia	is	given	in	Figure	3‐4	 (a)	 [91].	During	a	2‐hour	 interval,	 the	given	building	 load	 follows	a	pattern	similar	to	Figure	3‐4	(b),	which	is	extracted	from	Figure	3‐4	(a)	for	the	interval	from	2	pm	to	4	pm.	To	demonstrate	the	proposed	approach,	herein	it	 is	assumed	that	TSSE	is	executed	every	60	seconds.	The	smart	meters	are	assumed	to	measure	every	15	minutes	[12].	Moreover,	to	emulate	 a	 realistic	 scenario	 as	 explained	 in	 [12],	 [107],	 the	 smart	 meters	 are	 not	synchronized,	 implying	that	at	any	given	time	during	the	simulation	some	measurements	have	just	been	updated	while	the	others	are	out‐of‐date	by	up	to	15	minutes	as	explained	in	chapter	2.	0 5 10 15 200.50.550.60.650.70.750.80.850.90.951Time (hour)Load (pu)14 14.5 15 15.5 160.820.840.860.880.90.920.940.960.981Time (hour)Load (pu)(a) (b)	Figure	3―4	Typical	24	hour	load	profile	normalized	with	respect	to	the	peak	value	based	on	historical	data	recorded	at	The	University	of	British	Columbia	[91];	(b)	Recorded	2‐hour	fragment	of	load	profile	from	2pm	to	4pm.		59 The	system	is	simulated	for	a	2‐hour	interval	using	one‐minute	time	steps.	At	each	step	of	the	 simulation,	 first	 the	 loads	 are	 updated	 according	 to	 Figure	 3‐4	 (b).	 Afterwards,	 a	 3‐phase	unbalanced	optimal	power	flow	[108]	is	performed	in	order	to	obtain	the	generators’	dispatch	 and	 get	 a	 reference	 solution	 in	 accordance	 to	 the	 status	 of	 all	 loads.	 After	 that,	Gaussian	noise	is	added	to	the	calculated	values	of	active	and	reactive	powers	and	voltage	magnitudes	in	order	to	emulate	the	measurements	errors.	Since	the	smart	meters	are	not	synchronized,	 only	 some	 of	 their	measurements	 are	 up‐to‐date	 at	 any	 given	 step	 of	 the	simulation,	while	the	rest	are	out‐of‐date	as	explained	in	[12].	Two	 methods	 are	 implemented	 for	 comparison.	 The	 procedure	 of	 calculating	 the	 TSSE	results	at	each	time	step	using	the	traditional	and	proposed	methods	is	depicted	in	Figure	3‐5.	 In	 the	 first	 method,	 herein	 referred	 to	 as	 the	 Traditional	 TSSE	 approach,	 TSSE	 is	executed	 by	 itself	 without	 any	 extra	 information	 from	 the	 distribution	 system.	 In	 the	second	 method,	 referred	 to	 as	 the	 Proposed	 TSSE	 approach,	 both	 DSSE	 and	 TSSE	 are	executed	according	to	the	methodology	presented	in	Section	2.2.	3.4.1 Restoring	observability		Before	 performing	 TSSE	 in	 a	 system,	 the	 observability	 of	 the	 system	 should	 be	 checked	since	 measuring	 and	 communication	 problems	 may	 render	 parts	 of	 the	 system	unobservable	on	a	regular	basis	 [103].	 In	Figure	3‐3	(a),	 the	measurement	devices	 in	 the	transmission	system	are	presented.	Based	on	this	allocation	of	measurement	devices,	 the	observability	 analysis	 [13]	 of	 the	 system	 depicted	 in	 Figure	 3‐3	 (a)	 shows	 that	 the	measurements	of	two	device	are	critical	in	this	system;	the	device	at	bus	9	and	the	device		60 between	 bus	 4	 and	 bus	 7.	 The	 absence	 of	 any	 other	 measurement	 will	 not	 stop	 the	execution	of	TSSE;	however,	if	any	of	the	two	aforementioned	devices	fail,	the	gain	matrix	in	(2‐11)	will	become	singular	and	the	TSSE	execution	will	stop.		Update loads according to their profilesRun Optimal Power Flow for the entire systemAdd noise to electric values of the transmission system to emulate the measurements of RTUsExecute Traditional TSSE based on the objective function in (2-9) and calculate the resultsAdd noise to electric values of distribution systems to emulate the AMI measurementsExecute DSSE for each distribution systemCalculate the feeder head measurements and their variances according to section 3.2Simulating the Traditional TSSESimulating the Proposed TSSEUpdate some smart meters’ measurement signals according to chapter 2 to simulate asynchronicity of the metersAdd noise to electric values of the transmission system to emulate the measurements of RTUsExecute the Proposed TSSE based on the objective function in (3-6) and calculate the results Figure	3―5	Procedure	of	calculating	the	results	of	Proposed	and	Traditional	TSSE	at	each	step	of	the	simulation.	To	demonstrate	the	improvement	of	observability,	 in	the	first	scenario,	 it	 is	assumed	that	the	measurement	device	of	bus	9	and/or	its	communication	system	has	failed.	Since	this	is		61 a	critical	measurement,	the	transmission	system	is	not	fully	observable	and	TSSE	cannot	be	successfully	executed.	However,	since	smart	meters	are	installed	at	each	load	in	Fig	3‐3	(b)	and	reclosers	have	voltage	magnitude	and	power	measurement	devices,	 the	DSSE	can	be	executed	 successfully	 at	 any	 time.	 Based	 on	 DSSE	 results,	 the	 voltage	 magnitude	 and	injected	 active	 and	 reactive	 powers	 at	 the	 distribution	 system	 feeder	 head	 and	 their	variances	are	calculated	according	to	section	3.2.	As	the	feeder	head	is	connected	to	bus	9	as	shown	in	Fig	3‐3	(b),	following	the	proposed	approach	in	(3‐10),	the	calculated	powers	at	the	feeder	head	will	restore	the	observability	of	the	transmission	system.		In	the	second	scenario,	it	is	assume	that	the	measurement	device	between	bus	4	and	bus	7	and/or	its	communications	system	has	failed.	Unlike	scenario	one,	in	this	case,	DSSE	based	calculation	 of	 none	 of	 the	 distributions	 systems	 will	 be	 a	 replacement	 for	 the	 missing	measurements.	 However,	 a	 great	 property	 of	 the	 proposed	 TSSE	 is	 that,	 even	 in	 this	scenario,	 observability	 can	 be	 restored.	 As	 Figure	 3‐6	 shows,	 the	 injected	 power	 to	 the	region	encompassing	bus	4	to	bus	7	transmission	line	is	available	by	the	red	measurement	devices.	 Given	 the	 DSSE	 results	 in	 PQ	 buses	 4,	 5,	 8,	 9,	 13,	 and	 14,	 and	 assigning	 the	generator	in	bus	2	as	the	slack	bus,	the	injection	power	at	all	buses	is	either	measured	or	calculable	 in	 the	 depicted	 region.	 In	 a	 system	 where	 all	 of	 the	 injection	 powers	 are	available,	conventional	load	flow	can	be	executed	and	bus	voltages	can	be	calculated.	This	means	 that	 the	 system	 is	observable	and	 the	proposed	TSSE	can	obtain	 the	 system	state	and	maintain	the	observability	of	the	system	despite	that	a	critical	measurement	has	failed.		62  Figure	3―6	IEEE	14‐bus	transmission	system	with	a	faulty	critical	measurement	device.	The	subsystem	which	has	a	faulty	measurement	device	is	depicted.	Red	measurement	devices	show	the	injection	power	from	the	rest	of	the	system	to	the	subsystem.	3.4.2 Improving	numerical	conditioning	In	 the	 next	 set	 of	 studies	 it	 is	 assumed	 that	 all	 measurement	 devices	 are	 functioning	properly.	 To	demonstrate	 one	of	 the	 additional	 benefits	 of	 incorporating	DSSE	 results	 in	TSSE,	the	condition	numbers	of	the	TSSE	gain	matrix	for	the	two	benchmark	systems	have	been	calculated	during	the	2‐hour	study	interval.	Since	the	condition	numbers	also	change	with	the	measurements,	the	calculated	maximum,	minimum,	and	mean	condition	numbers	obtained	 during	 the	 2‐hour	 interval	 are	 summarized	 in	 Table	 3‐1.	 As	 Table	 3‐1	 shows,		63 employing	 the	 DSSE	 results	 as	 additional	 measurements	 (by	 2	 orders	 of	 magnitude)	improves	the	condition	numbers	for	both	benchmark	systems.	In	practice,	one	can	further	improve	 the	 numerical	 characteristics	 using	 numerical	 conditioning	 methods	 such	 as	orthogonal	 transformation	 [13].	 Here,	 to	 show	 the	 impact	 of	 the	 proposed	 method	 on	numerical	characteristics	exclusively,	other	numerical	methods	are	not	added	to	the	TSSE	formulation.	Table	3―	1	Statistical	Results	of	Calculating	the	Condition	Number	of	TSSE	for	100	Runs	Benchmark	system	based	on	the	IEEE	14‐bus	transmission	system	Method	 Min	 Mean	 Max	Traditional	TSSE	 3.6423e+007	 2.8646e+011	 8.4949e+11	Proposed	TSSE	 8.0149e+006	 1.5899e+09	 2.3343e+09	Benchmark	system	based	on	the	IEEE	57‐bus	transmission	system	Method	 Min	 Mean	 Max	Traditional	TSSE	 2.7879e+09	 5.7949e+011	 9.1302e+011	Proposed	TSSE	 3.9346e+08	 6.2415e+09	 6.5149e+09	 The	 number	 of	 iterations	 to	 achieve	 a	 converged	 solution	 is	 another	 indicator	 of	 the	numerical	 stability	 of	 the	 TSSE	 problem.	 Figure	 3‐7	 shows	 the	 recorded	 number	 of	iterations	it	took	to	converge	in	each	case	study.	In	the	first	study	with	a	smaller	IEEE	14‐bus	benchmark	system	[see	Figure	3‐7	(a)],	the	proposed	approach	reduces	the	number	of	iterations	 from	a	 range	of	 4	 to	8	 to	 a	 range	of	 4	 to	5	 iterations.	 Similarly,	 Figure	3‐7(b),	shows	 the	recorded	number	of	 iterations	 for	 the	 IEEE	57‐bus	 transmission	system.	Here,		64 the	 difference	 between	 Traditional	 and	 Proposed	 TSSE	 approaches	 is	 even	 more	pronounced.	 As	 it	 can	 be	 seen	 in	 Figure	 3‐7(b),	 the	 number	 of	 iterations	 using	 the	Traditional	TSSE	often	between	10	and	20,	and	there	are	two	points	where	the	method	fails	to	 converge	 after	 150	 iterations.	 However,	 the	 Proposed	 TSSE	 approach	 has	 always	converged	and	remained	within	a	reasonable	range	of	4	to	8	iterations.	0 0.5 1 1.5 244.555.566.577.58Time (hour)Number of Iterations  Traditional TSSEProposed TSSE 	65 0 0.5 1 1.5 2020406080100120140160Time (hour)Number of Iterations  Traditional TSSEProposed TSSENot Converged Figure	3―7	Number	of	iterations	taken	to	achieve	a	converged	TSSE	solution	for:	(a)	the	IEEE	14‐bus	transmission	system;	and	(b)	the	IEEE‐57	bus	transmission	system.	The	total	and	average	numbers	of	iterations	for	these	two	cases	(over	the	two	hours)	are	summarized	 in	Table	3‐2.	As	 it	 can	be	seen	 from	Table	3‐2,	 the	Traditional	TSSE	method	required	 3,678	 and	 8,864	 iterations	 for	 the	 small	 and	 large	 benchmark	 systems,	respectively,	 whereas	 the	 Proposed	 TSSE	 method	 took	 3,363	 and	 3,429	 iterations,	respectively,	 for	 the	 same	 two	 systems.	 On	 the	 other	 hand,	 the	 size	 of	 the	 optimization	problem	in	(10)	is	increased	by	about	60%	because	of	the	additional	measurements	at	the	PQ	 buses,	 which	 imposes	 extra	 computations.	 Therefore,	 it	 is	 possible	 that	 the	computational	burden	is	increased	in	total.	In	practice,	utilities	such	as	BC	Hydro	often	use	pseudo	measurements	for	all	loads.	The	proposed	method,	once	implemented,	would	only	replace	 pseudo	 measurements	 with	 feeder	 head	 estimations.	 Thus,	 practically,	 the	computational	burden	would	not	increase	as	a	result	of	the	proposed	method.		66 Table	3―2	Number	of	Iterations	Taken	by	the	Traditional	and	Proposed	TSSE	Methods	Over	2	Hour	Interval.	Benchmark	system	based	on	the	IEEE	14‐bus	transmission	network	Method	 Total	 Average	Traditional	TSSE	 613	 5.11	Proposed	TSSE	 560	 4.67	Benchmark	system	based	on	the	IEEE	57‐bus	transmission	network	Method	 Total	 Average	Traditional	TSSE	 1477	 12.31	Proposed	TSSE	 619	 5.16		3.4.3 Impact	on	accuracy	of	TSSE	Taking into consideration more measurements (from DSSE) with higher accuracy as well as an improved condition number should further improve the accuracy of TSSE. To examine the impact of the Proposed TSSE approach on TSSE accuracy, the total error of the estimated voltages over all buses is calculated at every time step as     busnoi actiestierror _1 2vv  (3‐15)where	 busno_ 	 is	 the	 number	 of	 buses,	 and	 estiv 	 and	 activ 	 are	 the	 estimated	 and	 exact	(errorless)	 voltages,	 respectively,	 at	 the	 i th	 bus.	 Each	 voltage	 variable	 in	 (3‐15)	 is	 a	complex	number	in	per	unit.	The	accumulative	error	in	(3‐15)	over	the	2	hour	simulation	interval	is	referred	to	as	the	total	error	of	the	estimation.		67 Figures	3‐8(a)	and	3‐8(b)	demonstrate	the	calculated	error	achieved	by	the	Traditional	and	Proposed	TSSE	approaches	compared	to	the	traditional	TSSE	for	the	IEEE	14‐bus	and	57‐bus	 systems,	 respectively.	 In	 Figure	 3‐8(a),	 it	 is	 clear	 that	 the	 error	 has	 significantly	decreased	both	 in	 terms	of	 its	 value	at	 every	 step	as	well	 as	 its	 average	over	 the	2‐hour	interval.	 Similar	 observations	 are	 made	 in	 Figure	 3‐8(b)	 for	 the	 larger	 system.	 These	improvements	 are	 due	 to	 a	 number	 of	 factors	 such	 as	 using	 DSSE	 results	 and	 better	numerical	conditioning	of	the	problem.	However,	Figure	3‐8(b)	also	shows	several	spikes	in	 the	 error	 of	 the	 Proposed	 TSSE	 method,	 which	 is	 the	 result	 of	 using	 out‐of‐date	measurements	used	in	DSSE.	0 0.5 1 1.5 200.020.040.060.080.10.12Time (hour)Error of SE 	68 0 0.5 1 1.5 200.050.10.150.20.250.30.350.40.45Time (hour)Error of TSSE  Traditional TSSEProposed TSSE Figure	3―8	Total	error	of	TSSE	evaluated	over	all	buses	at	each	time	step	and	its	average	for	the	2‐hour	study	interval	with	smart	meters	measurements	updating	every	15	minutes	for:	(a)	the	IEEE	14‐bus	transmission	system;	and	(b)	the	IEEE	57‐bus	transmission	system.	The	 accuracy	 of	 DSSE	 depends	 on	 many	 factors	 including	 the	 variation	 of	 loads,	 the	accuracy	of	the	smart	meters,	as	well	as	the	rate	at	which	the	smart	meters	measurements	are	 updated.	 To	 study	 the	 significance	 of	 smart	 meters	 measurement	 updating	 rate,	another	simulation	is	executed	on	the	IEEE	57‐bus	test	system	assuming	that	their	rate	is	increased	 from	15	 to	 30	minutes.	 The	 results	 are	 presented	 in	 Figure	3‐9.	As	 this	 figure	shows,	a	reduction	of	the	sampling	rate	noticeably	increases	the	error	of	the	estimation	at	certain	 times.	 The	 reason	 is	 that	 some	 loads	 may	 vary	 significantly	 between	 two	subsequent	 measurement	 updates	 of	 the	 smart	 meters,	 thereby	 yielding	 larger	 errors	between	these	two	points.	The	probability	and	magnitude	of	this	variation	increases	with	the	duration	of	the	sampling	interval.	However,	it	is	important	to	point	out	that	the	system		69 still	 converges	 for	 all	 steps,	 and	 the	 numerical	 conditioning	 is	 still	 improved.	 Figure	 3‐9	shows	that	it	is	beneficial	for	utilities	to	invest	on	increasing	the	resolution	of	smart	meter	measurement	updates	in	their	system.	0 0.5 1 1.5 200.10.20.30.40.50.60.7Time (hour)Error of TSSE  Traditional TSSEProposed TSSE Figure	3―9	Total	error	of	TSSE	evaluated	over	all	buses	at	each	time	step	and	its	average	for	the	2‐hour	study	interval	when	the	smart	meters	measurements	are	updated	every	30	minutes	for	the	IEEE	57‐bus	transmission	system.	3.4.4 Facilitating	bad	data	identification	and	accuracy	improvement	Improving	 the	 identification	 of	 anomalies	 in	 TSSE	 is	 another	 important	 aspect	 of	 the	Proposed	TSSE	approach.	The	effectiveness	of	bad	data	identification	methods	depends	on	many	factors,	including	the	level	of	redundancy	and	the	accuracy	of	the	measurements.	In	order	to	demonstrate	this	contribution,	in	this	section,	TSSE	of	the	IEEE	14‐bus	benchmark	system	 at	 2	 pm	 is	 considered.	 To	 emulate	 a	 bad	 datum,	 the	 active	 power	 injection		70 measurement	at	bus	10,	which	is	a	non‐critical	measurement,	is	distorted	to	contain	10%	error.	 This	 error	 could	 be	 a	 result	 of	 false‐data	 injections	 or	 technical	 problems.	 The	calculated	 initial	 normalized	 residuals	 (i.e.,	 before	 bad	 data	 detection)	 of	 the	 active	 and	reactive	power	 injections	at	bus	10	using	 the	Traditional	and	Proposed	TSSE	approaches	are	summarized	in	Table	3‐3.	Table	3―3	Normalized	Residuals	of	TSSE	of	the	IEEE	14‐bus	Transmission	System	for	Bus	10	before	Bad	Data	Detection	Bus	 Type	of	Measurement	Normalized	Residuals	Traditional	TSSE	 Proposed	TSSE	10	 active	power	 2.2905	 7.4819	10	 reactive	power	 0.7462	 0.8543	 Based	 on	 the	 normal	 distribution	 characteristics,	 a	 threshold	 of	 2.57	 is	 assumed	 to	determine	the	presence	of	bad	data	with	95%	confidence	[7].		As	it	can	be	seen	in	Table	3‐3,	with	the	Proposed	TSSE	approach,	the	residual	of	the	active	power	measurement	of	bus	10	is	greater	than	2.57.	However,	with	Traditional	TSSE,	the	normalized	residual	is	below	the	assigned	threshold.	The	reason	is	that	the	proposed	method	provides	more	redundancy	to	identify	 bad	 data.	 At	 the	 same	 time,	 the	 measurement	 of	 reactive	 power	 (which	 is	 not	corrupted)	is	correctly	classified	as	regular	data.	The	benefit	of	removing	bad	data	is	shown	in	Table	3‐4,	wherein	the	total	error	[see	(16)]	of	Proposed	TSSE	is	shown	to	decrease	from	0.0393	 to	 0.0346	 when	 the	 erroneous	 power	 measurement	 is	 removed.	 Table	 IV	 also	demonstrates	 the	 improvements	 in	 accuracy	 obtained	 by	 using	 Proposed	 TSSE	 before	removing	 any	 bad	 data	 in	 comparison	 to	 Traditional	 TSSE.	 The	 proposed	 approach	 also		71 enables	 detection	 of	 bad	 data	 in	 buses	 with	 critical	 active	 and	 reactive	 power	 injection	measurements.	In	Traditional	TSSE,	the	residuals	of	critical	measurements	are	always	zero,	whereas	 the	 proposed	 approach	 renders	 those	 measurements	 non‐critical	 by	 providing	redundant	measurements	resulting	in	non‐zero	residuals.		Table	3―4	Total	Error	of	TSSE	for	the	IEEE	14‐bus	Transmission	System	with	Bad	data	at	Bus	10	and	Its	Elimination	Traditional	TSSE	 Proposed	TSSE	0.0588	No	Bad	Data	Identification	 Applying	Bad	Data	Identification	and	Elimination	0.0393	 0.0346			72 CHAPTER	4: BAD	 DATA	 IDENTIFICATION	AND	ELECTRICITY	THEFT		4.1 Modeling	Theft	as	Bad	Data	Sometimes	in	power	systems,	typically	at	the	distribution	level,	there	are	a	number	of	users	who	are	stealing	electricity	by	bypassing	their	measurement	devices.	A	simplified	diagram	depicting	 connection	 of	 smart	 meters	 at	 the	 distribution	 feeder	 for	 modeling	 electricity	theft	and	bad	data	 is	shown	in	Figure	4‐1.	Specifically,	 in	 figure	4‐1(a),	 load	2	 is	partially	bypassing	 its	 smart	 meter	 and	 stealing	 electricity.	 Therefore,	 the	 quantities	 which	 are	reported	by	 the	 smart	meter	will	 become	erroneous.	 From	 the	point	 of	 view	of	 received	information,	this	case	is	analogous	to	a	situation	where	there	is	no	manipulation	with	the	metering	device,	but	the	meter	is	malfunctioning,	as	depicted	in	Figure	4‐1(b).	It	this	case,	there	is	no	bypassing	(stealing)	at	load	2,	but	its	smart	meter	is	malfunctioning	and	it	may	be	reporting	 the	same	quantity	as	 in	Figure	4‐1(a).	Sometimes,	despite	no	bypassing,	 the	user	 tampers	 the	meter	 to	 report	 less	 consumption	 and	 steal	 electricity.	 These	 cases	 of	tampering	the	meters	are	also	analogous	to	cases	of	malfunctioning	devices.	In	both	cases	(theft	and/or	meter	malfunction),	the	delivered	power	from	the	system	to	the	load	and	the	voltage	at	the	load	are	the	same	since	this	measurement	does	not	impact	the	operation	of	the	physical	system.		73 Smart MeterLoad 2Distribution FeederLoad 4Smart MeterLoad 3Smart MeterLoad 1Smart MeterMalfunctioning Device Reading the Same Power as in Case (a)Bypassing Branch (electricity theft)Smart MeterLoad 2Distribution FeederLoad 4Smart MeterLoad 3Smart MeterLoad 1Smart Meter(a) (b) 	Figure	4―1	A	simplified	diagram	depicting	connection	of	smart	meters	for	modeling	electricity	theft	as	bad	data:		(a)	load	2	is	partially	bypassing	its	smart	meter	to	commit	electricity	theft;	and	(b)	smart	meter	of	load	2	is	malfunctioning.		The	measurements	 of	malfunctioning	 devices	 are	 referred	 to	 as	 bad	 data.	 Therefore,	 an	algorithm	 to	 identify	 bad	 data	would	 also	 help	 to	 locate	 the	 loads	which	 are	 potentially	stealing	 electricity.	 In	 order	 to	 distinguish	 between	 the	 bad	 data	 due	 to	 device	malfunctioning	 or	 due	 to	 the	 theft,	 three	 general	 criteria	 could	 be	 considered.	 The	 first	criterion	 is	 recognition	 of	 obvious	 bad	 data.	 If	 bad	 data	 are	 obvious,	 such	 as	 negative	measured	voltage	magnitude	values	or	the	measurements	that	are	clearly	out	of	range,	it	is	more	 likely	 that	 the	 device	 is	 actually	 malfunctioning	 (as	 the	 thieves	 would	 generally	manipulate	 their	 meters	 in	 a	 non‐obvious	 way	 [16]).	 The	 next	 criterion	 is	 when	 the	reported	 measured	 power	 is	 more	 than	 its	 actual	 value.	 These	 cases	 are	 certainly	 not	instances	of	theft.	However,	determining	the	user	in	these	cases	is	important	and	helpful	to	utilities	since	the	user	is	being	billed	more	than	they	are	consuming.	Sometimes,	a	device	is		74 malfunctioning	 due	 to	 a	 temporary	 problem	 such	 as	 telemetry	 failures.	 Usually,	 in	 these	cases,	the	device	only	malfunctions	for	a	short	interval.	However,	a	tampered	or	bypassed	device	would	consistently	transfer	bad	data	for	an	extended	period.	In	this	chapter,	in	the	first	 stage,	 the	 problem	 of	 bad	 data	 identification	 to	 locate	 the	 two	 types	 of	 theft	 points	which	 result	 from	 meter	 tempering	 and	 smart	 meter	 bypassing	 is	 formulated	 as	 an	optimization	problem.	4.2 Problem	Formulation	of	Bad	Data	Identification	The	normalized	residual	method	for	bad	data	identification	has	been	presented	in	Section	3.3.3.	This	method	 is	most	 efficient	 to	 identify	 single	bad	data	 in	 a	 set	of	measurements.	However,	for	multiple	bad	data,	 it	could	classify	some	good	measurements	as	bad	data	or	miss	bad	data	[25].	In	this	section,	a	method	to	identify	multiple	bad	data	is	presented.	Identification	 of	 multiple	 bad	 data	 is	 treated	 as	 an	 optimization	 problem	 based	 on	 the	Decision	Theory	approach	[70].	For	a	set	of	 m 	measurements,	the	decision	as	to	whether	the	 i th	measurement	is	good	or	bad	is	represented	using	a	binary	variable,	 id .	Based	on	 id ,	a	measurement	considered	as	good	or	bad	data	as	follow;	0id ,		 if	the	 i th	measurement	is	bad	data	1id ,	 	 if	the	 i th	measurement	is	good	data.	A	decision	vector	is	defined	as	a	combination	of	decisions	(binary	variables)	on	a	set	of	m 	measurements,	as	  mdddd ,...,,, 321d .	Thus,	if	the	system	is	fully	observable,	 m2 possible		75 decision	vectors	exist.	Each	decision	vector	referred	to	as	a	plausible	for	the	observed	data	if,	after	the	removal	of	the	measurements	that	have	been	labeled	as	bad	data,	no	more	bad	data	is	detected.		In	order	 to	detect	whether	or	not	bad	data	exists	 in	a	set	of	measurements,	a	hypothesis	test		should	be	used	[69].	The	DSSE	objective	function	as	defined	in	Chapter	2	is		  mi iiizJ12)( x .	(4‐1)As	 (4‐1)	 shows,	 )(xJ 	 is	 the	summation	of	 the	square	of	 m 	normally	distributed	random	valuables.	Let	 n 	denote	the	number	of	state	variables	(which	is	the	size	of	 x 	vector).	Given	that	m 	measurements	are	used	in	(4‐1),	according	to	the	hypothesis	test,	 )(xJ is	a	random	variable	 which	 follows	 the	 chi‐square	 distribution	 with	 nm  	 degrees	 of	 freedom	 [13],	denoted	as	 2 nm .	For	example,	Figure	4	–	2	shows	the	probability	density	function	(pdf)	of	the	chi‐square	distribution	with	8	degrees	of	freedom	i.e.,	 8 nm .	The	null	hypothesis	is	defined	 as	 the	 assumption	 that	 there	 are	 no	 bad	 data	 in	 the	 set	 of	 measurements.	Therefore,	as	the	only	source	of	error	to	measurements	would	be	the	Gaussian	noise,	 the	error	of	 the	 i th	measurement	 in	 (4‐1)	 follows	a	normal	distribution	with	mean	equal	 to	zero	 and	 variance	 equal	 to	 i .	 The	 considered	 test	 verifies	 whether	 or	 not	 the	 null	hypothesis	is	a	correct	assumption.	To	do	so,	the	value	of	 )(xJ 	is	compared	against	a	pre‐calculated	threshold,	which	is	depicted	in	Figure	4	–	2	as	C .	This	threshold	is	defined	in	a		76 manner	that	if	 )(xJ 	is	greater	than	it,	the	null	hypothesis	is	rejected,	and	if	 )(xJ 	is	smaller	than	it,	the	null	hypothesis	is	accepted.			Figure	4―2	Probability	density	function	of	the	chi‐square	distribution	with	8	degrees	of	freedom.	The	value	of	threshold	C 	corresponds	to	the	confidence	level	in	hypothesis	test,	denoted	as	 .	For	example,	 if	hypothesis	 test	 is	 set	 to	be	correct	 for	95%	of	 the	 time	( =0.95),	 the	threshold	C 	would	be	calculated	from	the	following;	   2.2.)(95.022120 nmetwithd nmnmC 	(4‐2)where	   	 is	 the	  2 nmpdf  ;	and	  	 is	 the	Gamma	function	[25].	According	to	hypothesis	test,	the	null	hypothesis	is	accepted	{meaning	that	the	measurements	  mzzz ,...,, 21 	in	(4‐1)	are	considered	as	good	data}	only	if	  xJ 	is	less	than	the	threshold	C .			77 The	decision	vector	d 	contains	the	information	on	which	measurements	are	considered	to	be	 bad.	 By	 removing	 bad	 measurements	 from	 (4‐1),	 there	 will	 be	 remainedm 	 remaining	measurements,	 denoted	 as	  dz .	 Then,	 DSSE	 is	 executed	 using	 the	 remaining	measurements	  dz ,	 the	result	of	which	is	the	corresponding	state	variables	  dx 	and	the	value	of	objective	function	   dxJ .	The	remaining	 remainedm 	(good)	measurements	are	used	in	 (4‐2),	 based	 on	 nmremained  	 degrees	 of	 freedom,	 to	 calculates	 the	 corresponding	threshold	  dC .	 Then,	 the	 value	 of	   dxJ 	 is	 re‐compared	 with	  dC 	 to	 complete	 the	Hypothesis	test	process	as	    ddx CJ  ,	 	  dz 	contains	some	measurements	that	are	bad	data	    ddx CJ  ,	 	 All	measurements	in  dz 	are	assumed	as	good	data	If	  dz 	passes	the	hypothesis	test,	the	decision	vector	d 	is	considered	plausible.	Generally,	more	 than	 one	 plausible	 decision	 vector.	 Therefore,	 a	 plausible	 decision	 vector	 with	maximum	likelihood	should	be	found.		Let	 ip 	denote	the	probability	of	measurement	 i 	to	be	good,	and	 iq 	denote	the	probability	of	it	to	be	bad	data.	A	decision	vector	separates	the	measurements	into	good	and	bad	data.	If	 G 	 is	 the	 set	 of	 good	measurements,	 and	 B 	 is	 the	 set	 of	 bad	measurements,	 then	 the	probability	of	a	given	decision	vector	d 	would	be		78   BGdiiii qpP .	 (4‐3)Generally,	it	is	assumed	that	all	measurement	devices	have	the	same	reliability.	Therefore,	all	measurements	are	assumed	have	the	same	 ip 	and	 iq .	Given	that	 ip 	is	close	to	one	[25],	we	have	                miiimiiiiiiii dqqdqqpP1111loglog1loglogloglogBBGd .	 (4‐4)In	 (4‐4),	 the	 log 	 function	 is	 used	 to	 simplify	 the	 equation.	 The	 decision	 vector	with	 the	most	 likelihood	maximizes	   dPlog .	 Therefore,	 the	 solution	which	maximizes	   dPlog 	would	also	minimize	the	objective	function	 )(d as	        miimii dMinimizedqPMinimize111 1)(1loglog)( ddd .	(4‐5)This	means	that	the	most	likely	decision	vector	is	the	one	which	results	in	identifying	the	minimum	number	of	bad	measurements.		However,	 a	 decision	 vector	 is	 considered	 unacceptable	 if	 the	 resulting	 system	 is	 not	observable.	Also,	it	must	pass	the	hypothesis	test.	Therefore,	the	identification	of	multiple	bad	data	problem	is	formulated	as		79        ddxdCJobservableisSystemtosubjectdMinimizemii :11	(4‐6)4.3 Solution	Method	Various	 optimization	 techniques	 have	 been	 proposed	 to	 solve	 optimization	 problems	including	 linear	 programming	 (LP),	 and	 its	 hybrid	 versions,	 Newton‐like	 methods,	nonlinear	programming	(NLP),	quadratic	programming	(QP),	interior	point	method	(IPM),	sequential	 unconstrained	 minimization	 [109]‐[111],	 etc.	 Most	 of	 these	 classical	optimization	methods	 are	 limited	 to	 objective	 functions	 and	 constraints	 that	 are	 convex,	differentiable,	 and	 continuous,	 and	may	 depend	 on	 specific	 functions	 and/or	 constraints	[112].	Due	to	the	nature	of	these	methods,	they	might	converge	to	local	solutions	and	fail	to	achieve	the	global	optimum	for	non‐convex	problems	with	complicated	constraints	[113].	For	 example,	 it	 is	 very	 difficult	 to	 incorporate	 the	 observability	 constraint	 in	 (4‐6)	 into	classic	optimization	methods.	A	new	category	of	optimization	tools	including	evolutionary	algorithms	 (EAs)	 [114]‐[117],	 Simulated	 Anealing	 (SA)	 [118],	 Artificial	 Neural	 Network	(ANN)	 [119]‐[122],	 Tabu	 Search	Algorithm	 (TSA)	 [123],	 dual‐type	methods	 [124],	 [125],	mean	field	theory	[126],	ordinal	optimization	theory	[108],	etc.	have	been	proposed.	In	this	chapter,	a	new	solution	for	multiple	bad	data	identification	problem	based	on	shuffled	frog	leaping	 algorithm	 (SFLA)	 optimization	 is	 proposed.	 The	 SFLA	 is	 a	 meta‐heuristic	optimization	 method	 based	 on	 observing	 and	 modeling	 behavior	 of	 frogs.	 The	 SFLA	combines	 the	 benefits	 of	 the	 genetic‐based	 mimetic	 algorithms	 (MAs)	 and	 the	 social		80 behavior‐based	 particle	 swarm	 optimization	 (PSO)	 algorithm	 [127],	 [128].	 Since	 this	algorithm	has	not	been	applied	in	multiple	bad	data	identification	problem	in	the	past,	to	demonstrate	 the	 capabilities,	 the	 results	of	 it	 are	also	 compared	 to	GA	and	PSO	methods	which	have	been	previously	applied	to	solve	this	problem.		4.3.1 Shuffled	frog	leaping	algorithm	In	SFLA,	there	is	a	population	of	possible	solutions	defined	by	a	set	of	frogs	that	is	divided	into	 subgroups	 called	 “memeplexes,”	 each	 performing	 a	 local	 search.	 At	 first,	 an	 initial	population	 of	 F frogs	 is	 created	 randomly	 within	 the	 feasible	 search	 space.	 For	 an	 m 	variable	optimization	problem,	the	 i th	frog	is	represented	as	  mi xxx ,...,, 21X .	The	fitness	of	each	frog	is	 its	value	of	the	objective	function.	After	calculating	the	fitness	of	the	frogs,	they	are	sorted	in	a	descending	order	according	to	their	fitness.	Then,	the	whole	population	of	 F 	 frogs	 is	 divided	 into	 S 	 memeplexes,	 each	 containing	 SF 	 number	 frogs.	 In	 this	procedure,	the	first	frog	moves	to	the	first	memeplex,	the	second	frog	moves	to	the	second	memeplex,	and	so	on	all	 the	way	until	 the	 thS 	 frog	moves	to	the	 thS memeplex.	Then,	 the	 thS 1 	frog	goes	to	the	first	memeplex,	  thS 2 frog	goes	to	the	second	memeplex,	and	so	on	until	all	of	the	frogs	are	distributed	in	memplexes.	Within	each	memeplex,	the	frogs	with	the	best	and	worst	fitness	are	labeled	as	 bX 	and	 wX ,	respectively.	Also,	the	frog	with	the	best	global	fitness	is	denoted	as	 gX .	Then,	in	each	 memeplex,	 a	 procedure	 is	 applied	 to	 improve	 the	 frog	 with	 the	 worst	 fitness	 as	follows:		81    wbi XXRandL  2,0 .	 (4‐7)iwwNEW LXX  .	 (4‐8)where	  2,0Rand 	is	a	vector	of	random	numbers	between	0	and	2,	 iL 	is	the	leap	size,	and	NEWwX is	the	new	position	of	the	worst	frog	after	leaping.	If	this	procedure	does	not	result	a	better	solution,	the	calculations	in	(4‐7)	and	(4‐8)	are	repeated	with	replacement	of	 bX 	by	gX .	These	calculations	constitute	to	the	next	attempt.	If	no	improvement	is	achieved	in	this	case	 either,	 then	 a	 new	 frog	 is	 randomly	 generated	 within	 the	 feasible	 search	 space	 to	replace	the	worst	frog.	These	calculations	are	repeated	for	a	specific	number	of	iterations	for	each	memeplex.	Afterwards,	to	ensure	global	exploration,	the	memeplexes	are	shuffled,	and	the	frogs	are	sorted	and	distributed	among	memeplexes	again	[129].	The	local	search	within	memeplexes	and	the	shuffling	continue	until	convergence	criteria	are	satisfied.	4.3.2 Satisfying	the	constraints	A	 common	 approach	 to	 satisfy	 constraints	 in	 optimization	 problems	 is	 to	 modify	 the	objective	 function	 using	 penalty	 factors	 [115],	 [130].	 Since	 the	 goal	 of	 optimization	 is	 to	minimize	the	objective	function,	violation	of	constraints	in	(4‐6)	is	formulated	as	additional	terms	 to	 the	objective	 function	using	penalty	 factors.	Therefore,	 the	objective	 function	 in	(4‐6)	is	modified	as			82     ObservePlausemiidMinimize   11d .	 (4‐9)where Plause 	and	 Observe 	are	the	plausibility	and	observability	penalty	terms	defined	as:		              leunobservabissystemtheifkobservableissystemtheifCJifCJkCJifObservePlause2100 ddxddxddx.	where	 1k 	and	 2k 	are	the	penalty	factors.	(4‐10)4.3.3 Implementation	of	SFLA	as	the	proposed	solution	To	use	SFLA	for	multiple	bad	data	identification,	each	frog	corresponds	to	a	decision	vector.	In	a	system	with	 n 	state	variables	and	mmeasurements,	in	order	to	maintain	observability,	at	 most	 nm  	 measurements	 could	 be	 eliminated.	 Therefore,	 the	 defined	 frog	 in	 the	optimization	 process	 is	 a	 nm  ‐dimensional	 vector.	 Arrays	whose	 value,	 l ,	 is	 a	 number	from	1	 to	 m 	 indicates	elimination	of	 the	 l th	measurement.	Since	 nm  	 is	 the	maximum	number	 of	measurements	 to	 be	 eliminated,	 some	 arrays	may	 not	 indicate	 elimination	 of	any	 measurements.	 The	 arrays	 whose	 values	 are	 greater	 than	 m or	 less	 than	 zero	 are	interpreted	as	not	eliminating	any	measurements.	Therefore,	 the	boundaries	on	an	array	are	 set	 to	  mm 2, 	 interval	 to	 enable	 SFLA	 to	 explore	 all	 possibilities	 including	 not	eliminating	any	measurements.	During	 the	optimization	process,	whenever	a	new	 frog	 is	generated,	 each	 array	 of	 it	 is	 checked	 to	 see	 whether	 it	 is	 within	 the	  mm 2, 	 interval		83 boundaries.	If	an	array	violates	a	boundary,	its	value	is	adjusted	to	be	equal	to	the	violated	boundary.		Since	SFLA	was	originally	designed	for	continuous	search	spaces,	some	modifications	are	required	for	a	discrete	search	space	since	the	optimization	output	must	be	a	vector	(frog)	with	integer	values.	Here,	in	the	considered	implementation,	the	search	space	is	considered	continuous	and	the	arrays	of	frogs	are	not	defined	as	integer	values.	However,	to	evaluate	the	 fitness	 of	 a	 frog,	 first	 the	 values	 of	 arrays	 are	 temporarily	 rounded	 to	 their	 nearest	integer.	Then	the	objective	function	corresponding	to	the	frog	is	evaluated.	Afterwards,	the	arrays	of	the	frog	will	be	reverted	to	the	original	values	before	rounding.	Therefore,	next	population	of	frogs	is	generated	from	the	original	values	of	arrays	(before	rounding),	and	the	rounded	values	are	only	used	to	evaluate	the	fitness	of	frogs	based	on	(4‐9).	Figure	4‐4	summarizes	the	optimization	process.			84 StartInitialize: F=Frogs population size S=Number of memeplexesMem_iter=Number of iterationsGenerate frogs randomlyCalculate the fitness frogs based on (4-9)Sort frogs in descending orderDivide frogs into S memeplexesFirst memeplex: i=1First iteration: j=1Determine Xb, Xw and XgApply equations (4-6) and (4-7)Is the the worst frog imprved?Apply equations (4-7) and (4-8) replacing Xbby XgNoIs the the worst frog imprved?Noj=Mem_iter?Next memeplex: i=i+1i=S?Shuffle the memeplexesConvergence criteria satisfied?Determine XgEndGenerate a random frogYesYesYesYesYesNoNoNoj=j+1	Figure	4―3	The	optimization	process	using	SFLA.	4.3.4 Case	study	To	demonstrate	the	proposed	methodology,	the	IEEE	123‐bus	distribution	system	[100]	as	depicted	in	Figure	4‐4	is	considered.	Among	the	modifications,	the	lines	in	blue	are	added		85 to	create	some	loops	in	the	system	with	single	phase	loads.	The	buses	are	numbered	in	a	manner	that	the	difference	between	bus	numbers	is	proportional	to	the	distance	between	two	buses	in	general.	For	the	purpose	of	this	study,	a	total	of	eight	points	of	theft	have	been	assumed,	 which	 are	 highlighted	 as	 read	 dots	 in	 Figure	 4‐5.	 Voltage	 magnitude	measurements	 of	 smart	meters	 are	 not	 used	 in	 this	 stage.	 The	 amount	 of	 stolen	 energy	varies	from	one	theft‐load	point	to	another.	Without	loss	of	generality,	the	percent	of	stolen	power,	 which	 is	 not	measured	 by	 the	 corresponding	 smart	meter	 at	 each	 theft	 point,	 is	summarized	in	Table	4‐1.	For	example,	the	total	consumption	of	load	7	in	Table	4‐1	will	be	the	measured	power	multiplied	by	 7.011 .			Figure	4―4	Considered	IEEE	123	distribution	systems	depicting	eight	loads	with	electricity	theft.		86 	Table	4―1	Percentage	of	the	Unmeasured	Consumption	at	Each	Theft	Point	Theft	load	point	 7	 16	 41	 62	 69	 70	 71	 84	 85	Percentage	of	consumption	that	is	not	reported	in	measurement	 70	 20	 95	 35	 55	 40	 40	 50	 60		In	 addition	 to	 SLFA,	 genetic	 algorithm	 (GA)	 [131]	 and	 PSO	 [132]	 have	 also	 been	implemented	to	solve	the	multiple	bad	data	identification	problem.	The	number	of	frogs	in	SFLA,	 particles	 in	 PSO,	 and	 chromosomes	 in	 GA	 is	 set	 to	 60.	 In	 SLFA,	 the	 number	 of	memeplexes	is	set	to	6.	The	confidence	level	parameter	in	(4‐2)	is	set	to	95%.	The	penalty	factors	 1k 	and	 2k 	in	(4‐10)	are	set	to	1	and	10,	respectively.		The	results	of	the	simulation	are	shown	in	Table	4‐2.	As	the	Table	4‐2	shows,	both	the	GA	and	PSO	methods	 found	only	 five	 theft	points	 (missed	theft	 four	points!)	while	satisfying	the	 observability	 constraint.	 As	Table	 4‐2	 shows,	 the	 identified	 theft	 points	 in	 these	 two	methods	are	different	while	 their	DSSE	objective	 function,	  xJ ,	 has	 the	same	value.	The	reason	is	that	eliminating	the	measurements	of	one	of	the	three	theft	points,	69,	70,	or	71	will	 render	 the	 other	 two	 as	 critical	measurements.	 As	 a	 result,	 the	 residual	 of	 the	 two	remaining	 measurements	 in	 DSSE	 would	 be	 zero.	 Therefore,	 whether	 theft	 point	 70	 is	eliminated	in	GA	or	theft	point	69	is	eliminated	in	PSO,	the	WLS	objective	function	would	have	the	same	value.	This	means	that	all	of	the	frogs	which	satisfy	the	constraints	in	(4‐6)	at	the	end	of	the	optimization	should	be	determined,	and	the	measurements	they	identify	should	be	considered	as	suspicious	theft	point.	Among	the	three	applied	methods,	the	SFLA		87 happens	 to	be	 the	most	 successful	one	by	 finding	six	 theft	points	and	missing	only	 three	points,	 while	 satisfying	 the	 observability	 constraint.	 The	 reason	 that	 these	 three	 theft	points	 are	 not	 identified	 is	 that	 they	 become	 critical	measurement	 after	 eliminating	 the	previously	 identified	 theft	points.	Therefore,	 their	 residual	becomes	zero	and	they	would	not	have	any	 impact	on	   dxJ ,	which	makes	 them	unidentifiable.	Moreover,	 if	 the	 theft	amount	is	very	small	and	temporary,	it	is	possible	not	to	be	identified	even	when	there	is	redundancy	of	measurements.	Table	4―2	Results	of	Multiple	Bad	Data	Identification	Solution	Achieved	by	Different	Methods	Solution	Method	Identified	Theft	Points		Unidentified	Theft	Points	   dxJObservability	Constraint	Status	GA	 7,	41,	62,	70,	85	 16,	69,	71,	84	 46.725	 Satisfied	PSO	 7,	41,	62,	69,	85	 16,	70,	71,	84	 46.	725	 Satisfied	SFLA	 7,	16,	41,	62,	69,	85	 70,	71,	84	 29.094	 Satisfied		4.4 Discrepancies	between	Estimated	and	Measured	Voltages	As	 observed	 in	 Section	 4.3.4,	 the	 capability	 of	 the	 proposed	method	 to	 identify	 the	 theft	loads	 is	 limited	to	availability	of	adequate	redundant	measurements,	and	that	 the	critical	measurements	are	generally	difficult	to	identify	if	there	is	a	theft	there	or	not.	To	the	best	of	our	knowledge,	there	is	no	solution	to	identify	bad	data	for	critical	measurements	[13].	At	the	same	time,	since	the	measurement	devices	in	parts	of	distribution	system	may	not	be	sufficient	 for	 adequate	 redundancy,	 the	 number	 of	 identifiable	 bad	 data	 as	 theft	 is	 also	limited.			88 In	this	section,	voltage	magnitudes	measured	by	smart	meters	will	be	used	 in	the	second	stage	 of	 identification	 of	 theft	 points.	 After	 the	 execution	 of	 DSSE	 accompanied	 with	elimination	of	bad	data,	the	voltage	magnitude	of	loads	is	estimated	as	a	part	of	the	system	state.	In	order	to	extend	the	capability	of	the	method	proposed	in	Section	4.2,	a	criterion	is	proposed	 to	 identify	 critical	 bad	measurements	 based	 on	 the	 discrepancies	 between	 the	estimated	and	measured	voltages.	The	goal	of	this	criterion	is	to	find	the	acceptable	range	of	 discrepancy	 between	 the	 estimated	 and	 measured	 voltages	 for	 each	 load.	 If	 the	discrepancy	between	the	estimated	and	measured	voltages	violates	the	proposed	criterion,	then	electricity	theft	at	that	point	may	be	suspected.		As	explained	in	Section	3.2,	the	covariance	matrix	of	the	estimated	state	variable,	 xR ,	is	the	inverse	 of	 the	 gain	matrix.	 The	 covariance	matrix,	 xR ,	 can	 be	 partitioned	 into	 four	 sub‐matrices	 corresponding	 to	 the	 covariance	 of	 the	 estimated	 voltage	 angles	 and	 voltage	magnitudes	as	 maganglemagmagangleanglexxxxx RRRRR ,,.	(4‐11)where	 superscripts	 angle 	 and	 mag refer	 to	 angles	 and	 magnitudes,	 respectively.	 The	vector	of	standard	deviation	of	the	estimated	voltage	magnitudes	is	the	square	root	of	the	diagonal	elements	of	the	voltage	magnitude	covariance	sub‐matrix	in	(4‐12).	Therefore,	 magest diag xRσ  .	 (4‐12)	89 where	 estσ 	 is	 the	vector	of	 the	standard	deviation	of	estimated	voltage	magnitudes.	Next,	consider	the	discrepancy	between	the	estimated	and	measured	voltage	magnitudes	as	   magestmagmmagestmagactualmagmmagactualmagestmagmmagdiseeeVeVVVV 	 (4‐13)where	 magdisV 	 is	 the	 discrepancy	 between	 measured	 and	 estimated	 voltage	 magnitudes;	magactualV 	 is	 the	 actual	 value	 of	 voltage	 magnitudes	 with	 no	 error;	 magmV 	 and	 magme 	 are	 the	measured	voltage	magnitudes	and	 their	measurement	errors,	 respectively;	and	 magestV 	 and	mageste 	are	the	estimated	voltage	magnitudes	and	the	estimation	errors,	respectively.	Let	 2mσ 	denote	the	measurement	error	variances.	Given	that	 estσ 	 is	calculated	in	(4‐12),	the	mean	and	standard	deviation	of	voltage	magnitudes	discrepancy	is	     222EEEestmdismagestmagmmagdisσσσ0eeV .	 (4‐14)where	  magdisVE 	and	 disσ 	are	the	mean	and	standard	deviation	of	 the	discrepancy	between	the	 measured	 and	 estimated	 voltage	 magnitudes,	 respectively.	 Thus,	 the	 discrepancy	between	 the	measured	 and	 estimated	 voltage	magnitudes	 should	 be	within	 the	 range	 of	 disdis σσ 3,3 	with	 the	probability	 of	 99.7%	as	 a	 property	of	 normally	 distribute	 random	variables	 [13].	 In	 the	 next	 subsection,	 the	 voltage	magnitudes	 discrepancy	 for	 the	 same	system	 of	 Section	 4.3.4	 is	 calculated	 to	 find	 the	 measured	 voltage	 magnitudes	 which	violated	the	proposed	criteria.		90 4.4.1 Case	study	In	 this	 case	 study,	 it	 is	 assumed	 that	 the	 same	 theft	 points	 7,	 16,	 41,	 62,	 69,	 and	 85	 are	eliminated	as	a	result	of	multiple	bad	data	identification	method.	The	only	theft	points	left	are	 70,	 71,	 and	 84	 with	 bad	 data	 according	 to	 SFLA	 results	 in	 Table	 4‐2.	 The	 DSSE	 is	executed	for	the	system	of	Figure	4‐5.	Table	 4‐3	 shows	 the	measured	 and	 estimated	 voltage	magnitudes	 of	 the	 system.	 In	 this	table,	 the	 loads	 65,	 69,	 70,	 70,	 83,	 84,	 and	 85	 are	 identified	 as	 loads	which	 violated	 the	criteria	 proposed	 in	 Section	4.4.	 Specifically,	 the	 points	65	 and	83	 are	 identified	 as	 theft	loads	mistakenly.	To	explain	this	miss‐identification,	one	should	notice	that	the	proposed	method	has	3	sources	of	inaccuracy.	The	first	source	is	that	the	parameters	of	the	network	have	 some	 error,	 as	 explained	 in	 Section	 2.3.	 Secondly,	 the	 off‐diagonal	 elements	 of	 the	covariance	 matrix	 in	 the	 approximation	 in	 (4‐12)	 are	 neglected.	 Thirdly,	 the	 proposed	method	 assumes	 that	 the	 voltage	magnitude	 of	 a	 load	 is	most	 sensitive	 to	 its	 active	 and	reactive	powers	which	 is	 less	accurate	 for	users	at	 the	end	of	a	 feeder	 [133].	The	reason	that	points	65	and	83	are	 impacted	by	 these	 inaccuracies	 is	 that	 they	are	 located	 in	very	close	physical	distance	from	the	points	with	bad	data.	Thus,	their	measured	and	estimated	voltage	magnitudes	 are	 close	 to	 those	 points.	 Therefore,	 distinguishing	 them	 from	 theft	points	 is	 difficult.	 It	 should	 be	 noted	 that	 the	 proposed	method	 is	 narrowing	 down	 the	suspicious	points	of	theft	to	a	very	small	set	of	loads	which	is	feasible	for	utilities	to	inspect,	even	when	few	extra	points	are	identified.		91 It	 is	 important	 to	mention	that	 identification	of	bad	measurements	at	critical	points	does	not	have	any	 impact	on	 the	accuracy	of	DSSE	 results.	The	 reason	 is	 that	DSSE	 cannot	be	executed	 after	 removing	 a	 critical	 bad	 data	 and	 no	 further	 estimation	 is	 obtainable.	However,	once	the	identified	meters	are	inspected	and	the	ones	with	bad	data	are	fixed,	the	accuracy	of	state	estimation	will	be	improved.	Table	4―3	Comparison	between	Measured	and	Estimated	Voltage	Magnitudes	Discrepancies	with	3σ	Load#	 Measured	 Estimated	Absolute	of	Residual	3σ	 Load# Measured Estimated	Absolute	of	Residual	3σ	1  1.0371  1.0371  0.0000  0.0073 44  0.9928  0.9906  0.0022  0.0076 2  1.0323  1.0376  0.0053  0.0073 45  0.9915  0.9905  0.0010  0.0075 3  1.0373  1.0373  0.0000  0.0073 46  0.9928  0.9906  0.0022  0.0076 4  1.0316  1.0308  0.0007  0.0073 47  1.0079  1.0065  0.0014  0.0075 5  1.0300  1.0292  0.0007  0.0073 48  0.9922  0.9900  0.0022  0.0077 6  1.0295  1.0287  0.0007  0.0073 49  0.9918  0.9896  0.0022  0.0077 7  1.0287  1.0280  0.0007  0.0073 50  1.0189  1.0175  0.0014  0.0075 8  0.9809  0.9803  0.0006  0.0073 51  1.0109  1.0109  0.0001  0.0076 9  0.9850  0.9847  0.0003  0.0078 52  0.9992  0.9970  0.0022  0.0077 10  1.0250  1.0247  0.0003  0.0078 53  1.0184  1.0169  0.0014  0.0075 11  1.0270  1.0212  0.0058  0.0078 54  1.0110  1.0110  0.0001  0.0076 12  1.0246  1.0242  0.0003  0.0078 55  0.9988  0.9966  0.0022  0.0077 13  0.9971  0.9913  0.0058  0.0078 56  1.0021  1.0030  0.0009  0.0075 14  1.0127  1.0127  0.0000  0.0074 57  1.0110  1.0109  0.0001  0.0076 	92 Load#	 Measured	 Estimated	Absolute	of	Residual	3σ	 Load# Measured Estimated	Absolute	of	Residual	3σ	15  1.0235  1.0222  0.0013  0.0074 58  0.9983  0.9961  0.0022  0.0077 16  1.0233  1.0230  0.0003  0.0079 59  1.0021  1.0030  0.0009  0.0075 17  1.0296  1.0295  0.0001  0.0074 60  1.0210  1.0209  0.0001  0.0076 18  1.0181  1.0165  0.0016  0.0075 61  0.9983  0.9961  0.0022  0.0077 19  1.0293  1.0292  0.0001  0.0074 62  1.0263  1.0271  0.0009  0.0075 20  1.0218  1.0281  0.0063  0.0079 63  1.0028  1.0068  0.0040  0.0073 21  1.0247  1.0246  0.0001  0.0075 64  0.9985  0.9979  0.0006  0.0079 22  1.0101  1.0079  0.0022  0.0076 65  0.9979  1.0059  0.0080  0.007923  1.0095  1.0073  0.0022  0.0076 66  0.9927  0.9927  0.0000  0.0073 24  1.0093  1.0072  0.0022  0.0076 67  0.9985  0.9979  0.0006  0.0079 25  1.0093  1.0072  0.0022  0.0076 68  0.9991  0.9954  0.0037  0.0079 26  1.0089  1.0067  0.0022  0.0076 69  0.9963  1.0047  0.0085  0.007927  1.0085  1.0085  0.0000  0.0078 70  0.9976  1.0068  0.0092  0.007928  1.0068  1.0041  0.0027  0.0078 71  0.9913  1.1003  0.1090  0.007929  1.0046  1.0040  0.0006  0.0078 72  1.0147  1.0146  0.0001  0.0076 30  1.0003  1.0002  0.0001  0.0076 73  0.9986  0.9964  0.0022  0.0077 31  1.0027  1.0004  0.0022  0.0077 74  1.0127  1.0128  0.0001  0.0075 32  1.0089  1.0088  0.0001  0.0076 75  1.0127  1.0127  0.0001  0.0075 33  1.0027  1.0092  0.0066  0.0075 76  1.0121  1.0129  0.0008  0.0075 34  1.0002  1.0001  0.0001  0.0076 77  1.0154  1.0154  0.0001  0.0076 35  1.0015  0.9993  0.0022  0.0077 78  0.9995  0.9973  0.0022  0.0077 	93 Load#	 Measured	 Estimated	Absolute	of	Residual	3σ	 Load# Measured Estimated	Absolute	of	Residual	3σ	36  0.9990  0.9967  0.0022  0.0077 79  1.0141  1.0141  0.0001  0.0076 37  1.0047  1.0032  0.0015  0.0075 80  1.0099  1.0115  0.0016  0.0075 38  1.0205  1.0204  0.0001  0.0076 81  1.0167  1.0166  0.0001  0.0076 39  1.0004  0.9982  0.0022  0.0077 82  1.0004  0.9982  0.0022  0.0077 40  1.0011  1.0017  0.0006  0.0075 83	 1.0011	 1.0090	 0.0079	 0.007741  1.0208  1.0207  0.0001  0.0076 84	 1.0045	 1.0123	 0.0078	 0.007542  0.9995  0.9973  0.0022  0.0077 85	 1.0039	 1.0123	 0.0083	 0.007643  1.0199  1.0185  0.0014  0.0075	The	proposed	method	has	been	tested	on	various	distributions	of	theft	points	throughout	the	 network	 and	 similar	 performance	 of	 the	 proposed	 case	 study	 has	 been	 observed.	 In	practice,	it	is	likely	that	utilities	would	have	an	assessment	of	suspicious	regions	for	theft	in	their	distribution	systems.	Therefore,	the	theft	points	location	would	be	system	dependent.			94 CHAPTER	5: SUMMARY	OF	CONTRIBUTIONS	AND	FUTURE	WORKS	5.1 Conclusions	and	Contributions	The	main	objective	of	this	thesis	was	to	address	challenges	in	using	AMI	in	order	to	obtain	an	 accurate	 distribution	 system	 state	 estimation,	 and	 utilize	 its	 results	 to	 enhance	 the	transmission	 system	 state	 estimation.	 Furthermore,	 as	 an	 application	of	DSSE,	 electricity	theft	identification	method	was	also	proposed.	Based	on	the	contributions	presented	in	this	thesis,	all	three	original	objectives	have	been	achieved	as	summarized	below.	Objective	1	In	 the	 first	 objective,	 the	 challenge	 of	 using	 AMI	 with	 asynchronous	 measurements	 has	been	 solved	by	proposing	a	method	based	on	modeling	 load	variations	during	any	given	interval	of	the	day.	In	Chapter	2,	the	total	error	of	a	power	measurement	signal	is	assumed	to	consist	of	two	terms:	i)	the	measurement	error	due	to	imprecision	of	the	meter	device,	which	 is	 modeled	 as	 a	 normally	 distributed	 random	 variable;	 and	 ii)	 the	 load	 variation	between	 the	 measurement	 time	 and	 DSSE	 execution.	 A	 statistical	 analysis	 on	 load	variations	 has	 been	 conducted.	 According	 to	 this	 analysis,	 it	 is	 shown	 that	 the	 load	variation	can	be	represented	as	a	random	variable	with	normal	distribution,	and	its	error	increases	as	the	time	passes	from	the	sampling	instance	and	measurement	becomes	out‐of‐date.	 The	DSSE	 formulation	 has	 been	modified	 to	 incorporate	 the	 proposed	 approach	 of		95 appropriately	modifying	the	variances	in	the	estimation	procedure	as	to	give	more	weight	to	 the	 fresh	measurements	and	 reduce	 the	weight	of	 the	old	measurements,	 accordingly.	This	modification	and	methodology	is	compatible	with	commercial	packages,	which	makes	it	implementable	in	industry.		It	is	explained	that	by	using	smart	meters,	the	distribution	system	would	be	observable	and	DSSE	can	be	successfully	executed.	The	proposed	DSSE	was	tested	against	traditional	DSSE	on	 IEEE	 13‐bus	 and	 IEEE	 123‐bus	 distribution	 systems	 assuming	 that	 smart	 meters’	measurements	are	updated	every	15	minutes.	One	of	 the	criteria	was	 that	 the	developed	method	should	be	applicable	 for	both	radial	and	meshed	networks.	Therefore,	 IEEE	123‐bus	 system	 was	 modified	 to	 include	 loops.	 According	 to	 the	 simulation	 results,	 the	proposed	 method	 is	 significantly	 better	 than	 the	 traditional	 method	 and	 improved	 the	accuracy	of	DSSE	by	14%	to	16%	for	the	considered	IEEE	test	systems.	Furthermore,	 the	robustness	of	this	method	is	verified	against	highly	fluctuating	unpredictable	load	profiles.	Moreover,	as	the	noise	generated	is	a	random	phenomenon,	the	simulation	is	repeated	for	10000	times	and	the	robustness	of	the	proposed	method	and	its	consistency	in	improving	the	accuracy	of	the	estimation	is	demonstrated.	Objective	2	The	second	objective	of	this	thesis	was	to	enhance	the	transmission	state	estimation	using	AMI	 that	 is	 becoming	 available	 at	 the	 distribution	 level.	 The	 proposed	 TSSE	 method	 is	based	on	 incorporating	the	results	of	DSSE	 into	the	measurement	set	of	 the	transmission	system.	Specifically,	PQ	bus	voltage	magnitudes	and	active	and	reactive	powers	estimated		96 by	DSSE	at	the	feeder	heads	are	incorporated	into	TSSE	in	the	form	of	measurements	at	the	corresponding	 PQ	 buses.	 The	 weights	 of	 these	 additional	 measurements	 are	 calculated	from	the	gain	matrix	in	DSSE	as	a	part	of	the	method	proposed	in	Chapter	3.	Four	areas	of	improvements	 on	 two	 benchmark	 systems	 based	 on	 the	 IEEE	 14‐bus	 and	 IEEE	 57‐bus	transmission	systems	that	are	connected	to	the	IEEE	13‐bus	distribution	system	at	their	PQ	buses	have	been	investigated.		Firstly,	 it	 is	 shown	 that	 an	 unobservable	 transmission	 system	 (due	 to	 lack	 of	measurements,	 or	 measurement	 and	 communication	 failures)	 can	 become	 observable	using	 the	 additional	 DSSE	 results	 according	 to	 the	 proposed	 approach.	 In	 particular,	 2	scenarios	are	presented	in	which	a	critical	measurement	is	not	available	and	the	IEEE	14‐bus	transmission	system	is	unobservable.	Then,	 for	each	scenario,	 it	 is	explained	that	the	proposed	method	provides	adequate	measurements	which	restore	the	observability	of	the	system.	Secondly,	as	TSSE	typically	suffers	from	ill‐conditioning,	the	condition	numbers	of	the	two	benchmarks	 are	 presented	 in	 a	 simulation.	 The	 results	 show	 that	 the	 proposed	 TSSE	method	 improved	 the	 condition	 number	 by	 2	 orders	 of	magnitude	 in	 comparison	 to	 the	tradition	TSSE	method.	Moreover,	 the	 number	 of	 iterations	which	 takes	TSSE	 to	 achieve	convergence	 is	 compared	 between	 the	 proposed	 TSSE	 and	 the	 traditional	 TSSE.	 It	 is	showed	that	the	proposed	TSSE	converges	with	60%	faster	convergence	rate.	Therefore,	as	the	proposed	TSSE	expands	the	size	of	measurements	by	approximate	60%,	it	is	concluded		97 that	the	computational	burden	imposed	by	the	proposed	approach	is	almost	compensated	by	its	convergence	rate.	Since	the	proposed	TSSE	increases	the	redundancy	of	the	system,	it	facilitates	the	bad	data	identification	methods.	 In	 a	 comparison	 to	 traditional	 TSSE,	 in	 a	 case	 where	 traditional	TSSE	 fails	 to	 identify	 a	 bad	 datum,	 it	 is	 demonstrated	 that	 the	 proposed	 TSSE	 method	increases	 the	 capability	 of	 the	 normalized	 residual	 based	bad	data	 identification	method	and	identifies	the	bad	datum.	The	proposed	TSSE	resulted	in	better	numerical	conditioning,	higher	rate	of	successful	bad	data	 identification	 and	 its	 subsequent	 elimination,	 accurate	 DSSE	 calculation	 due	 to	advance	 metering	 infrastructure,	 all	 of	 which	 positively	 contributes	 to	 the	 improved	resulted	 accuracy	 of	 the	 proposed	 TSSE.	 However,	 it	 is	 explained	 that	 improvement	 of	accuracy	also	depends	on	the	resolution	of	smart	meters	measurements.	Simulation	results	showed	 that	 when	 the	 resolution	 of	 smart	 meters	 measurements	 decreases,	 the	improvement	of	accuracy	of	the	proposed	TSSE	decreases	too.	Objective	3	The	 third	 objective	 was	 to	 propose	 a	 methodology	 to	 identify	 the	 theft	 points	 in	distribution	 systems.	 In	 Chapter	 4,	 the	 idea	 of	 modeling	 electricity	 theft	 as	 bad	 data	 is	presented.	A	two	stage	method	is	proposed	to	identify	theft	points	in	a	system.	In	the	first	stage,	 assuming	 adequate	 redundancy	 of	measurements,	multiple	 bad	 data	 identification	problem	is	using	a	heuristic	optimization	method	is	proposed	to	identify	theft	points.	In	the		98 second	 stage,	 in	 order	 to	 identify	 the	 theft	 points	 without	 redundant	 measurements,	 a	criterion	 is	 designed	 based	 on	 the	 discrepancy	 of	 measured	 and	 estimated	 voltage	magnitudes.		The	 results	 of	 the	 proposed	method	 are	 demonstrated	 on	 the	 IEEE	123‐bus	 distribution	system.	It	is	shown	that	in	the	first	stage,	some	of	the	theft	points	are	identified.	However,	theft	 points	 without	 redundant	 measurements	 are	 missed.	 In	 the	 second	 stage,	 the	proposed	 method	 identifies	 the	 theft	 point	 without	 redundant	 measurements.	 It	 is	explained	that,	in	the	process	of	identifying	theft	points,	the	proposed	method	may	identify	some	points	mistakenly	as	 theft	points.	However,	 it	 is	discussed	 that	 the	 identification	of	suspicious	smart	meters,	even	combined	with	some	misidentifications,	is	beneficial	since	it	is	feasible	for	utilities	to	inspect	a	number	of	locations	which	is	small	in	comparison	to	the	entire	distribution	system.		5.2 Potential	Impacts	of	Contributions	The	 problem	 of	 non‐synchronized	 AMIs	 measurements	 in	 some	 cases	 has	 discouraged	utilities	 from	 investing	 on	 DSSE.	Without	 DSSE,	 most	 of	 the	 applications	 in	 distribution	systems	 are	 not	 implementable.	 For	 instance,	 real‐time	 power	 quality	measures	 such	 as	Voltage/Var	optimization	(VVO)	are	not	practiced	(or	are	poorly	practiced)	due	to	lack	of	monitoring	mechanisms	in	place.	The	 proposed	 DSSE	 is	 reliable	 and	 can	 be	 relatively	 easy	 implemented	 in	 industrial	programs,	 and	 therefore	 may	 encourage	 utilities	 to	 use	 non‐synchronized	 AMI		99 measurements	in	order	to	estimate	the	state	of	the	distribution	system.	This	would	not	be	an	expensive	investment	for	utilities	since	the	proposed	DSSE	is	compatible	with	existing	commercial	 packages.	 As	 a	 result,	 a	 better	monitoring	 of	 the	 system	would	 be	 achieved.	This	 is	 a	 fundamental	 step	 in	 controlling	 distribution	 systems	 and	 practice	 advanced	applications	in	DMS.	Presently,	 the	 design	 of	 smart	 meters	 has	 been	 mainly	 focused	 on	 dynamic	 billing	applications.	As	smart	meters	may	be	used	in	real‐time	operation	of	power	systems,	in	the	future,	 the	 manufacturers	 may	 redesign	 smart	 meters	 to	 provide	 additional	 metering	functions.	Operation	 of	 transmission	 systems	 at	 lower	 voltage	 level	 often	 suffers	 from	unobservability.	The	proposed	TSSE	method	renders	these	systems	observable.	This	would	enable	 the	 operators	 to	 have	 a	 better	 insight	 about	 the	 system	 state.	 Also,	 real‐time	applications	such	as	optimal	power	flow	would	include	a	larger	portion	of	the	system	and	result	in	more	comprehensive	solutions.	Also,	improvements	in	numerical	conditioning	and	accuracy	 of	 the	 system	 would	 have	 a	 direct	 impact	 on	 the	 accuracy	 of	 important	applications	 such	 as	 transient	 and	 voltage	 stability	 analysis,	 since	 these	 applications	 are	based	 on	 TSSE	 results.	 Therefore,	 the	 performance	 of	 real‐time	 applications	 would	 be	improved,	and	the	control	decisions	are	less	likely	to	be	problematic.		Finally,	the	proposed	theft	identification	method	based	on	DSSE	results	is	a	novel	attempt	at	 identifying	 the	 manipulated	 measurements.	 Electricity	 theft	 has	 greatly	 impacted	 the	revenue	of	utilities,	and	the	proposed	method	has	a	potential	to	reduce	this	damage.	Also,		100 sometime	 smart	 meters	 may	 expand	 the	 vulnerability	 of	 metering	 infrastructure,	 and	because	 of	 that	 utilities	may	 be	 disincentivized	 from	 deployment	 of	more	 smart	meters.	Consequently,	 the	 implementation	 of	 applications	 such	 as	 dynamic	 pricing	 and	 demand	response	 management,	 which	 require	 smart	 meters,	 may	 be	 hindered.	 However,	 the	proposed	 method	 will	 be	 a	 positive	 influence	 towards	 installing	 more	 smart	 meters	 by	reducing	the	risk	to	energy	theft.	5.3 Future	Work	One	possible	improvement	in	the	DSSE	method	proposed	in	Chapter	2	could	be	considering	various	 statistical	 distributions	 and	 finding	 a	 model	 which	 represents	 the	 load	 more	accurately	 than	 the	 normal	 distribution	 based	 models.	 In	 the	 next	 step,	 combining	 the	chosen	 distribution	 with	 normally	 distributed	 measurement	 noise	 should	 yield	 more	accuracy.	 Moreover,	 trying	 other	 state	 estimation	 techniques	 such	 as	 nonlinear	 Kalman	Filter	 could	 result	 in	 improvements	 of	 the	 results.	 Another	 area	 of	 research	 could	 be	studying	the	impact	of	modeling	loads	as	dependent	random	variables	which	may	conform	to	similar	profiles.	As	mentioned	in	Chapter	3,	if	the	resolution	of	smart	meters’	measurements	is	very	low,	the	DSSE	results	will	not	be	accurate	enough	to	improve	the	accuracy	of	TSSE	when	fast	load	variation	 occurs.	 To	 find	 a	 method	 that	 may	mitigate	 this	 problem	 could	 be	 a	 research	topic.	For	example,	developing	accurate	load	forecasting	methods	and	incorporating	them	into	 the	 SE	 framework	 could	 improve	 the	 results.	 The	 reliability	 of	 the	 developed	forecasting	 methods	 should	 be	 studied.	 It	 is	 important	 to	 notice	 that	 in	 real‐time		101 applications,	 the	 reliability	 of	 the	 methods	 used	 is	 more	 important	 than	 it	 is	 in	 off‐line	applications	such	as	planning.	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