New Hierarchical Signal Processing for Tractable Power Flow …dkundur/pub_pdfs/SriKunTSIPN19.pdf · 2020. 1. 21. · 86 IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER

86 IEEE TRANSACTIONS ON SIGNAL AND INFORMATION PROCESSING OVER NETWORKS, VOL. 5, NO. 1, MARCH 2019

Hierarchical Signal Processing for Tractable PowerFlow Management in Electric Grid Networks

Pirathayini Srikantha , Member, IEEE, and Deepa Kundur , Fellow, IEEE

Abstract—Rapid advancements in smart grid technologies havebrought about the proliferation of intelligent and actuating powersystem components such as distributed generation, storage, andsmart appliance units. Capitalizing fully on the potential benefitsof these systems for sustainable and economical power genera-tion, management, and delivery is currently a significant challengedue to issues of scalability, intermittency, and heterogeneity of theassociated networks. In particular, vertically integrated and cen-tralized power system management is no longer tractable for op-timally coordinating these diverse devices at large scale while alsoaccounting for the underlying complex physical grid constraints. Toaddress these challenges, we propose a hierarchical signal process-ing framework for optimal power flow management whereby thecyber-physical network relationships of the modern grid are lever-aged to enable intelligent decision-making by individual devicesbased on local constraints and external information. Decentral-ized and distributed techniques based on convex optimization andgame theoretic constructs are employed for information exchangesand decision-making at each tier of the proposed framework. Itis shown via theoretical and simulation studies that our techniqueallows for the seamless integration of power components into thegrid with low computational and communication overhead whilemaintaining optimal, sustainable, and feasible grid operations.

Index Terms—Adaptive signal processing, optimization, smartgrids.

I. INTRODUCTION

THE rapidly evolving nature of the power grid today hastransformed it into a highly diverse and dynamic net-work[1]. For example, the recent cyber-enablement of powersystem components, the rising penetration of sustainable Dis-tributed Generation (DG) sources, the commercialization ofsmart appliances and the introduction of competitive energymarkets promote a broad spectrum of extended grid capabilitiessuch as advanced monitoring/control and sustainable operations[37], [38]. However, several open challenges deter full realiza-tion of such functionalities.

The main difficulties stem from the lack of compatibility ofmodern power devices with aging legacy infrastructure that still

Manuscript received November 10, 2017; revised May 9, 2018 and June 25,2018; accepted July 8, 2018. Date of publication July 22, 2018; date of currentversion February 5, 2019. This work was supported by Natural Sciences andEngineering Research Council of Canada. The associate editor coordinating thereview of this manuscript and approving it for publication was Dr. D. Mandic.(Corresponding author: Pirathayini Srikantha.)

P. Srikantha is with the Department of Electrical and Computer Engineering,Western University, London, ON N6A 3K7, Canada (e-mail:,[email protected]).

D. Kundur is with the Department of Electrical and Computer Engineering,University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail:,[email protected]).

Digital Object Identifier 10.1109/TSIPN.2018.2858750

comprises a major part of the physical grid. For instance, theDistribution Network (DN), by design, transports power fromthe bulk grid to individual consumers. Now, with significantpenetration of DGs (e.g. solar roof-top panels, etc.) at closeproximity to consumers, power flow can conceivably occur inthe reverse direction during excess generation periods. As distri-bution lines are associated with low reactance to resistance ratio,voltage rise is more substantial and excessive reverse power flowcan then result in adverse equipment damages [2].

Another challenge is due to the vertically-integrated grid man-agement paradigm (still utilized by many system operators).Competitive energy markets resulting from deregulation havespurred the deployment of private Independent Power Plants(IPPs). Directly controlling an immense number of these in-termittent components in a centralized manner, as typical invertically-integrated processes, is costly and unscalable. With-out coordination in place, increasing congestion is evident inthe Transmission Network (TN) which transports bulk poweracross the grid with lines operating close to maximum limits[3]. On the other hand, upgrading the infrastructure to accom-modate these changes will result in exorbitant costs. Hence, itis necessary to employ alternative strategies to overcome theseissues.

In this paper, we propose a hierarchical network-basedsignal processing framework that harnesses the intelligenceand communication capabilities of actuating nodes to automatedecision-making based on external information and localfeasibility constraints. The structural characteristics of physicalpower networks residing at the transmission and distributionlevels along with the highly granular monitoring informationcontinuously generated in the grid are leveraged in the design ofsignals exchanged amongst these cyber-enabled nodes. Thesesignals encapsulate general trends in the system which allowindividual actuating nodes to adaptively and optimally respondto dynamically occurring changes while being attuned to localgrid constraints and feasibility requirements. Abstraction anddecoupling built into each tier of the proposed hierarchicalsystem allows for the plug-and-play integration of a largenumber of diverse entities across TN and DN levels of thepower grid in an optimal manner.

As such, our contributions are four-fold in this paper: 1)A novel hierarchical signal processing paradigm is presentedalong with a literature survey in this area; 2) A proof-of-conceptof how this framework can be effectively utilized for coordi-nation over short horizons in the electric grid at the transmis-sion and distribution network levels is presented using convex

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https://orcid.org/0000-0001-7453-2825https://orcid.org/0000-0001-5999-1847mailto:[email protected]:[email protected] advance �reakcnt @ne penalty -@M utoronto.camailto:[email protected] advance �reakcnt @ne penalty -@M utoronto.ca

SRIKANTHA AND KUNDUR: HIERARCHICAL SIGNAL PROCESSING FOR TRACTABLE POWER FLOW MANAGEMENT IN ELECTRIC GRID 87

optimization [30] and game theoretic techniques [31]; 3) Designof signals, information exchange paradigm and actuation deci-sions are presented for both the TN and DN levels; and 4) Resultsfrom theoretical performance analysis and practical simulationsare included along with comparison to existing state-of-the art.This paper is organized as follows. The Optimal Power Flow(OPF) formulation used by system operators for coordinatingthe grid is presented in Sec. II along with a discussion of the as-sociated challenges. This section is concluded with a literaturesurvey. In Sec. III, the hierarchical model is presented. In Sec. IVand V, decentralized and distributed coordination at the TN andDN levels are discussed. Mathematical tractability of the pro-posed framework is discussed in Sec. VI. Simulation results arepresented in Sec. VII. Conclusions follow in Sec. VIII.

II. OPTIMAL POWER FLOW AND CHALLENGES

The Optimal Power Flow (OPF) formulation is typically con-structed by the system operators to compute the minimal costconfiguration of participating active nodes (e.g. power con-sumers, DGs, etc.) that are subject to power flow, bus voltage,power demand and generation constraints [4]. The most generalform of the OPF problem is listed in POP F which is applicableto both TNs and DNs:

POP F : minP,Q,|V |,Θ

∑

i∈B

(fi(p

gi ) + fi(q

gi ) + fi(p

ri )

)

∀ i ∈ Bpi = p

gi − pdi + pri , qi = qgi − qdi (C0)

pi =∑

j∈Ni ,i|Vi ||Vj |

(gi,j cos(θi − θj ) + bi,j sin(θi − θj )

)(C1)

qi =∑

j∈Ni ,i|Vi ||Vj |

(gi,j sin(θi − θj )− bi,j cos(θi − θj )

)(C2)

pgi≤ pgi ≤ p̄gi , qgi ≤ q

gi ≤ q̄gi , pri ≤ pri ≤ p̄ri , V i

≤ |Vi | ≤ V̄i (C3)The OPF problem is constructed on a physical network graphdefined by buses B, generators G and admittances (i.e. weightof each line). The optimization variables of POP F consist ofthe real and reactive power injections along with bus voltagemagnitudes and angles which are represented by vectors P ∈R|B|, Q ∈ R|B|, |V | ∈ R|B| and Θ ∈ R|B| respectively. Realpower injection pi ∈ P at each Bus i ∈ B is further decomposedinto three parts: power injected by local generators pgi , actualbus power demand pdi and power demand reduction by flexiblelocal consumers pri . These result in pi = p

gi − pdi + pri where

pi ∈ P . Similarly, reactive power injection at Bus i ∈ B is de-fined as qi = q

gi − qdi where qi ∈ Q. The objective of POP F

is a summation of functions operating on real and reactivepower generation at each bus and demand reduction by flexibleconsumers.

Line (i, j) connects Buses i and j and transports power be-tween these. The associated bus admittance yi,j of (i, j) is de-fined by conductance gi,j and susceptance bi,j parameters whichcapture power transfer properties across Bus i and Bus j. Theset Ni is composed of all buses that are directly connected to

Bus i via a single line. Constraints 1 and 2 enforce power bal-ance at each bus for real and reactive power injections. Capacityconstraints and voltage stability requirements impose upper andlower limits on variables pgi , p

di , p

ri and Vi . Upper limits are

denoted by parameters p̄gi , p̄di , p̄

ri and V̄i respectively. Similarly,

lower limits are represented by the parameters pgi, pd

i, pr

iand V i

respectively. A realistic power system consists of a large numberof buses that are associated with highly fluctuating parametersand variables. Efficient processing of information and tailoredcommunications are necessary for optimal actuation which isthe focus of this paper.

A. Challenges

The main difficulty in directly solving POP F stems fromconstraints (C1) and (C2) (derived from Kirchhoff’s voltageand current laws) representing the net real and reactive powerinjections into Bus i. Quadratic and sinusoidal terms contributeto the non-convexity of these equality constraints. An optimiza-tion problem consisting of non-convex constraints is classifiedas NP-Hard. Problems falling in this class cannot be solved ina tractable manner especially for large systems [5]. Moreover,the underlying structure of the graph representing the physi-cal system is a large and complex network consisting of mixedtopologies (i.e. mesh and/or tree) and significant coupling forwhich analysis is not straightforward.

These difficulties are further exacerbated with the blurringof traditionally well-defined roles associated with power con-sumers and generators. Consumers are now actively generatingsustainable power via roof-top solar panel and micro-wind tur-bine DGs. Power loads are no longer static as consumers haveinherent demand flexibility that can be effectively leveraged us-ing Energy Management Systems (EMSs) to promote sustain-able energy consumption. Deregulation has enabled privatelymanaged IPPs to competitively participate in active generation.This movement has introduced a large number of optimiza-tion variables and highly fluctuating parameters that have ledto significant unpredictability in the grid and frequent systemcongestions.

B. Literature Review

Existing literature pertaining to optimal power flow man-agement is extensive. Generally, these techniques apply someform of simplification to the original OPF listed in POP F toovercome the challenges outlined above. The simplified prob-lem is then solved using one of centralized, decentralized andhierarchical grid network management techniques. This hierar-chical signal processing technique essentially summarizes thepertinent information about lower tiers and abstract unnecessarydetails which will otherwise add significant complexity to theactuation process.

1) OPF Simplification Techniques: The simplest approachin the literature entails completely eliminating the problem-atic constraints from the optimization formulation [6]. As theseconstraints capture the underlying physical grid properties, ig-noring these will not be appropriate for real-time power supplyand demand matching in today’s aggressively utilized powernetworks. Linearization methods are also investigated widely in


the literature. A commonly used technique applies a series ofsimplifications to eliminate reactive power flows and bus voltagemagnitudes from the OPF. This method also excludes importantvariables capturing the underlying physical system attributes.Other related work such as reference [7] applies double approx-imation where the cosine terms in the power balance equationsare first replaced with quadratic terms. Then, another relaxationis applied to the resulting quadratic equality constraints. The re-sulting formulation consists of quadratic inequality constraintsthat are convex but computationally expensive than linear con-straints. Reference [8] linearizes these quadratic terms via theloss factor-based linearization method based on sensitivity anal-ysis. Other relaxation techniques such as cone and semi-definiterelaxations are applied instead to obtain a tight convex repre-sentation of the original feasible set formed by (C1)–(C3) [4].These result in exact solutions for radial power networks (e.g.DNs) under some mild conditions [9]. These conditions, how-ever, fail in mesh topologies which typically represent TNs [10].To render convex relaxations exact in these systems, the authorsof [11] propose the strategic addition of Flexible AlternatingCurrent Transmission System (FACTS) devices. As these de-vices are costly, this is an expensive approach. Another optioninvolves applying heuristics which are commonly used to solvenon-convex optimization problems [12]. These provide no guar-antee with respect to the convergence rate and optimality of thecomputed solution.

2) Centralized Strategies: Centralized grid management in-volves the computation of optimal operating setpoints of actuat-ing grid entities by the system operator over long time horizons(e.g. once a day or every hour) using the original or simpli-fied version of the OPF [4]. Forecast models are leveraged tocompute system parameters (e.g. generation capacities and de-mands) over these extended periods. These models are howeverassociated with significant error margins especially due to thepresence of renewable resources and highly fluctuating powerconsumers in the system [13]. Smaller the time horizon con-sidered, the lower is the associated error of the prediction algo-rithms. However, with smaller coordination horizons, the systemoperator is subject to significant computational and communi-cation overheads entailed in frequently solving the selected OPFfor a large number of optimization variables.

3) Decentralized Strategies: In decentralized strategiesproposed in the literature, every actuating element typicallyparticipates in iteratively computing the optimal solution.Communication signals are typically exchanged betweenneighbouring nodes only. Thus, the central point of controlis completely eliminated in this grid coordination paradigm.Methods that include dual decomposition, consensus andmonotone operators are applied to a relaxed version of the OPFto compute the iterative decision-making by participating nodes[6], [14], [15], [38]. As the convergence rates of these strategiesare proportional to the number of agents participating in thecoordination process, greater the number of agents coordinatedthe more time is required for convergence. Other coordinationmechanisms that are based purely on local measurements aremyopic and examples include droop control employed bygeneration systems to adjust to instantaneous fluctuations in fre-quencies and power consumption [3]. Although control at this

level addresses transient disturbances, these are not designedfor optimality due to limitations in the information available.

4) Hierarchical Strategies: As managing power flow acrossthe entire grid at high granularity is an intractable task, propos-als such as reference [17] have divided the coordination processinto general tiers such as tertiary, secondary and primary con-trol layers. The computational methods evoked at each layer arebased on either centralized or decentralized methods outlinedin the above. For instance, in practical settings, coordination atthe tertiary level entails planning and for this only real powerbalance is considered (i.e. reactive power and voltages are ig-nored). The control horizon in this tier can be as long as oneday and central optimization is utilized here. In the secondarytier, central computations are conducted over hourly intervalsusing dynamic security assessment to compute the operating set-points of generation systems [18]. In the primary control level,droop control mechanisms are utilized by large synchronousgeneration sources to automatically adapt to fluctuations. Theafore-mentioned hierarchical coordination system is ideal fora grid consisting of predictable generation systems and powerconsumers. This is, however, no longer the case as today’s gridconsists of a large number of highly fluctuating actuating en-tities. Recent work on hierarchical approaches for power flowmanagement utilize constructs such as clustering and partition-ing to decompose the power grid into regions consisting ofpower elements that have significant coupling amongst one an-other [19], [20]. Local coordination is conducted within thesepartitioned regions and then combined in the subsequent tier.As the grid topology is no longer static due to the presenceof renewable generation, electric vehicles and smart breakers,these static analyses may not account for the dynamic nature oftoday’s grid. Other hierarchical methods proposed for managingsmall microgrid communities cannot be applied to manage anentire power grid [21], [22].

III. HIERARCHICAL MODEL

There are three main challenges associated with optimallycoordinating the modern power grid:

1) Presence of a large number of optimization variables andnon-convex constraints;

2) Unpredictability of fluctuating actuating elements; and3) Congestions due to power infrastructure limitations.Effectively overcoming these issues entails the incorporation

of abstraction and independent decision-making. For this,we leverage a hierarchical steady-state power managementapproach illustrated in Fig. 1 composed of three tiers. Thesethree tiers are inspired by the primary, secondary and tertiarygrid management paradigms commonly adopted by gridoperators [4]. Main differences between the proposed andexisting frameworks stem from the granularity of coordination,signals exchanged between actuating entities and incorporationof physical power flow constraints. Decision-making at everytier by each node depends only on:

1) Local aggregate conditions of components managed in thesubsequent tier; and

2) External information encapsulating the general state ofthe system with respect to optimality.


Fig. 1. Hierarchical Power Flow Management.

TABLE IHIERARCHICAL COORDINATION SUMMARY

The large grid coordination problem is now divided into man-ageable sub-problems and details of external entities are hidden.Hence, the number of variables handled by each node is a smallsubset of the original problem. The coordination horizon con-sidered at each tier is divided into various scales ranging froma few minutes to seconds with granularity increasing from topto bottom of the hierarchy. This division of the coordinationhorizon reduces error margins associated with using predictionmodels over long forecast periods. Moreover, nodes at each tierutilize communication paradigms that are uniquely suited forthe underlying physical requirements and associated resourceconstraints. Table I presents a summary coordination exactedacross the three tiers with details listed in the following.

A. Coordination at Tier 1

The lowest tier (Tier 1) consists of individual power con-sumers, Independent Power Plants consisting of renewable gen-eration and synchronous generation systems. It is important tonote that coordination at this tier is local to individual con-sumer and generation entities. The coordinating agents consistof EMSs that reside within consumer premises and power con-trollers in generation systems. These agents residing in Tier1 will optimally actuate based on local feasibility (e.g. con-sumer comfort, local appliance statuses, local feeder bus volt-ages, and generation capacities) and the general signal broad-cast by the corresponding managing entity (e.g. IPP Agent, DN

Agent, Plant Agent) in Tier 2. To determine local feasibility, ifthere is more than one node in the system (e.g. multiple windturbines in wind farms, consumers in DNs), nodes will commu-nicate with one another as necessary within their local system(e.g. local elements in the wind farm, local consumers in theDN). The time scale for coordination is in the range of secondsto accommodate for the constant flux in operating conditionsof local loads and generation entities. The managing entity inTier 2 will broadcast a general signal every one second andlocal aggregate measurements are reported to the managing en-tity every one second. As point-to-point communication withindividualized signals is not established by the managing entityto every consumer and generation system, computational andcommunication overheads are significantly reduced. Aggregatemeasurements can be readily reported by local data concen-trators commonly present in the grid as part of the AdvancedMetering Infrastructure (AMI) by aggregating sensor measure-ments generated by abundantly deployed Phasor MeasurementUnits (PMUs) [4].

B. Coordination at Tier 2

Tier 2 consists of agents (e.g. IPP, DN or Plant agents) rep-resenting individual systems such as synchronous generationplants, IPPs and DNs that directly connect to buses interfacingwith the TN. Hence, there can be multiple IPP, DN and Plantagents in Tier 2 (depending on the number of these systems


present in the grid). These agents manage the overall power in-jection of the individual systems that are directly coordinatedby these. For example, power injections by synchronous gener-ation plants and IPPs are managed by the associated individualplant authorities whose control entities serve as the Plant or IPPagents. Real power injection by a DN is indirectly managed bythe associated Electric Power Utility (EPU) which serves as aDN agent. These agents will iteratively compute general signalsevery second and broadcast these to their associated controlelements using general signals received from Tier 3 and aggre-gate information conveyed from Tier 1. The main objective ofthese agents is to coordinate the aggregate behaviour of theirlocal systems within a control horizon of 60 seconds based onthe requirements conveyed by the general signals relayed fromTier 3. The general signals relayed from Tier 3 are transmittedevery ten minutes and the Tier 2 agents will reformulate theiroptimization problem every ten minutes based on these generalsignals. Information is not exchanged between agents residingin Tier 2. Tier 2 agents will also forecast aggregate parametersof the Tier 1 system (e.g. wind farm, DN) at every ten minuteinterval based on information relayed by the data concentratorsin Tier 1 and send this to the managing entity in Tier 3. Sincethe forecast interval is very short (i.e. ten minutes) and is forindividual systems composed of small number of entities, errormargins will be very low.

C. Coordination at Tier 3

Tier 3 consists of agents representing each bus in the TN.TN agent i representing Bus i, communicates with TN agentsrepresenting neighbouring buses to iteratively compute net realpower injection pi in an iterative manner based on ten minuteforecasts of demand/supply sent by local agents in Tier 2 andneighbouring TN constraints in Tier 3. Information exchangebetween neighbouring buses takes place every 10 milliseconds(to account for latencies in communication) [1] so that all theTN agents will converge to an optimal equilibrium within tenminutes. This optimal power injection computed by every TNagent is then relayed to local Tier 2 agents (i.e. DN, IPP and/orPlant agents). The coordination horizon of ten minutes in Tier 3is significantly more granular than day-ahead unit commitmentwhich is typically engaged for coordination at the TN level [4].This granularity enables active and adaptive response by activenodes to short-term volatilities in the TN. Moreover, new TNnodes can be added to the system in a plug-and-play manner asthere is no overhead associated with centralized administration.This model is well-suited for today’s evolving landscape ofcompetitive markets and deregulations.

D. Summary

The hierarchical signal processing and actuation setup sum-marized in the above allows for the complete abstraction ofindividual power component operations in a complex powernetwork from a system-wide controller. At each tier, the agentsutilize aggregate information and signals exchanged with neigh-bouring or managing entities to make local decisions. Thismarkedly reduces communication overhead otherwise needed toforge individual communication links with all actuating nodes.

Moreover, as computations are divided and parallelized amongsttiers and nodes, computational overhead is also greatly dimin-ished. Although hierarchical management is not a unique con-cept (e.g. reference [6]), our work is a novel departure due tothe granularity of optimization, scalability and the integrationof various communication topologies based on the underlyingphysical constraints of the system. The primary, secondary andtertiary (PST) grid management paradigm utilized widely todayto coordinate grid operations has some parallels to Tiers 1, 2 and3 respectively in the proposed hierarchical framework [4]. Onecommon feature of these frameworks is the manner in which en-tities are organized within each levels. This is essentially basedon the ownership and management structure of power system.For instance, in the PST framework, EPUs are organized basedon geographical contexts and deliver power to directly to con-sumers. System operators deal with balancing power demandwith supply at the TN level.

However, there are some marked differences. At the ter-tiary control level, EPUs purchase power from system operatorsbased on day-ahead markets. Thus, the system operators utilizeforecasts of demand and supply over very large time horizons(i.e. 24 hour period). In the secondary layer, spot-markets dealwith hourly changes and the associated problem formulationsincorporate only real power balance and bus angles. Reactivepower and bus voltage magnitudes are ignored. Generation set-points are computed and communicated to generators at hourlyintervals which are maintained at the primary level. Controllersin individual generators adapt to instantaneous fluctuations inpower demands by utilizing droop control techniques to main-tain the system frequency at nominal levels and this is entirelybased on local information. The tertiary layer results in signif-icant error margins due to forecast errors, the secondary layerneglects important grid variables such as reactive power and busvoltage magnitudes that will lead to inefficiencies and finally theprimary level leads to completely myopic decisions that do notaccount for the conditions of other entities in the system. In orderto ensure that the system functions within acceptable stabilitylimits in the event of unexpected events, contingency analysesare conducted by system operators and these cannot effectivelyaccount for fluctuations inherent in DGs. Moreover, actuationof flexible consumers is not included in this model.

In our framework, we incorporate power flow constraintsin all tiers. In Tier 3, TN nodes exchange information withneighbours to compute physically viable setpoints. In Tier 1,DN nodes exchange information with neighbouring nodes toensure that the impending switch satisfies physical constraints.Tier 2 bridges Tier 3 and Tier 1 via generalized broadcast signals.The next differentiating attribute is that the largest coordinationhorizon is associated with Tier 3 which is ten minutes in length.This results in a significant reduction of error/inefficiencies dueto forecast uncertainties.

E. Proof-of-Concept

Next, as a proof-of-concept, we will demonstrate in the re-mainder of this paper how the hierarchical signal processingmodel outlined in the above can be applied for practical coordi-nation in the power grid. For Tier 3 coordination, we adopt the


OPF formulation introduced in reference [23] which applies aseries of transformations to obtain a close approximation of theoriginal non-convex power flow constraints via a set of linearequality and inequality constraints without eliminating impor-tant features of the problem such as reactive power flow andbus voltages in the TN. Although the number of constraints willincrease with this technique, these linear constraints capture thebasic nuances of non-convex terms up to a fairly high level ofaccuracy and allow for greater computational efficiency [23].Moreover, this technique has been recommended in referencessuch as [10] as a viable alternative for modelling the TN. Themain difference between our representation of the approximatedpower flow constraints and that proposed in [23] is our use ofbus admittances instead of line admittances. We assert that busadmittances allow for the incorporation of realistic effects onpower flow due to transformers, shunt admittances and linecharging susceptances. We then design decentralized coordina-tion for Tier 3 by applying ADMM to the afore-mentioned OPFmodel. ADMM is typically applied to relaxed OPF formulationsassociated with computational overheads stemming from thepresence of quadratic inequality constraints constructed for theDN with radial connection topology (e.g. [15]) or non-convexOPF (e.g. [24]) which may not converge. Our novel contribu-tion in the design of decentralized Tier 3 coordination is that weconstruct the optimal coordination problem at the transmissionnetwork level so that it is decomposable and separable at eachactuating node even in the presence of cycles/loops which aretypical in mesh networks while accounting for reactive powerand bus voltage magnitude constraints. Signals designed usingADMM now encapsulate trends from the perspective of everycyber-enabled node arranged in a mesh configuration regardingoptimality and feasibility in the transmission network. Thesesignals that are generated and exchanged amongst actuatingnodes are used to iteratively improve local actuation. These it-erative revisions converge to the optimal solution even in thepresence of loops due to the convexity of the decomposed prob-lem. Moreover, important grid characteristics are retained in thelocal decision-making process. Accuracy of solutions in prac-tical Tier 3 systems obtained via our method are presented inSec. VII.

For Tier 2 coordination, we utilize decomposition and pop-ulation game theory to design general signals that will be uti-lized by nodes located in the DN, IPPs and/or large scale plantsto make local actuation decisions. In the literature, distributedmethods based on sub-gradient, consensus and game theoreticapproaches have been proposed (e.g. [14], [25], [26]). Mainchallenges associated with these are the speed of convergenceand scalability. As per our discussion on Tier 2 coordination, acompletely decentralized solution is not necessary as aggregatemeasurements can be utilized to construct general signals thatguide individual entities in Tier 1. We utilize population gametheory for designing general signals. The incremental impactof actuation by individual active nodes enables the applicationof population game theory in designing generalized broadcastsignals that iteratively guide these nodes with minimizing thesystem cost in a distributed manner. The dynamics resultingfrom these strategy revisions are guaranteed to converge to theoptimal configuration. Local feasibility checks by the revising

node ensure that DN bus voltage and apparent power flow con-straints are met by impending changes in actuation. This differsfrom our earlier work in [27], [28] as we design signals basedon Tier 3 and aggregate conditions in Tier 1. Response by indi-vidual elements in Tier 1 is based on signals transmitted fromTier 2 and local feasibility conditions.

In summary, the proposed signal processing framework al-lows for the coordination of diverse actuating elements residingin the complex and highly coupled electrical network. Structuralfeatures of the underlying physical networks along with highlygranular monitoring information generated in the grid are lever-aged to design and refine signals constructed and exchanged inthe power system across the transmission, distribution and localactuation tiers.

IV. COORDINATION IN TIER 3

The design of decentralized information exchange betweenTN agents for iteratively determining optimal power injection inTier 3 where agents are arranged in a mesh topology as typicalin the TN is detailed in the following.

A. TN Line Characteristics

Main attributes of the TN that affect power flow are lineadmittances yli,j = g

li,j + ib

li,j , line charging susceptances y

ci,j ,

shunt admittances ysi,i and transformer tap ratios |Ti,j | and phaseshifts si,j (i.e. Ti,j = |Ti,j |∠s◦i,j ) which are combined to formthe Y-bus matrix consisting of the following elements:

yi,i =∑

j∈Ni

2yli,j + yci,j

2|Ti,j |2 + ysi,i , yi,j =

−yli,jIi,j T ∗i,j + Ij,iTi,j

SetNi consists of all buses that are adjacent to Bus i (i.e. directlyconnected to Bus i via a line) and Ii,j is an indicator function thatreturns 1 if line i↔ j is in the direction of stepped-down voltageand 0 otherwise. |Ti,j | and si,j for lines without transformersare set to 1 and 0 respectively. Each element of the Y -bus matrixyi,j can be separated into real and reactive components whichare referred to as bus conductance gi,j and bus susceptancebi,j respectively. These line properties, connecting bus anglesand voltages dictate the net real (pi) and reactive (qi) powerinjections into each Bus i as listed in (C1) and (C2) ofPOP F andthese are non-convex constraints. The following approximationsare applied to transform these into a set of convex constraints.

B. Approximation of Power Flow Constraints

First, consider the real power flow constraint (C1). Real powerflow is influenced significantly by differences in bus anglesrather than bus voltage magnitudes. For this reason, the voltagemagnitude variables |Vi | and |Vj | are replaced by constants |V ti |and |V tj | that can either be bus voltage magnitude setpoints orthe current voltage magnitude measured prior to a change inthe system. This eliminates the voltage magnitude variables in(C1). Next, the sin(θi − θj ) term is replaced by θi − θj as thephase angle differences are very small (close to zero) and thesine function is approximately linear close to the origin. Finally,the cosine term is approximated by a set of linear inequal-ity constraints as outlined in reference [23]. We do not apply


Fig. 2. Approximation of cos(θ) via Linear Inequalities.

first-order Taylor series expansion to approximate this functionas the cosine term is not linear around the origin. A new variableˆcosi,j subject to the following n + 2 constraints is introduced

to replace the cos(θi − θj ) term:ˆcosi,j ≥ 0 (C1′)ˆcosi,j ≤ −sin(ak )(θi − θj − ak ) + cos(ak ) ∀ k ∈ n (C2′)ˆcosi,j = ˆcosj,i (C3′)

where ak = − π2 + πn+1 k. The inequalities are linearly depen-dent on the bus angles θi and θj . The third constraint ensuresthat the ˆcos terms are the same for both directions of the line.

Fig. 2 illustrates how the positive region enclosed by thecos(θ) function is approximated by (C1’–C3’) when n = 9. Themaximum value ˆcosi,j can take after being subjected to theseinequalities results in a close approximation of cos(θi − θj ).Hence, in the minimizing objective, an additional term − ˆcosi,jis added to obtain this close approximation. Combining all ofthese steps results in:

pgi + pri − pdi =

∑

j∈Ni ,i|V ti ||V tj |

(gi,j ˆcosi,j + bi,j (θi − θj)

)(C4′)

(C1’)–(C4’) along with the additional − ˆcosi,j term in the ob-jective form the linear approximation of (C1).

Next, the reactive power flow constraint (C2) is linearized byapplying another set of approximations. As reactive power flowsignificantly depends on changes in voltage magnitudes, it isnecessary to take these into account in the approximation. Toenable this, the bus voltage |Vi | is separated into two components|Vi | = |V ti |+ φi . |V ti | is the target voltage magnitude and φirepresents the deviation of the actual voltage magnitude fromthe target. |V ti | is known in advance and is therefore a constant.φi is a new variable. Substituting these into the quadratic voltagemagnitude terms in (C2) results in:

|Vi ||Vj | = |V ti ||V tj |+ φi |V tj |+ φj |V ti |+ φiφjThe first term in the above relation is constant. This termmultiplies into gi,j sin(θi − θj ) + bi,j cos(θi − θj ). Applying

the linear approximation of the sine and cosine terms resultsin |V ti ||V tj |(gi,j (θi − θj ) + bi,j ˆcosi,j ). However, the remainingthree terms are still problematic as these consist of non-linearcomponents such as φi(sin(θi − θj )), etc. First-order linearterms in the Taylor series expansion of these nonlinear termsreplace these where φi , φj and θi − θj are evaluated around 0as these are expected to be close to 0. These changes result inthe following linearized approximation of (C2):

qgi − qdi =∑

j∈Ni ,i

[|V ti ||V tj |(gi,j (θi − θj )− bi,j ˆcosi,j

)

− (φi |V tj |+ φj |V ti |)bi,j]

(C5′)

C. Approximation Errors

Three main approximations are made in the process outlinedabove. First, sin(θi − θj ) is replaced with θi − θj . As the phaseangle differences between TN buses are typically close to 0, thelinear term adequately approximates the sine term around theorigin. Next, the cosine term is replaced with a new optimizationvariable ˆcosi,j which is subject to linear inequality constraints.As evident in Fig. 2, when the number of line segments increases,the greater is the accuracy of representation of the cosine term.We utilize 21 line segments in our practical studies presented inSec. VII to approximate the cosine term. The − ˆcosi,j term isadded to the objective function of the linearized OPF to obtaina close approximation of the original cosine function. Althoughthis term can introduce an offset to the optimal value of theobjective function, values taken by ˆcosi,j ∈ [0, 1] is negligiblein comparison to the values taken by the original objective func-tion f of the OPF. Finally, the remaining non-linear terms inthe power balance equations are approximated using the first-order Taylor series expansion around the origin with error thatis quadratic (in the worst case) with respect to the deviationof independent variables from the origin. As this deviation isexpected to be minimal, this error is inconsequential.

D. Separable Formulation

After applying the aforementioned transformations, the origi-nal power flow equations can now be approximated by the linearconstraints (C1′)–(C5′). PcOP F is convex as all constraints arenow linear and the objective is convex [4]:

PcOP F : minP,Q,Φ ,Θ , ˆcos

∑

i∈B

(fi(p

gi) + fi(q

gi) + fi(p

ri))−

∑

i↔jˆcosi,j

s.t. (C1′)–(C5′), (C3), ∀ i, j ∈ BAlthough this problem can be solved exactly by commerciallyavailable solvers, it is difficult to centrally coordinate fluctuatingrenewable generation entities and consumer demands at a highlygranular level while ensuring that the system remains withinsecure operational limits [18].

A decentralized approach, whereby all the active power in-jecting buses in the system iteratively contribute to the op-timal solution by performing individual computations basedon locally changing demand and generation requirements,will alleviate significant overhead otherwise incurred throughcentral coordination. Moreover, this also allows every active


participant to adapt to changes in the system in an automatedand cost effective manner. The decentralized algorithm pro-posed in this paper depends on information exchanges betweenneighbouring buses. Hence, the OPF must be formulated so thatit is separable amongst all the buses in the system and each TNagent representing a bus depends only on information updatessent by neighbouring buses to solve its local problem. Each TNagent performs iterative updates of its local solution by lever-aging on ADMM. Convergence to optimality is guaranteed forthis method as the problem is convex [29].

1) Background on ADMM: Next, a general overview ofADMM is presented. The method by which PcOP F is refor-mulated so that ADMM can be applied is discussed thereafter.

ADMM is an operator splitting method that can be applied toproblems with the following general form [30]:

PADM M : minx∈X ,y∈Y

f(x) + g(y)

Ax = By.

The optimization variables x ∈ Rb and y ∈ Rc are each sub-ject to individual constraints represented by convex sets X andY . The objective is a summation of two terms that are convexfunctions of x and y respectively. These variables are coupledto each other by p equality constraints that are succinctly repre-sented by constant matrices A ∈ Rp×b and B ∈ Rp×c . Withoutthe equality constraints, PADM M would be completely sepa-rable and the optimal solutions x∗, y∗ can then be computedindividually. In order to enable the decomposition of PADM M ,the augmented Lagrangian Lρ(x, y, ν) is constructed as follows:

Lρ(x, y, ν) = f(x) + g(y) + νT(Ax−By) + ρ2 ||Ax−By||22

where the first two functions are from the objective inPADM M ,ν ∈ Rp is the vector of Lagrangian multipliers associated witheach equality constraint in PADM M , the last term is the aug-mentation term that enforces strict convexity when ρ > 0.Lρ(x, y, ν) is used to update x, y, and ν individually in aniterative manner. At the kth iteration, the following updates areapplied in series:

xk+1 = argminxLρ(x, yk , νk )

yk+1 = argminyLρ(xk+1 , y, νk )

νk+1 = νk + ρ (Axk+1 −Byk+1)where the first update is applied to x when y and ν are fixed tovalues computed from the previous iteration. The next updateis applied to y where xk+1 is the value computed immediatelybefore and νk has been computed in the previous iteration.ν is a dual variable and is updated using a sub-gradient likemethod where x and y are set to xk+1 and yk+1 computed in thecurrent iteration. It is evident that at each iteration, three sets ofinformation exchanges are necessary to update the primal anddual variables in a decentralized manner.

2) Converting Convex OPF Into the ADMM Form: PcOP Fcan also be converted into the form listed in PADM M towhich decentralized updates outlined above can be applied by

individual buses aided by local information exchanges. For this,new variables are first introduced that serve as perspectives ofvariables belonging to neighbouring buses and this notion of per-spective variables was first introduced in [15]. Consensus is thenestablished between these and the original variables. Our workdiffers vastly from reference [15] as we are applying ADMMat the TN level which is of a mesh topology that can consist ofcycles whereas reference [15] applies ADMM to the DN whichis a radial/tree structure with no loops. We consider the businjection model to represent the network topology of the TNwhereas reference [15] utilizes the branch-flow model to repre-sent the DN. The constraints considered in this paper are basedon linear approximations of the AC power balance constraintswhich is well-suited at the TN level [10]. On the other hand,reference [15] utilizes second-order cone relaxations which iswell suited in the context of DN not TN. Thus, the problemformulation (e.g. variables and constraints), signals exchangedand the topology under consideration differs significantly fromprior art.

Upon examining PcOP F , it is clear that constraints(C1′)–(C5′) associated with each bus depend on variables suchas θ and φ of neighbouring buses. (C3) represents constraintsthat are completely local to Bus i and therefore are separable. Torender constraints (C1′)–(C5′) separable as well, new variablesare introduced which will be referred to as perspective variables.TN agent i will maintain a set yi consisting of variables whichare local perspectives of certain optimization variables of localand neighbouring buses as follows:

yi = {pgi,i (y ) , pri,i (y ) , qgi,i (y ) , θ(y )i,i , φ(y )i,i , φ(y )j,i , θ(y )j,i ,ˆcos(y )i,j ∀ j ∈ Ni}.

The first subscript index in each variable denotes the perspectiveof and the second index indicates the perspective from. For ex-ample, θj,i is the perspective of the voltage angle of Bus j fromBus i. Perspectives of local variables are maintained in order toallow for complete separability of the y variables. As TN agenti will not have direct access to optimization variables of neigh-bouring nodes, perspective variables serve as local substitutesof the actual variables used in constraints (C1′)–(C5′). Now,TN agent i can evaluate these constraints locally. (C1′)–(C5′)are reformulated with the perspective variables as follows whichform the constraint set Yi :

− ˆcos(y)i,j ≤ 0 ˆcos(y )i,j ≤ −sin(ak )(θ(y )i,i − θ(y )j,i − ak )

+ cos(ak ) ∀ k ∈ npgi,i (y ) + pgi,i (y ) − pdi,i(y )

=∑

j

|V ti ||V tj |(gi,j ˆcosi,j + bi,j (θ

(y )i,i − θ(y )j,i )

)qgi,i

(y )

− qdi,i(y )

=∑

j

[|V ti ||V tj |(gi,j (θ

(y )i,i − θ(y )j,i )− bi,j ˆcosi,j

)

− (φ(y )i,i |V tj |+ φ(y )j,i |V ti |)bi,j].


Next, the original optimization variables are maintained in setxi for each TN agent i as follows:

xi = {pgi (x) , pri (x) , qgi (x) , θ(x)i , φ(x)i , ˆcos(x)i,j ∀ j ∈ Ni}.These are subject to the local separable constraints from (C3)which form the feasible set Xi :

pig≤ pgi (x) ≤ p̄ig , pir ≤ pri

(x) ≤ p̄ir , qig ≤ qgi

(x) ≤ q̄ig ,

V i ≤ φ(x)i + |V ti | ≤ V̄iAs a final step, it is necessary to establish a consensus betweenthe perspective and original variables. For instance, the perspec-tive variable θ(y )j,i must be equal to the original variable θ

(x)j . All

the consensus constraints associated with TN agent i are listedin the following:

pgi(x) = pgi,i

(y ), pri

(x) = pri,i(y ) , qgi

(x) = qgi,i(y )

θ(x)i = θ

(y )i,j , φ

(x)i = φ

(y )i,j ∀ j ∈ (Ni , i)

ˆcos(x)i,j = ˆcos(y )j,i , ˆcos

(x)j,i = ˆcos

(y )i,j ∀ j ∈ Ni

which can be generally expressed as Aixi = Biyi . Combiningall of the afore-mentioned transformations results in:

PdOP F : minxi ∈Xi ,yi ∈Yi ∀i∈B

∑

i∈B

(fi(p

gi

(x)) + fi(qgi

(x)) + fi(pri(x))

)

Aixi = Biyi ∀ i ∈ Bwhich is equivalent to the original formulation PcOP F . It isalso clear that this problem has the same structure as PADM M .Hence, ADMM can now be applied to PdOP F for decentralizedcomputation of local power injection by every active bus in theTN. First, the augmented Lagrangian is constructed:

Lρ(x, y, ν) =∑

i∈B

(fi(p

gi

(x)) + fi(qgi

(x)) + νpi,i(pgi

(x) − pgi,i (y ))

+ νri,i(pri(x) − pri,i (y )) + νqi,i(qgi (x) − qgi,i (y )) +

∑

j∈Ni

[νcosi,j ( ˆcos

(x)i,j

− ˆcos(y )i,j ) + νcosj,i ( ˆcos(x)i,j − ˆcos(y )j,i )]+

∑

j∈Ni ,i

[νθi,j (θ

(x)i − θ(y )i,j )

+ νφi,j (φ(x)i − φ(y )i,j )

]+

ρ

2(pgi

(x) − pgi,i (y ))2 +ρ

2(qgi

(x) − qgi,i (y ))2

+ρ

2(pri

(x) − pri,i (y ))2 +ρ

2

∑

j∈Ni ,i

[(θ(x)i − θ(y)i,j)2 + (φ(x)i − φ(y)i,j)2

]

+ρ

2

∑

j∈Ni

[( ˆcos(x)i,j − ˆcos(y)i,j)2 + ( ˆcos(x)i,j − ˆcos(y)j,i)2 − 2 ˆcos(x)i,j /ρ

]).

It is evident from this construction that each TN agent i isassociated with a set of Lagrangian multipliers νi :

νi = {νpi,i , νri,i , νqi,i , νθi,j , νφi,j , νcosi,j , νcosj,i ∀ j ∈ Ni}.Lρ(x, y, ν) can now be decomposed into individual sub-problems that can be solved by each bus. Iterative updates takingplace every 0.1s (communication delay is typically in the rangeof microseconds [16]) are applied to local variables xi , yi and νi

Algorithm 1: Decentralized Coordination at the TN Levelfor TN Agent i.

Initialize: xi ← 0, yi ← 0, νi ← 0, k ← 0, rk+1i ← 2�r ,sk+1i ← 2�s , V ti ← V ciwhile rk+1i > �

r and sk+1i > �s do

xk+1i ← argminxi ∈Xi

Liρ(xi, yk , νkn )- Broadcast to all Ni the computed xk+1iyk+1i ← argmin

yi ∈YiLiρ(xk+1n , y, νkn )

- Broadcast to all Ni computed yk+1iνk+1i ← νki + ρ(xk+1i − yk+1i )- Broadcast to all Ni computed νk+1i- Compute residuals: rk+1i ← ||Aixk+1i −Biyk+1i ||,sk+1i ← || − ρATi Bi(yk+1i − yki )||- k ← k + 1

end while

in series where variables currently not being updated are fixedto values evaluated immediately before.

These updates form the decentralized algorithm listed inAlg. 1 for coordination of active nodes in Tier 3. Updates willbe terminated once the primal and dual residuals r and s arebelow the threshold �r and �s . The primal residual is the normof the difference between the perspective and original variablesand this is represented as rk+1i = ||Aixk+1i −Biyk+1i ||. Dualresidual is expressed as sk+1i = || − ρATi Bi(yk+1i − yki )|| andis associated with dual feasibility [30].

V. COORDINATION IN TIER 2 AND TIER 1

Tier 2 consists of agents that bridge co-ordination betweenthe TN and elements (e.g. generation, DNs, etc.) that deliveror consumer power. Tier 1 is composed of actual nodes suchas synchronous generation systems, IPPs and DNs that receivesignals from Tier 2 agents to make local actuation decisions.Each one of these entities coordinate local power injections toachieve psi = p

gi + p

ri computed by TN agent i in Tier 3 to which

these are directly connected to over the subsequent 10 minuteperiod. Coordinating power injections for large systems such asthe DN that consists of thousands of residential consumers witha wide variety appliance mix, load curtailment preferences andsustainable micro-DGs is more challenging.

We consider real power actuation in Tier 1 (assume that re-active power supply is readily available from the main grid)for balanced single phase systems. In order to abstract fromthe actual elements coordinated in Tier 1, it is assumed that allTier 1 entities have the same set of strategies S (power injectionor power demand curtailment) at their disposal. These strate-gies are considered to be power changes with fixed magnitudes.Since revisions are made at random time instances, the impactof these are incremental from a system-wide perspective. Forconsistency, if a node has greater capacity than the strategiesat hand, virtual agents are introduced. This allows us to applypopulation game theory regardless of the variations in the ac-tuation capacities of individual nodes. For instance, if a large


synchronous plant has a large generation capacity, then virtuallocal entities are created that can actuate from one of S strate-gies. The number of virtual agents created is p̄/max(S) wherep̄ is the maximum power actuation capacity. The signal com-puted by agent i in Tier 2 is constructed by first formulating thegeneral problem for local coordination:

(P iT 2) minx∈�

fio(x) =n∑

j=1

Cj .(m.yj .xj )2

s.t.

n∑

j=1

m.yj .xj = psi

where xj is the proportion of agents in Tier 1 using strategy j,Cj is the cost of strategy j, n is the total number of strategiesin S and yj is the power actuated in strategy j and the couplingconstraint dictates that the total power actuated (obtained fromdata concentrator in Tier 1) is equal to the power injection valuepsj computed by the corresponding TN agent in Tier 3. The costsignal Fj for strategy j is selected to be the gradient of theLagrangian of PT 2 :

Fj (x) = K.m.yj (2.Cjm.yj .xj + ν∗)As this is a convex optimization problem, the above cost

signal transmitted by an agent in Tier 2 to local entities in Tier 1will converge to zero given that there exists sufficient resourcesin the system to cumulatively achieve psi [28]. Upon receivingthis cost signal, Tier 1 entities will randomly select a time torespond using the probability ρm,k (F ):

ρi,j (F (x), x) =[Fi − Fj ]+

n.xi

This is known as the projection revision in population gametheory. The resulting system dynamic is:

ẋi =n∑

j=1

ρj,i .xj − xi.n∑

j=1

ρi,j

where the first term is the rate at which agents switch to strategyi and the second term is the rate at which agents leave strategy i.This state dynamic is guaranteed to converge to the optimal so-lution as a Lyapunov function exists for the system dynamic dueto the construction of the problem. As the cost function F as-signed to the strategies is the gradient of the Lagrangian ofPT 2 ,the Lagrangian (i.e. potential function) is the Lyapunov functionof the system dynamic [31]. When there exists sufficient systemresources, the incremental effect of individual agents acting ina large population will result in the system dynamic ẋi withguaranteed convergence to the optimal solution of PT 2 .

Large populations of entities is natural in DNs and IPPs andthis can be artificially constructed for large synchronous plantsusing virtual agents. Prior to revising local actuation, each agentin Tier 1 will ensure that this change in strategy heeds localgeneration capacity, available power injection within the localfeeder and/or comfort requirements. This feasibility check isconducted by measuring the available power limits in the linesthat connect the actuating entity to the main feeder. Voltage andpower flow in the distribution network can be modelled using

Algorithm 2: Coordination in Tiers 1 and 2.At the beginning of every one minute cycle, initialize:t← 0Each agent j in Tier 1 selects sj ∈ S and exponentiallydistributed tjr (time for strategy revision)while

∑nj=1 m.yj .xj �= pis do

Tier 2 managing entity broadcasts cost signal to localTier 1 entities.

Each agent j in Tier 1 responds as follows:if tr ≤ t then

k ← sj and compute ρk,m (F ) ∀m ∈ SUpdate sj based on ρk,m (F ) and local feasibilityCompute tnewr using exponential distributiontr ← t + tnewr

end ifend while

the branch-flow model [11]:

Sij = Si +∑

h∈Ci(Shi − zhiI2hi) (1)

V̂i − V̂j = 2Re(zijSij )− |zij |2I2ij (2)

I2ij =|Sij |V̂i

(3)

where Sij = pij + jqij is the complex power flow in line ij, jis the complex number

√−1, Si is the complex power injectionat bus i, V̂i is the complex bus voltage magnitude squared, zijis the line impedance and Ci is a set representing the immediatedescendants of node i from the perspective of the feeder node.The non-linear terms are associated with the resistive powerlosses zhiI2hi . Ignoring these will result in a conservative ap-proximation of voltage drop and power flow in the DN lineswith negligible error [11]. During each strategy revision, therevising agent communicates the impending change in powerinjection to its ancestral nodes which will factor in the changein power flow to compute the approximate change in bus volt-age magnitudes and power flow. If there is a violation in voltagemagnitude limits or power flow, this is communicated to therevising node which will then halt the strategy revision. If noviolations are encountered, then the agent will proceed with thestrategy change. The complexity of this feasibility check is aconstant which is the height of the radial network representingthe DN.

If these conditions are not met, the strategy revision willnot take place. These computations within Tier 1 and 2 levelsare summarized in Alg. 2. System convergence based on thispopulation game theory setup for Tiers 1 and 2 is independentof the number of Tier 1 agents participating in the system [31]and this will be shown under practical settings in Sec. VII.

VI. MATHEMATICAL TRACTABILITY

In this section, a discussion on the mathematical tractabil-ity of the proposed hierarchical grid management frameworkis presented. In Table II, the computational complexities of the


TABLE IICOMPARISON OF COMPUTATIONAL COMPLEXITY

proposed hierarchical method, centralized method applied tothe linearized power-flow constraints presented in constraints(C1’) (C5’), (C3) and centralized optimization applied to theoriginal non-convex optimization problem in POP F are com-pared. The Big-O notation is utilized to measure the asymptoticcomputational complexity for the three methods at the TN andDN levels separately and then these are combined.

ADMM is utilized in the proposed framework for coordina-tion at the TN level. If the number of nodes at the TN level is nand the optimization problem is convex, then the number of iter-ations required for the ADMM algorithm to converge isO(n) asestablished by reference [15]. At the DN level, broadcast signalsare computed using population game theoretic constructs whichare independent of the number of nodes present in the system[31]. Hence, the computational complexity at this level is O(k)where k is a constant that is independent of the nodes present inthe DN. Thus, the combined computational complexity for theoverall proposed framework is O(kn) = O(n).

Applying a centralized solver, such as the interior pointmethod, to compute the optimal solution of a quadratic programwith convex constraints results in computational complexity thatis polynomial in the size of the problem [4]. This implies that thecomputational complexity at the TN level containing n nodesis O(np) and at the DN level containing m nodes is O(mp).Combining the computational complexity at both levels resultsin O((mn)p). It is clear that the computational complexity as-sociated with the centralized solver is much more concentratedthan the proposed hierarchical framework.

Finally, solving the original problem POP F for the DN andTN in a central manner is NP-hard due to the presence of a mul-titude of non-convex constraints that are associated with powerflow balance at both the TN and DN levels. Thus, it is clearfrom these results that the hierarchical framework divides themonolithic and complex problem at hand into simpler manage-able components that can be solved in a tractable manner bycombining distributed and decentralized approaches.

VII. IMPLEMENTATION

In this section, the proposed hierarchical signal-processingframework is studied under realistic circumstances via practicalsimulations.

A. Tier 3 Coordination

In this section, the performance of the decentralized algorithmoutlined in Alg. 1 for the Tier 3 level is evaluated in IEEE 39-bus, IEEE 118-bus and Polish 2736-bus systems. All line andbus parameters are obtained from MATPOWER. V t is set to1 p.u. and the initial values of the optimization variables are set

Fig. 3. Convergence in the IEEE 118-bus System.

TABLE IIIRELATIVE RMSE OF OPTIMIZATION VARIABLES

TABLE IVREAL POWER INJECTIONS IN IEEE 118-BUS SYSTEM

to 0. First, the impact of ρ on the convergence of the solutionto optimality is evaluated in the IEEE 118-bus system in Fig. 3for real power injection P . It is evident that the proportion ofthe computed value of Psim with respect to the actual solutionPact varies significantly for different values of ρ. Based on theseresults, we have selected ρ = 100 in the remaining simulationsto ascertain reduced iterations.

Next, the relative root mean square error of optimizationvariable x is averaged across the buses according to xrmse =( 1|B|

∑i∈B(

xip −xiaxia

)2)1/2 where xip is the result of the proposedalgorithm and xia is the actual optimal solution obtained usingthe solver provided by MATPOWER. Results are presented inTable III for x = P and x = |V | in the IEEE 39-bus and IEEE118-bus systems. Moreover, Table IV contains specific compar-ison of power generation computed in the IEEE 118-bus systemusing the proposed decentralized technique and exact resultsobtained from MATPOWER in MegaWatts (MW) for the ACmodel. It is clear from these results that our decentralized pro-posal results in minor deviations that can be readily offset byancillary services available in the TN.

Results for the Polish 2736-bus system are not includedin Table III as the size of the system and non-linearities inthe original OPF formulation can result in inaccuracies in the


Fig. 4. Residual for the Polish 2736-bus System.

solution computed by the built-in MATPOWER solver. Instead,the primal residual obtained with our proposal is plotted forthe Polish 2736-bus system in Fig. 4. It is clear that the resid-ual decreases rapidly initially with minor differences later onas expected with the ADMM algorithm. Convergence for thePolish 2736-bus system occurs within 100 iterations. Commu-nication delay incurred in signal exchanges is approximately10 ms [16]. Thus, convergence will be achieved within 1 minutein the Polish 2736-bus system. As the coordination interval is10 minutes in Tier 3, there is a tolerance of communicationdelays in signal exchanges up to 9 minutes which accounts foruncertainties and synchronization issues of signals exchanged.The number of iterations needed for the IEEE 39-bus system andIEEE 118-bus system to reach the same residual is 20 and 28iterations respectively. As expected, the size of the system hasan impact on the convergence time of the system being coordi-nated. Theoretically, the convergence time of the ADMM algo-rithm is linear with respect to the size of the system for convexformulations [15].

Next, ADMM is compared to other decentralized techniquessuch as dual decomposition (DD) and consensus algorithms usedin the literature. DD can be applied to constrained optimizationproblems (e.g. [6], [14]). If the objective is separable, then thedual problem can be divided into sub-problems that can be paral-lelized between multiple agents. Updates using the sub-gradientmethod are used to improve the overall solution. However, forguaranteed convergence, the original problem must be strictlyconvex. This is not the case for PdOP F as the objective is not afunction of V and Θ. We applied the DD technique to PdOP Ffor many different constant step-size selected in the range of[10−5 , 103] and were not able to observe convergence to the op-timal solution as expected. The simulations had in fact divergedfrom optimality. With ADMM, this issue is eliminated by theaugmented Lagrangian term that enforces strong convexity inthe problem [30]. Consensus methods are also widely used inthe literature to parallelize computations (e.g. [26]). However,the convergence speed is related to the second smallest eigen-value of the Laplacian matrix representing the communicationnetwork topology. This, in the worst case, is proportional toO(n3) as indicated by reference [32]. On the other hand, theconvergence of the ADMM algorithm is linearly proportionalto the size of the problem (i.e. O(n)) [15]. Hence, even thoughthis is a proof-of-concept, superior convergence properties ofthe ADMM method renders it suitable for decentralized man-

TABLE VCOMPARISON OF DECENTRALIZED TECHNIQUES

Fig. 5. Distributed Coordination in Danish DN.

Fig. 6. Effect of no Coordination.

agement of buses in the TN as summarized in Table V. Thisremoves the need for a central coordinating entity at the TNlevel.

B. Tier 1 and 2 Coordination

Based on the optimal power injection values computed inTier 3, Tier 1 and Tier 2 coordination is exacted according toAlg. 2. To illustrate the effectiveness of Alg. 2 even in the pres-ence of a large number of entities in Tier 1, the first result inthis section illustrated in the second subplot of Fig. 5 depictsthe rapid convergence of actuating DGs and flexible consumerstotalling hundred entities residing in a Danish DN system withparameters obtained from [33] over one optimization interval(i.e. 1 minute) based on optimal power injection value com-puted in Tier 3. The EPU is presiding as the managing entityin Tier 2. Fig. 6 illustrates the effect on bus voltages with nocoordination scheme in place. Maximum voltages of buses inthe DN are exceeding the 1.1 p.u. threshold. With coordination,


this is not the case as illustrated in Fig. 5. Flexible consumersalso conduct feasibility checks prior to making a demand cur-tailment decision. It is clear from Fig. 5 that the system ofactuating components are able to achieve the power injectionsetpoint computed in Tier 3 exponentially fast and maintainvoltages across the buses within feasible limits.

Thus, these results illustrate effectiveness of the hierarchicalsignal processing framework presented in this paper in allowingfor the seamless, plug-and-play integration and coordination ofdiverse entities across the power grid.

C. Performance Comparison With State-of-the-Art

A large number of proposals in the existing literature aim tosolve the OPF problem. In these proposals, the OPF is solvedin its original form (i.e. non-convex) via heuristics or via con-vex relaxations which are applied for tractability. In the originalnon-convex form, although all original variables and constraintsare accounted for, no guarantees on optimality and convergencecan be established due to the presence of non-convex constraints.Iterative methods such as sub-gradient methods (e.g. dual de-composition), consensus methods and operator splitting meth-ods (e.g. ADMM) fail as these algorithms can diverge due to thenon-convexities in the problem formulation [24], [30]. Central-ized solvers using heuristic techniques (e.g. genetic algorithm,simulated annealing, etc.) have been presented in the literature[34]. These result in sub-optimal solutions with no guaranteeson the performance or convergence [12]. Moreover, these entailsignificant tuning of parameters based on the characteristics ofthe underlying system which can result in additional overhead.Finally, myopic techniques (e.g. droop control [35], machinelearning [36], etc.) where nodes make decisions using only localmeasurements result in inefficiencies as the state of the externalenvironment is not accounted in these decisions.

When relaxations are applied to the OPF, non-convex con-straints are eliminated. For instance, DC power flow is heavilyutilized by system operators in today’s grids for planning pur-poses [4]. This formulation eliminates reactive power and busvoltage magnitudes which are important considerations in theintegration of renewables and variable loads. Thus, the trade-off for simplicity is essentially the practicality of the OPF for-mulation. Other types of relaxations such as semi-definite andsecond-order cone relaxations have been proposed for the DN[9]. These are well-suited for DNs that have tree topology andnot for TNs which are associated with mesh structure [10]. Lin-ear approximations of AC constraints presented in reference[23], which has been applied in this paper, retains all importantvariables and is applicable to high-voltage power networks withmesh topology like the TN [10].

Applying iterative methods to strictly convex optimizationproblems will guarantee convergence [26]. However, sub-gradient and consensus methods entail extensive parametertuning (e.g. step-size) and are associated with polynomialconvergence rates O(np) [6]. ADMM is associated with linearconvergence rateO(n) with no need for fine-grained parametertuning. Moreover, the underlying problem at hand is not re-quired to be strictly convex for convergence (convexity suffices).Myopic decisions are independent of the OPF structure as these

TABLE VICOMPARISON OF OPTIMAL PERFORMANCE

strive to maintain feasibility not optimality. A summary of theperformance of various techniques applied in the state-of-the-artfor different types of OPF formulations is presented in Table VI.

VIII. FINAL REMARKS

In this paper, a hierarchical signal processing framework ispresented which allows for the effective and sustainable coordi-nation of a diverse set of power components across the moderngrid while ensuring that the grid limits imposed on the underly-ing complex physical network structure are taken into account.With abstraction and decomposition built into decision-makingat each tier of the proposed framework, every actuating nodeis able to adapt to changing conditions across the entire grid.This ability to respond in an adaptive manner not only enhancesthe resilience of the grid but also enables a large number ofheterogeneous power components to seamlessly interact withthe grid in a unified and non-disruptive manner. In future work,we hope to investigate myopic decision-making based solelyon local measurement signals to address potentially disruptivetransient conditions in microgrid settings into our hierarchicalframework.

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Pirathayini Srikantha received the B.A.Sc. degreein systems design engineering and the M.A.Sc. de-gree in electrical and computer engineering in 2009and 2013, respectively, both from the Universityof Waterloo, Waterloo, ON, Canada, and the Ph.D.degree from University of Toronto, Toronto, ON,Canada, in 2017. She is currently an Assistant Pro-fessor with the Department of Electrical and Com-puter Engineering, Western University, London, ON,Canada. Her main research interests include includelarge-scale optimization and distributed control for

enabling adaptive, sustainable, and resilient power grid operations. Her workhas been published in premier smart grid journal and conference venues. She isa certified Professional Engineer (P.Eng.) in Ontario. Her research efforts havereceived recognitions that include the best paper award (IEEE Smart Grid Com-munications) and runner-up best poster award (ACM Women in Computing).She is also actively involved in professional and social activities. She was theSession Chair and Technical Program Committee member in IEEE conferences.She is a reviewer in numerous IEEE transactions journals.

Deepa Kundur (F’15) was born in Toronto, Canada.She received the B.A.Sc., M.A.Sc., and Ph.D. degreesall in electrical and computer engineering in 1993,1995, and 1999, respectively, from the University ofToronto, Toronto, ON, Canada. She is currently theChair of the Division of Engineering Science and aProfessor and the Director of the Centre for Power &Information, The Edward S. Rogers Sr. Departmentof Electrical & Computer Engineering, University ofToronto. . From January 2003 to December 2012, shewas a faculty member with the Department of Elec-

trical & Computer Engineering, Texas A&M University, and from September1999 to December 2002, she was a faculty member with the Department ofElectrical & Computer Engineering, University of Toronto. She is an author ofmore than 200 journal and conference papers. Her research interests lie at theinterface of cyber security, signal processing, and complex dynamical networks.

She is also a recognized authority on cyber security issues and has appearedas an expert in popular television, radio, and print media. She has participated onseveral editorial boards and is currently on the Advisory Board of IEEE SPEC-TRUM. She is currently the General Chair for the 2018 GlobalSIP Symposiumon Information Processing, Learning and Optimization for Smart Energy In-frastructures and the TPC Co-Chair for IEEE SmartGridComm 2018. Recently,she was also also the Symposium Co-Chair for the Communications for theSmart Grid Track of ICC 2017, the General Chair for the Workshop on Com-munications, Computation and Control for Resilient Smart Energy Systems atACM e-Energy 2016, the General Chair for the Workshop on Cyber-PhysicalSmart Grid Security and Resilience at GLOBECOM 2016, the General Chairfor the Symposium on Signal and Information Processing for Smart Grid In-frastructures at GlobalSIP 2016, the General Chair for the 2015 InternationalConference on Smart Grids for Smart Cities, the General Chair for the 2015Smart Grid Resilience (SGR) Workshop at IEEE GLOBECOM 2015, and theGeneral Chair for the IEEE GlobalSIP’15 Symposium on Signal and Informa-tion Processing for Optimizing Future Energy Systems.

Professor Kundur’s research has received best paper recognitions at numerousvenues including the 2015 IEEE Smart Grid Communications Conference, the2015 IEEE Electrical Power and Energy Conference, the 2012 IEEE CanadianConference on Electrical & Computer Engineering, the 2011 Cyber Security andInformation Intelligence Research Workshop, and the 2008 IEEE INFOCOMWorkshop on Mission Critical Networks. She has also been the recipient ofteaching awards at both the University of Toronto and Texas A&M University.She is a Fellow of the Canadian Academy of Engineering.

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