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Two-Timescale Stochastic Dispatchof Smart Distribution Grids

Luis M. Lopez-Ramos, Member, IEEE, Vassilis Kekatos, Senior Member, IEEE,Antonio G. Marques, Senior Member, IEEE, and Georgios B. Giannakis, Fellow, IEEE

Abstract—Smart grids should efficiently integrate stochasticrenewable resources while effecting voltage regulation. Energymanagement is challenging since it is a multistage problemwhere decisions are not all made at the same timescale andmust account for the variability during real-time operation. Thejoint dispatch of slow- and fast-timescale controls in a smartdistribution grid is considered here. The substation voltage, theenergy exchanged with a main grid, and the generation schedulesfor small diesel generators have to be decided on a slow timescale;whereas optimal photovoltaic inverter setpoints are found on amore frequent basis. While inverter and looser voltage regulationlimits are imposed at all times, tighter bus voltage constraintsare enforced on the average or in probability, thus enablingmore efficient renewable integration. Upon reformulating thetwo-stage grid dispatch as a stochastic convex-concave problem,two distribution-free schemes are put forth. An average dispatchalgorithm converges provably to the optimal two-stage decisionsvia a sequence of convex quadratic programs. Its non-convexprobabilistic alternative entails solving two slightly different con-vex problems and is numerically shown to converge. Numericaltests on real-world distribution feeders verify that both schemesyield lower costs over competing alternatives.

Index Terms—Multistage economic dispatch, voltage regula-tion, stochastic approximation, convex-concave problem.

I. INTRODUCTION

With increasing renewable generation, energy managementof power distribution grids is becoming a computationallychallenging task. Solar energy from photovoltaic (PV) unitscan change significantly over one-minute intervals. The powerinverters found in PV units can be commanded to curtailactive power generation or adjust their power factor withinseconds [1], [2]. At a slower timescale, distribution gridoperators exchange energy with the main grid hourly or ona 10-minute basis, and may experience cost penalties upondeviating from energy market schedules [3]. Moreover, voltageregulation equipment and small diesel generators potentially

Manuscript submitted August 3, 2016; revised December 2, 2016; acceptedJanuary 11, 2017. Date of publication DATE; date of current version DATE.Paper no. TSG.01019.2016.

This work was supported by the Spanish Ministry of Education FPU GrantAP2010-1050; CAM Grant S2013/ICE-2933; MINECO Grant TEC2013-41604-R; and NSF grants 1423316, 1442686, 1508993, and 1509040.

L.-M. Lopez-Ramos and Antonio G. Marques are with the Dept. ofSignal Theory and Communications, King Juan Carlos Univ., Fuenlabrada,Madrid 28943, Spain. V. Kekatos is with the ECE Dept., Virginia Tech,Blacksburg, VA 24061, USA. G. B. Giannakis is with the Digital Tech-nology Center and the ECE Dept., University of Minnesota, Minneapolis,MN 55455, USA. Emails: luismiguel.lopez@urjc.es, kekatos@vt.edu, anto-nio.garcia.marques@urjc.es, georgios@umn.edu.

Color versions of one or more of the figures is this paper are availableonline at http://ieeexplore.ieee.org.

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installed in microgrids respond at the same slower timescale.As a result, comprehensive designs to optimize such diversetasks call for multistage smart grid dispatch solutions.

Spurred by demand-response programs and the use of PVinverters to accomplish various grid tasks [4], single-stagedispatch schemes for distribution grids have been an activearea of research. Power inverters can be controlled usinglocalized rules for voltage regulation, see e.g., [5], [6], [7],[8]. Assuming two-way communication between buses andthe utility operator, dispatching a distribution system can beposed as an optimal power flow (OPF) problem. Centralizedschemes use nonlinear program solvers [9]; or rely on convexrelaxations of the full AC model of balanced [10], [11],or unbalanced grids [12]. Distributed solvers with reducedcomputational complexity have been devised in [13], [14],[15].

Nevertheless, the efficient and secure operation of dis-tribution grids involves decisions at different timescales. Adynamic programming approach for a two-stage dispatch issuggested in [10]: The taps of voltage regulators are set ona slow timescale and remain fixed for consecutive shortertime slots over which elastic loads are dispatched; yet theflexibility of loads is assumed known a priori. Alternatively,centrally computed OPF decisions can be communicated tobuses at a slow timescale, while on a faster timescale, PVpower electronics are adjusted to optimally track variationsin renewable generation and demand [16], [17]. Relying onapproximate grid models and ignoring the effect of uncertaintyon the dispatch of slow-responding units, the latter schemesyield a partially decentralized real-time allocation of the powerflows across fast-responding units.

Multistage dispatching under uncertainty is routinely usedin transmission systems and microgrids [18]. Robust ap-proaches find optimal slow-timescale decisions for the worst-case fast-timescale outcome; see [19] and references therein.To avoid the conservativeness of robust schemes, probabilisticapproaches postulate a probability density function (pdf) fordemand, wind generation, and system contingencies to findday-ahead grid schedules [20], [21]. The risk-limiting dispatchframework adjusts multistage decisions as the variance ofthe random variables involved decreases while approachingactual time [3]. Decisions can be efficiently calculated onlyfor convenient pdfs for a network-constrained risk-limitingdispatch and under congestion assumptions [22]. As a thirdalternative, sample average approximation (SAA) approachesyield optimal slow-timescale decisions using samples drawnfrom the postulated pdf; see e.g., [23], [19]. Recent works

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impose limits on the probability of undesirable events, eitherrelying on convex approximation of chance constraints [24], orvia the (sample-based) scenario approximation approach [25]to reduce computations; e.g., [26].

Focusing to distribution grids, PV inverters could be over-loaded sporadically in time and across buses to accommodatesolar fluctuations and prevent overvoltages [27]. The spa-tiotemporal overloading of power system components (suchas inverters, bus voltages, line flows) could thus constitutean additional means for integrating renewables in smartgrids. Nonetheless, ensuring that overloading occurs sparinglycouples decisions across time. The single-stage scheme of[28] finds optimal PV setpoints while limiting time averagesof overloaded quantities. The latter approach has been alsoadopted in [29] for dispatching a transmission system in aday-ahead/real-time market setup under load shedding.

Jointly dispatching slow- and fast-timescale grid resourcesunder average or probabilistic constraints over fast-timescaledecisions is considered here. Our contribution is three-fold.First, Section III formulates a two-stage grid dispatch asa convex-concave problem: The expected cost over a slowcontrol period is minimized, while looser voltage limits aresatisfied at all times and tighter voltage limits are enforcedon the average or in probability. Second, upon adapting thestochastic saddle-point approximation scheme from [30], theprovably convergent algorithm in Sec. IV provides optimalslow-timescale decisions for the average-constrained formu-lation. Different from SAA approaches, this stochastic ap-proximation (SA) scheme processes random samples one ata time to improve computational efficiency. Third, in the caseof non-convex probabilistic constraints, an algorithm solvingtwo similar convex problems for each second stage is putforth in Sec. V. Although the expected cost enjoys zero-duality gap [31], the overall two-stage dispatch is not convex-concave, which explains why the algorithm’s performance isvalidated numerically. Both schemes require only samples ofloads and solar generation (rather than their joint pdfs), andcan rely either on an approximate, or a convexified grid model.Numerical tests using the linearized distribution flow model on56- and 123-bus feeders corroborate the validity of our findingsin Sec. VI.

Regarding notation, lower-(upper-)case boldface letters de-note column vectors (matrices), with the only exception of thepower flow vectors, which are uppercase. Calligraphic lettersare used to denote sets. Symbol > denotes transposition, while0 and 1 are the all-zeros and all-ones vectors of appropriatedimensions. The indicator function 1· equals 1 when itsargument is true, and 0 otherwise. A diagonal matrix withthe entries of vector x on its main diagonal is denoted bydg(x). The operator [·]+ projects its argument onto the positiveorthant; E[·] denotes expectation and Pr· probability.

II. PROBLEM FORMULATION

Consider a distribution grid whose energy needs are pro-cured by distributed renewable generation, distributed con-ventional (small diesel) generators, and the main grid. Thedistribution grid operator aims at serving load at the min-imum cost while respecting voltage regulation and network

constraints. Energy is exchanged with the main grid at whole-sale electricity prices through the feeder bus. To effectivelyintegrate stochastic renewable generation, the focus here is onshort-term grid dispatch. To that end, the distribution grid isoperated at two timescales: a slower timescale correspondsto 5- or 10-min real-time energy market intervals, while theinverters found in PVs are controlled at a faster timescale ofsay 10-sec intervals. One period of the slower timescale iscomprised by T faster time slots indexed by t = 1, . . . , T .

The grid is operated as a radial network with N + 1 busesrooted at the substation bus indexed by n = 0. The distributionline feeding bus n is also indexed by n for n = 1, . . . , N . Letpn,t and qn,t denote respectively the net active and reactivepower injections at bus n and slot t; the N -dimensional vectorspt and qt collect the net injections at all buses except forthe substation. Diesel generators are dispatched at the slowertimescale to generate pd throughout the subsequent T slotsat unit power factor. During slot t, PVs can contribute solargeneration up to prt that is modeled as a random process.Smart inverters perform active power curtailment and reactivepower compensation by following the setpoints prt and qrtcommanded by the utility operator. Load demands plt andqlt are also modeled as random processes. To simplify theexposition, (plt,q

lt) are assumed inelastic and known at the

beginning of slot t; although elastic loads can be incorporatedwithout any essential differences. The operator buys a powerblock pa0 from the main grid at the slow timescale, which canbe adjusted to p0,t := pa0 + pδ0,t in actual time.

Voltage regulation is effected by controlling (re)activepower injections at slot t. Let vn,t denote the squared voltagemagnitude at bus n and slot t, and vt the vector collectingvn,tNn=1. The substation voltage va0 is controlled at theslower timescale [10], while voltage magnitudes at all busesmust adhere to voltage regulation standards, e.g., ANSI C84.1and EN50160 in [32], [33]. These standards differentiatebetween a narrower voltage regulation range denoted here byVA in which voltages should lie most of the time; and a widerrange VB (with VA ⊂ VB) whom voltages should not exceedat any time. One of the goals of this work is to leverage thisflexibility to design dispatch schemes that: i) guarantee thatvoltages lie in VB at all times, while ii) they belong to VA ina stochastic fashion. To this end, two alternative schemes arepresented, the difference between them being how constraintii) is formulated. The first scheme guarantees that the averagevoltage lies in VA, whereas the second one maintains theprobability of under-/over-voltage at a specified low value.

A. Grid modelingTo account for voltage and network limitations, the distri-

bution grid is captured by the approximate linear distributionflow (LDF) model [34]. To briefly review this model, letr and x be accordingly the vectors of line resistances andreactances across lines. Define also the branch-bus incidencematrix A ∈ RN×(N+1) whose (i, j)-th entry is

Aij =

+1 , if j − 1 is the source bus of line i−1 , if j − 1 is the destination bus of line i0 , otherwise.

(1)

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Partition A into its first column and the reduced branch-busincidence matrix A as A = [a0 A]. Ignoring line losses, theLDF model asserts that the vectors of active and reactive linepower flows at time t can be approximated by

Pt = F>pt and Qt = F>qt (2)

where F := A−1. Moreover, the squared voltage magnitudescan be expressed as [34], [6]

vt = 2Rpt + 2Xqt + vd01 (3)

where R := Fdg(r)F> and X := Fdg(x)F>. The LDFmodel applies to both radial and meshed networks and, dif-ferent from the so termed DC power grid model, it does notignore line resistances [35]. It can be derived by assumingthat voltage magnitudes are close to unity and voltage angledifferences across neighboring buses are small. Alternatively,it can be obtained upon linearizing power injections at the flatvoltage profile [36].

Let us define the voltage regulation regions

VA := v : vA1 ≤ v ≤ vA1 (4a)VB := v : vB1 ≤ v ≤ vB1 (4b)

with vB ≥ vA and vB ≤ vA. Compliance with VA canbe imposed either on the average as Et [vt] ∈ VA, or inprobability as Prvt ∈ VA ≥ 1 − α for some small α.Either way, safe grid operation requires that vt ∈ VB at alltimes t. Within the optimization horizon, the random processesinvolved (demand and renewable generation) can be assumedergodic, i.e., their time averages converge to their ensembleaverages. For this reason, voltage constraints pertaining to VAwill be referred to as ergodic.

According to (2), if fn is the n-th column of F, the squaredpower flow on line n can be written as P 2

n,t = p>t fnf>n pt

and Q2n,t = q>t fnf

>n qt. Imposing the upper limit Sn on

the apparent flow on line n is thus expressed as the convexquadratic constraint

p>t fnf>n pt + q>t fnf

>n qt ≤ S

2

n. (5)

Although losses have been dropped in (2), upon assuming thatvoltage magnitudes are close to unity, active power losses canbe approximated as [37]

N∑n=1

rn(P 2n,t +Q2

n,t) = P>t dg(r)Pt + Q>t dg(r)Qt.

Using (2), the latter can be equivalently expressed as p>t Rpt+q>t Rqt, so the active power injection at the substation isapproximately

p0,t = −1>pt + p>t Rpt + q>t Rqt (6)

Regarding smart inverters, the tuple (prn,t, qrn,t), which de-

notes the power injection from the inverter located on bus nat slot t, should belong to the feasible set

Ωn,t :=

(prn,t, qrn,t) : 0 ≤ prn,t ≤ prn,t, (7a)

|qrn,t| ≤ φnprn,t, (7b)

(prn,t)2 + (qrn,t)

2 ≤ s2n

(7c)

that is random and time-variant due to the variability of prn,t.Constraint (7a) limits the active power generation accordingto the available solar power; constraint (7b) enforces thelower limit cos(arctan(φn)) on the power factor (lagging orleading); and (7c) limits the inverter apparent power.

B. Operation costs

If PV owners are compensated at price π for the activepower surplus they inject into the distribution grid, the relatedutility cost at slot t is CPV(prt ) := π>[prt −plt]+ with [·]+ :=max0, · applied entrywise on vector prt − plt. The dieselgeneration cost is represented by CD(pd). Regarding energytransactions with the main grid, the power block pa0 bought inadvance is charged at a fixed and known price β. Deviatingfrom pa0 by pδ0,t at slot t is charged at

Ct(pδ0,t) := γb[pδ0,t]+ − γs[−pδ0,t]+ (8)

for known prices (γb, γs). To avoid arbitrage, it is assumedthat 0 < γs < β < γb; see e.g., [3], [22]. Then,the deviation charge can also be expressed as Ct(pδ0,t) =maxγbpδ0,t, γspδ0,t, which is certainly convex [19].

C. Optimal grid dispatch

Depending on the way compliance with voltage regulationregion VA is enforced, two grid dispatch formulations aredeveloped next. Commencing with the average dispatch, theoptimal grid operation is posed as

P∗a := min CD(pd) + βpa0 +Et

[Ct(pδ0,t) + CPV(prt )

](9a)

s.to: pt = prt − plt + pd (9b)

qt = qrt − qlt (9c)

p0,t = pa0 + pδ0,t (9d)

p0,t ≥ −1>pt + p>t Rpt + q>t Rqt (9e)

p>t fnf>n pt + q>t fnf

>n qt ≤ Sn, ∀n ∈ N (9f)

pd ≤ pd ≤ pd (9g)

(prn,t, qrn,t) ∈ Ωn,t, ∀n ∈ N (9h)

v0 ≤ va0 ≤ v0 (9i)vt = 2Rpt + 2Xqt + va01 (9j)vt ∈ VB (9k)Et [vt] ∈ VA (9l)

over va0 , pa0 ,p

d, pt,qt,vt,prt ,qrt , p0,t, pδ0,tTt=1.

The slow-timescale variables va0 , pa0 ,pd are set in ad-vance, and remain fixed throughout the T subsequentcontrol slots over which the fast-timescale variablespt,qt,vt,prt ,qrt , p0,t, pδ0,tTt=1 are implemented. The lattervariables depend on the randomness of slot t as well as slow-timescale decisions.

Alternatively to (9), optimal grid operation can be posed as aprobabilistic dispatch that is identical to (9) with the exceptionthat (9l) is replaced by the probabilistic constraint

Prvt /∈ VA ≤ α (10)

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for some small parameter α > 0, say α = 0.05. The optimalcost for the probabilistic dispatch will be denoted by P∗p.

The objective function in (9a) involves the cost of en-ergy dispatched at the slow timescale plus the average fast-timescale energy management cost. Nodal (re)active powerbalance is ensured via (9b)–(9c). Constraint (9e) accounts foractive power losses. Since the cost in (9a) is non-decreasingwith respect to (pδ0,t, p

a0), relaxing (6) to the convex inequality

in (9e) does not incur loss of optimality. Constraint (9f) limitsline apparent power flows based on (5). Constraints (9i)–(9l) are voltage regulation constraints: In detail, (9j) relatessquared voltage magnitudes to power injections [cf. (3)]; (9i)constraints the substation bus voltage; and (9k) constraintsvoltages in VB . While (9l) maintains the average voltagemagnitudes in VA, its alternative in (10) limits the probabilityof voltage magnitudes being outside VA.

A pertinent question is which of the two proposed dispatchformulations is to be preferred. The probabilistic formulationis more sophisticated and aligned with voltage regulationstandards, emerging as the default option. However, as it willbe explained in Section V, enforcing even the single grid-level probabilistic constraint in (10) gives rise to a non-convexproblem, which comes with computational challenges. Theaverage dispatch does not suffer from these problems, whichcan be critical in scenarios where the duration of the slowperiod is short and the optimization has to be frequently re-run. Furthermore, when renewable generation and loads varyonly slightly during a slow period and/or local control loopsare in place, enforcing probabilistic guarantees may not bejustified and the simpler average constraints suffice.

D. Convexified AC grid model

Although (9) relies on the approximate LDF model, itcan be readily customized to the exact AC power flowmodel [34]. Upon introducing the optimization variable `t :=[`1,t . . . `N,t]

> with the squared line current magnitudes,constraints (9e)–(9f) should be substituted respectively by

p0,t ≥ −1>pt + 1>`t (11a)

P 2n,t +Q2

n,t ≤ Sn, ∀n ∈ N . (11b)

Constraint (9j) defining vt should be replaced by

vt = 2Fdg(r)Pt + 2Fdg(x)Qt + va01 (12)

and variable `t is linked to power flows and voltages throughthe additional constraints:

Pt = F>pt + F>dg(r)`t (13a)

Qt = F>qt + F>dg(x)`t (13b)

P 2n,t +Q2

n,t ≤ vπn,t`n,t, ∀n ∈ N (13c)

where πn is the parent bus of bus n. In fact, constraint (13c)constitutes a relaxation, since in the actual grid model it issatisfied with equality [10]. Nevertheless, the relaxation hasbeen shown to be exact in radial grids and under different con-ditions; see [38] for details. Critical for the ensuing sections isthat the differences between the formulation in (11)–(13) andthat for the LDF model pertain to the fast-timescale operation,

whereas the slow-timescale formulation and the constraintscoupling slow with fast timescale variables remain intact.

III. PROBLEM ANALYSIS

To facilitate algorithmic developments, the problem in (9) isexpressed in a compact form next. Collect the slow-timescalevariables in vector z> := [va0 , p

a0 ,p

d]; the fast-timescalevariables at slot t in y>t := [pt,qt,vt,p

rt ,q

rt , p0,t, p

δ0,t]; and

the random variables involved at slot t in ξ>t := [prt ,plt,q

lt].

The constraints in (9) can be classified into four groups:(i) Constraints involving fast-timescale variables only, such as(9c), (9f), (9h), and (9k), that will be abstracted as yt ∈ Yt.(ii) Constraints (9g) and (9i) that involve slow-timescalevariables only, and they will be denoted as z ∈ Z .(iii) The linear constraints (9b), (9d), and (9j), coupling slow-and fast-timescale variables as well as random variables. Theseconstraints are collectively expressed as Kz+Byt = Hξt forappropriate matrices K, B, and H.(iv) The ergodic constraints (9l) and (10) depend on thevoltage sequence vtTt=1, hence coupling decisions acrosstime. A substantial difference between (9l) and (10) is thatthe latter is a non-convex constraint.

If the exact grid model of Section II-D is used, the additionalvariables Pt, Qt, and `t are added, and set Yt in (i) is modifiedto incorporate (11)–(13). Under these considerations, the twodispatch problems can be compactly rewritten as

P∗(a,p) := minz,ytTt=1

f(z) +Et [gt(yt)] (14a)

s.to: z ∈ Z (14b)yt ∈ Yt ∀t (14c)Kz + Byt = Hξt ∀t (14d)Et [h(yt)] ≤ 0 (14e)

where f(z) := CD(pd) + βpa0 and gt(yt) := Ct(pδ0,t) +CPV(prt ). For the average dispatch, the optimal cost in (14) isP∗a and the function in (14e) is h(yt) = [vt−vA1, vA1−vt].For the probabilistic dispatch, the optimal cost is P∗p and thefunction in (14e) is h(yt) = 1vt /∈ VA − α.

The optimal values for the slow-timescale variables zmust be decided in advance. Once the optimal z is found,it remains fixed over the slow-timescale interval. The fast-timescale decisions yt(z) for slot t depend on z, while thesubscript t indicates their dependence on the realization ξt.Both the average and the probabilistic dispatch are stochasticprogramming problems with recourse [3]. Their costs can bedecomposed as P∗(a,p) = minz∈Z f(z) +G(a,p)(z), where theso termed expected recourse function is defined as

G(a,p)(z) := minyt∈Yt

Et [gt(yt)] (15a)

s.to: Kz + Byt = Hξt ∀t (15b)Et [h(yt)] ≤ 0. (15c)

Since problem (15) depends on z, its minimizer can be writtenas y∗t (z)Tt=1 and the recourse function as G(a,p)(z) =Et[gt(y

∗t (z))]. The ensuing two sections solve the average and

the probabilistic dispatches.

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IV. AVERAGE DISPATCH ALGORITHM

This section tackles problem (14) with the ergodic constraintin (14e), for which h(yt) = [vA1− vt,vt − vA1]. Althoughconvex, problem (14) is challenging due to the coupling acrossytTt=1 and between fast- and slow-timescale variables. Dualdecomposition is adopted to resolve the coupling acrossytTt=1. The partial Lagrangian function for (15) is

La(yt,ν) := Et

[gt(yt) + ν>h(yt)

](16)

with the entries of ν being the multipliers associated with theupper and lower per-bus constraints in (14e). The correspond-ing dual function is

Da(ν; z) := minyt∈Yt

La(yt,ν) (17)

s.to: Kz + Byt = Hξt ∀t.

Observe that after dualizing, the minimization in (17) is sep-arable over the realizations ξt. Precisely, the optimal fast-timescale variable for fixed (ν, z) and for a specific realizationξt can be found by solving:

y∗t (ν, z) ∈ arg minyt∈Yt

gt(yt) + ν>h(yt) (18a)

s.to: Kz + Byt = Hξt. (18b)

For future reference, let us also define λ∗t (ν, z) as the optimalLagrange multiplier associated with (18b). If ν is partitionedas ν> = [ν>,ν>] with ν corresponding to constraint Et[vt] ≥vA1 and ν to Et[vt] ≤ vA1, then (18) simplifies to

y∗t (ν, z) ∈ argmin Ct(pδ0,t) + CPV(prt ) + (ν − ν)>vt (19)

s.to: (9b)− (9f), (9h), (9j), (9k)

over pt,qt,vt,prt ,qrt , p0,t, pδ0,t

and can be solved as a convex quadratic program. If the relaxedAC grid model of Section II-D is used, then (19) becomesa second-order cone program (SOCP) which is also convex.Given the optimal pair (ν∗, z∗), the optimal fast-timescalevariables yt can be thus found for any ξt.

Back to finding the optimal primal and dual slow-timescalevariables, note that the dual problem associated with (17) is

ν∗ := arg maxν≥0

Da(ν; z). (20)

Duality theory asserts that (20) is a convex problem. Moreover,assuming a strictly feasible point exists for (15), strong dualityimplies that Ga(z) = Da(ν∗, z). Due to the latter, the originalproblem in (14) can be transformed to:

minz∈Z

f(z) +Ga(z) = minz∈Zf(z) + max

ν≥0Da(ν; z) (21a)

= minz∈Z

maxν≥0

fa(ν, z) (21b)

where the auxiliary function fa is defined as:

fa(ν, z) := f(z) +Da(ν; z). (22)

Being a dual function, Da(ν; z) is a concave function of ν.At the same time, Da(ν; z) is a perturbation function withrespect to z; and hence, it is a convex function of z [39].Recall that f(z) is a convex function of z too. Therefore,

Algorithm 1 Average Dispatch Algorithm (ADA)1: Initialize (z0,ν0).2: repeat for k = 0, 1, . . .3: Draw sample ξk.4: Find (y∗k(νk, zk),λ∗k(νk, zk)) by solving (18).5: Update (zk+1,νk+1) via (25).6: Compute sliding averages (zk, νk) through (26).7: until convergence of (zk, νk).8: Output z∗ = zk and ν∗ = νk.

the auxiliary function fa(ν, z) is convex in z and concave inν. Because of the randomness of ξt, function Da(ν; z) in(17) is stochastic. Consequently, problem (21b) is a stochasticconvex-concave saddle point problem [39], [30].

To solve (21b), we rely on the stochastic saddle-pointapproximation method of [30]. The method involves the sub-gradient of fa with respect to z, and its supergradient withrespect to ν. Upon viewing Da(ν, z) in (17) as a perturbationfunction of z, the subgradient of fa with respect to z is [39]

∂zfa = ∂zf(z) + K>Et[λ∗t (ν, z)]. (23)

By definition of the dual function, the supergradient of fa withrespect to ν is

∂ν fa = Et[h(y∗t (ν, z))]. (24)

The stochastic saddle point approximation method of [30]involves primal-dual subgradient iterates with the expectationsin (23)–(24) being replaced by their instantaneous estimatesbased on a single realization ξk. Precisely, the method involvesthe iterates over k:

νk+1 := [νk + dg(µk)h(y∗k(νk, zk))]+ (25a)

zk+1 := [zk − dg(εk)(∂zf(zk) + K>λ∗k(νk, zk))]Z (25b)

where the operator [·]Z projects its argument onto Z; andvectors µk = µ0/

√k and εk = ε0/

√k collect respectively

the primal and dual step sizes for positive µ0 and ε0. At everyiteration k, the method draws a realization ξk and solves (18)for the tuple (ξk,ν

k, zk) to acquire (y∗k(νk, zk),λ∗k(νk, zk))and perform the primal-dual updates in (25). The methodfinally outputs the sliding averages of the updates as:

zk :=(∑k

i=dk/2e zi/√i)/(∑k

i=dk/2e 1/√i)

(26a)

νk :=(∑k

i=dk/2e νi/√i)/(∑k

i=dk/2e 1/√i). (26b)

The proposed scheme converges to the value fa(ν∗, z∗) ob-tained at a saddle point (ν∗, z∗) asymptotically in the numberof iterations k [30, Sec. 3.1].

Upon convergence of the iterates in (26), the slow-timescalevariables z∗ have been derived together with the optimalLagrange multiplier ν∗ related to constraint (15c). The gridoperator can implement z∗, and the fast-timescale decisionsy∗t for a realization ξt can be found by solving (19). Theaverage dispatch algorithm (ADA) is summarized as Alg. 1.

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V. PROBABILISTIC DISPATCH ALGORITHM

The probabilistic version of problem (14) is considered next.Here, the ergodic constraint (14e) reads h(yt) = 1vt /∈VA − α. Despite the non-convexity of the probabilistic con-straint, (15) can still be solved optimally. However, optimalityfor (14) cannot be guaranteed. A heuristic solution is detailednext by adapting the solution of Sec. IV.

To that end, dual decomposition is used here as well. If ν isthe scalar Lagrange multiplier associated with constraint (14e),the partial Lagrangian function for (15) is now Lp(yt, ν) :=Et [gt(yt) + ν(1vt /∈ VA − α)]. The corresponding dualfunction, fast-timescale problem, and dual problem are definedanalogously to (17), (18), and (20). The indicator functionrenders Lp(yt, ν) non-convex. Surprisingly enough though,under the practical assumption that ξt follows a continuouspdf, problem (15) enjoys zero duality gap; see [31, Th. 1].

The additional challenge here is the non-convexity of theLagrangian minimization:

y∗t (ν, z) ∈ arg minyt∈Yt

gt(yt) + ν1vt /∈ VA (27)

s.to: Kz + Byt = Hξt.

Because the indicator function takes only the values 0, 1however, the solution to (27) can be found by solving a pairof slightly different convex problems. The first problem is

y∗t,A(z) ∈ arg minyt∈Yt

gt(yt) (28a)

s.to: Kz + Byt = Hξt (28b)vt ∈ VA (28c)

whereas the second problem ignores constraint vt ∈ VA as

y∗t,B(z) ∈ arg minyt∈Yt

gt(yt) (29a)

s.to: Kz + Byt = Hξt. (29b)

From the point of view of (27), if the voltages in y∗t,B(z)do not belong to VA, the solution to the second problemwill incur an additional cost quantified by ν. Observe thatneither problem (28) nor (29) depend on ν, while theircomplexity is similar to the one problem (18). Suppose that(28) and (29) have been solved and let λ∗t,A(z) and λ∗t,B(z)denote the optimal multipliers associated with (28b) and (29b),respectively. Then, problem (27) can be neatly tackled byidentifying two cases:

(c1) If gt(y∗t,A(z)) > gt(y∗t,B(z)) + ν, then y∗t,B(z) is

a minimizer of (27) as well and voltages are allowed tolie outside VA. In this case, set y∗t (ν, z) := y∗t,B(z) andλ∗t (ν, z) := λ∗t,B(z). This case includes instances whereproblem (28) is infeasible for which gt(y∗t,A(z)) =∞.

(c2) If gt(y∗t,A(z)) ≤ gt(y∗t,B(z)) + ν, then y∗t,A(z) min-

imizes (27) too and voltages lie within VA. In this case, sety∗t (ν, z) := y∗t,A(z) and λ∗t (ν, z) := λ∗t,A(z).

Case (c2) covers also instances where v∗t,B(z) happens tolie in VA. In these particular instances, y∗t,B(z) serves as aminimizer of (28) too. Then, it follows that gt(y∗t,A(z)) =gt(y

∗t,B(z)) ≤ gt(y

∗t,B(z)) + ν for ν ≥ 0. This implies that

one can solve (29) first and, if v∗t,B(z) ∈ VA, there is no needto solve problem (28).

Algorithm 2 Probabilistic Dispatch Algorithm (PDA)1: Initialize (z0, ν0).2: repeat for k = 0, 1, . . .3: Draw sample ξk.4: Find (y∗k,B(νk, zk),λ∗k,B(νk, zk)) by solving (29).5: Set y∗t (ν, z) := y∗t,B(z) and λ∗t (ν, z) := λ∗t,B(z).6: if v∗k,B(z) /∈ VA, then find y∗k,A(νk, zk) andλ∗k,A(νk, zk) by solving (28).

7: if gt(y∗t,A(z)) ≤ gt(y

∗t,B(z)) + ν, then set

y∗t (ν, z) := y∗t,A(z) and λ∗t (ν, z) := λ∗t,A(z).8: end if9: end if

10: Update (zk+1, νk+1) via (30).11: Compute sliding averages (zk, νk) through (26).12: until convergence of (zk, νk).13: Output z∗ = zk and ν∗ = νk.

To find the optimal slow-timescale variables under theprobabilistic dispatch, the stochastic primal-dual iterations ofSec. IV are adapted here as

νk+1 := [νk + µk(1v∗k(νk, zk) /∈ VA − α)]+ (30a)

zk+1 := [zk − dg(εk)(∂zf(zk) + K>λ∗k(νk, zk)]Z . (30b)

The probabilistic dispatch algorithm (PDA) is tabulated asAlg. 2. At every fast-timescale iteration, PDA solves (29)and possibly (28). Since the optimizations tasks (28)–(29) arestructurally similar to (18), PDA has at most twice the per-iteration complexity of ADA. Because function Gp(z) is notnecessarily convex, the iterates in (30) are not guaranteed toconverge to a minimizer of (14). The practical performance ofPDA in finding z∗ is numerically validated in Sec. VI.

VI. NUMERICAL TESTS

The proposed grid dispatches were tested on a 56-busSouthern California Edison (SCE) distribution feeder [11].5-MW PVs were added on buses 44 and 50; both with 6-MVA inverters enabling power factors as low as 0.83 (lead-ing or lagging) at full solar generation. The prices for theenergy exchange with the main grid were β = 37 $/MWh;γb = 45 $/MWh, and γs = 19 $/MWh. Diesel generatorswith capacity pdn = 0.5 MW were sited on buses 10, 18, 21,30, 36, 43, 51, and 55. The cost of diesel generation wasCD(pd) =

∑Nn=1(30pdn + 15(pdn)2) $/h with pd expressed

in MW. Apparent power flows were limited to 7 MVA. Thevoltage operation limits were set to vA = 0.982, vA = 1.022,vB = 0.972, and vB = 1.032, expressed in pu with respect toa voltage base of 12 kV. (Re)active nodal loads were Gaussiandistributed with the nominal load of the SCE benchmark asmean value, and standard deviation of 0.2 times the nominalload. The solar energy generated at each PV was drawnuniformly between 0.5 and 1 times the actual power PV rating.

The simulations presented next have been run using theLDF model. The LDF model is computationally less complexthan the relaxed AC grid model of Section II-D, which isadvantageous when many instances of the fast-variation scalehave to be solved. Our tests show the LDF model is 33%

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0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

0.05

0.1

0.15

0.2

0.25

I te rat i on index

pd[M

W]

0.5 1 1.5 2 2.5 3 3.5 4

x 104

0.97

0.975

0.98

0.985

0.99

0.995

1

Substationvoltage[p

u]

I te rat i on index0.5 1 1.5 2 2.5 3 3.5 4

x 104

−5

−4.9

−4.8

−4.7

−4.6

−4.5

pa 0[M

W]

0.5 1 1.5 2 2.5 3 3.5 4

x 104

−5

−4.5

Fig. 1. Convergence of primal variables for ADA: (top) diesel generation;(bottom) substation voltage v0 (left y-axis) and energy exchange pa0 (righty-axis). Sliding averages of optimization variables are depicted too.

faster, while it incurs 10% higher cost compared to the SOCPrelaxation. Such numbers are consistent with those observedfor other problems [6], [28]. In any case, the findings presentednext are valid for both models and also for cases where theLDF model is adopted only for finding the slow-timescalevariables, while the exact/relaxed AC grid model is employedduring the actual fast-timescale dispatch.

ADA was run with step sizes proportional to 1/√k with

initial values εv00 = 4·10−5, εp00 = 4·10−1, εpd0 = 6·10−3, andµ0 = 225, to account for different dynamic ranges. The iteratesfor primal and dual variables as well as their correspondingsliding averages are depicted in Figs. 1 and 2. Primal anddual slow-timescale variables hover in a small range whosewidth diminishes with time. Their sliding averages convergeasymptotically. The algorithm reaches a practically meaningfulsolution within 5,000 iterations. Buses 44 and 50 are prone toovervoltages since they host PVs, and buses 2 and 15 are proneto under-voltages; thus yielding non-zero dual variables for theaverage upper and lower voltage constraints, respectively.

PDA was tested using the same simulation setup for α =0.05 and µ0 = 1. Figure 3 shows the convergence of primaland dual variables, and the probability of voltages deviatingfrom VA. Since we know that the per-iteration computation ofPDA is at most twice that of ADA and the simulations showthat the number of iterations required for PDA and ADA issimilar, it then follows that the total computation time for PDAis at most twice that for ADA. Granted that the probabilisticconstraint in (10) applies collectively to all buses, the under-/over-voltage probabilities on a per-bus basis is depicted inFig. 4. The occurrences of overvoltage seem to be shared

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

2

4

6

8

10

12

14

I te rat i on index

ν

0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

0

1

2

3

4

5

6

7

I te rat i on index

ν

Fig. 2. Convergence of dual variables for ADA: (top) dual variables associatedwith average lower voltage limits for all buses; and (bottom) dual variablesassociated with average upper voltage limits for all buses. Sliding averagesof optimization variables are depicted too.

primarily among buses 40–56 which are neighboring to thePV buses 40 and 55. On the contrary, buses 10–16 beingelectrically far from both the substation and PVs, experienceunder-voltage with a small probability.

The effect of the average versus the probabilistic con-straint on voltage magnitudes was evaluated next. After slow-timescale variables z had converged, fast-timescale variablesyt were calculated for 6,000 instances of ξt using both ADAand PDA. The histograms of the voltage magnitudes on tworepresentative buses are presented in Fig. 5. Under PDA, theaverage voltage on bus 15 is slightly higher than the averagevoltage obtained by ADA. In exchange, the instantaneousvalue of the voltage on bus 15 stays within VA with higherprobability. A similar behavior is observed for the overvoltageinstances on PV bus 40.

ADA and PDA were finally compared to three alternativeschemes. The first two, henceforth called approximate averageand approximate probabilistic dispatches, obtained z by settingloads and solar generation to their expected values, whilevariables ν were calculated via dual stochastic subgradient,and ytTt=1 were found by solving either (18) or (27),depending on whether the setting is average or probabilistic.The third deterministic dispatch found z as the approximateschemes do, and ytTt=1 by enforcing vt ∈ VA at all times.Note that the three proposed alternatives provide feasiblesolutions satisfying voltage regulation constraints. The fivedispatches were tested under five scenarios: Scenario 1 isthe setup described earlier. Scenario 2 involved the tighter

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)

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Substation

voltage[p

u]

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x 104

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pa 0[M

W]

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x 104

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Prv

/∈VA

It er at ion index

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1

1.2

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ν

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x 104

0

0.2

0.4

0.6

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1

1.2

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1.6

1.8

2

Fig. 3. Convergence for PDA: (left) diesel generation; (middle) substation voltage (left y-axis) and energy exchange pa0 (right y-axis); and (right) dual variablerelated to probabilistic constraint (left y-axis) and under-/over-voltage probability (right y-axis). Sliding averages of optimization variables are shown too.

10 20 30 40 500

0.025

0.05

0.075

Bus index

Probability

α

Overvoltage

Undervoltage

Fig. 4. Per-bus probability of under-/over-voltages.

0.97 0.98 1 1.02 1.030

0.1

0.2

0.3

0.4

0.5

Relativ

eFrequency

Bus 15

Bus 40

0.97 0.98 1 1.02 1.03

0.1

0.2

0.3

0.4

0.5

Vol tage magni tude [pu]

RelativeFre

quency

Fig. 5. Histograms of voltage magnitudes on buses 15 and 40 under ADA(top) and PDA (bottom). Dashed lines show regulation limits VA and VB .

voltage limits vA = 0.992 and vA = 1.012. Scenarios 3,4, and 5 were generated by scaling the mean value and thestandard deviation for loads of scenario 1 by 0.5, 1.5, and 2,respectively. Figure 6 shows the expected operation costs forall five scenarios. ADA (PDA) yielded the lowest cost underall scenarios in the average (probabilistic) setting as expected.In all test cases, ADA yielded a slightly lower objective thanPDA for α = 0.05. The loss of optimality entailed by theapproximate average and probabilistic schemes is due to thesuboptimal choice of z. The deterministic scheme entailed

Scenario 1 Scenario 2 Scenario 3 Scenario 4 Scenario 5−180

−160

−140

−120

−100

−80

−60

−40

−20

0

20

Netw

ork

opera

tioncost

[$/h]

ADA (Algorithm 1)

PDA (Algorithm 2)

Approx. avg. dispatch

Approx. prob. dispatch

Deterministic dispatch

Fig. 6. Performance for ADA, PDA, approximate average, approximateprobabilistic, and deterministic scheme.

an additional loss of optimality by preventing the occasionalviolation of VA.

To gain insights on the algorithm scalability, numerical testswere also performed using the IEEE 123-bus feeder [40].PV systems were added at buses 92, 103, 119 and 122; anddiesel generators at buses 3, 7, 32, 37, 39, 44, 51, 54, 56,70, 74, 85, 92, 103, 119, and 122. Diesel generation costsand limits, and PV generation pdfs remained similar to theprevious test. The nominal (re)active loads were perturbedby zero-mean Gaussian random variables having a standarddeviation of 0.2 times the nominal value. The voltage operationlimits were set to vA = 0.992, vA = 1.012, vB = 0.982, andvB = 1.022 (pu). ADA was run with step sizes proportionalto 1/

√k, εv00 = 10−4, εp00 = 2 · 10−2, εpd0 = 10−3, and

µ0 = 400. Figs. 7 and 8 show the convergence of the primaland dual variables. For this larger feeder, the algorithm reachesa practically meaningful solution after around 10,000 iterationsand the average per-iteration computation increases by 90%.

VII. CONCLUSIONS

By nature of renewable generation, electromechanical com-ponent limits, and the manner markets operate, energy man-agement of smart distribution grids involves decisions at

IEEE TRANSACTIONS ON SMART GRID (TO APPEAR) 9

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W]

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u]

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x 104

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−2.2

−2

−1.8

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pa 0[M

W]

Fig. 7. Convergence of primal variables for ADA on the IEEE 123-bus feeder:(top) diesel generation; (bottom) voltage v0 (left y-axis) and energy exchangepa0 (right y-axis). Sliding averages of optimization variables are depicted too.

slower and faster timescales. Since slow-timescale controlsremain fixed over multiple PV operation slots, decisions arecoupled across time in a stochastic manner. To accommo-date solar energy fluctuations, voltages have been allowedto be sporadically overloaded; hence introducing couplingof fast-timescale variables on the average or in probability.Average voltage constraints have resulted in a stochasticconvex-concave problem, whereas non-convex probabilisticconstraints were tackled using dual decomposition and convexoptimization. Efficient algorithms for finding both slow andfast controls using only random samples have been put forth.Our two novel solvers converge in terms of the primal and dualvariables, and have attained lower operational costs comparedto deterministic alternatives. Although probabilistic constraintshave been applied grid-wise, voltages on individual busesremained within limits. Enforcing probabilistic constraints ona per-bus basis, developing decentralized implementations, andincluding voltage regulators are interesting research directions.

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[2] P. M. S. Carvalho, P. F. Correia, and L. A. Ferreira, “Distributed reactivepower generation control for voltage rise mitigation in distributionnetworks,” IEEE Trans. Power Syst., vol. 23, no. 2, pp. 766–772, May2008.

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0 0.5 1 1.5 2 2.5 3 3.5 4

x 104

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12

14

I te rat i on index

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0 0.5 1 1.5 2 2.5 3 3.5 4

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[10] M. Farivar, R. Neal, C. Clarke, and S. Low, “Optimal inverter VARcontrol in distribution systems with high PV penetration,” in Proc. IEEEPower & Energy Society General Meeting, San Diego, CA, Jul. 2012.

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[12] E. Dall’Anese, S. V. Dhople, and G. B. Giannakis, “Optimal dispatch ofphotovoltaic inverters in residential distribution systems,” IEEE Trans.Sustain. Energy, vol. 5, no. 2, pp. 487–497, Dec. 2014.

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flow in radial networks with distributed generation,” IEEE Trans. SmartGrid, 2016, to appear.

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[29] L. M. Lopez-Ramos, V. Kekatos, A. G. Marques, and G. B. Giannakis,“Microgrid dispatch and price of reliability using stochastic approx-imation,” in Proc. IEEE Global Conf. on Signal and Inform. Process.,Orlando, FL, Dec. 2015.

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Luis M. Lopez-Ramos (M’16) received the B.Sc.degree (with highest honors) in TelecommunicationsEngineering in 2010 from King Juan Carlos Uni-versity (URJC), Madrid, Spain; and the M.Sc. andPh.D. degrees in multimedia and communications in2012 and 2016, respectively, from URJC and CarlosIII University of Madrid, Spain. During the fall of2013 and 2014, he was a visiting scholar at the ECEDept. of the Univ. of Minnesota. In February 2017,he joined the Univ. of Agder, Grimstad, Norway asa post-doctoral research fellow. Dr. Lopez-Ramos’

research currently focuses on stochastic nonlinear programming, stochasticapproximation and signal processing techniques, and their applications inwireless networks and power grids.

Vassilis Kekatos (SM’16) obtained his Diploma,M.Sc., and Ph.D. in Computer Science and Engr.from the Univ. of Patras, Greece, in 2001, 2003,and 2007, respectively. He was a recipient of aMarie Curie Fellowship during 2009-2012. Duringthe summer of 2012, he worked for Windlogics Inc.After that, he was a research associate with the Dept.of Electrical and Computer Engr. of the Univ. ofMinnesota. During 2014, he stayed with the Univ.of Texas at Austin and the Ohio State Univ. as avisiting researcher, and he received the postdoctoral

career development award (honorable mention) by the Univ. of Minnesota.In August 2015, he joined the Dept. of Electrical and Computer Engr. ofVirginia Tech as an Assistant Professor. His research focus is on optimization,learning, and management of future energy systems. He is currently servingas an Associate Editor of the IEEE Trans. on Smart Grid.

Antonio G. Marques (SM’13) received theTelecommunications Engineering degree and theDoctorate degree, both with highest honors, from theCarlos III Univ. of Madrid, Spain, in 2002 and 2007,respectively. In 2007, he became a faculty of theDept. of Signal Theory and Communications, KingJuan Carlos Univ., Madrid, Spain, where he currentlydevelops his research and teaching activities as anAssociate Professor. From 2005 to 2015, he helddifferent visiting positions at the Univ. of Minnesota,Minneapolis. In 2015 and 2016 he was a Visiting

Scholar at the Univ. of Pennsylvania.His research interests lie in the areas of signal processing, communication

theory, and networking. His current research focuses on stochastic resourceallocation for wireless networks and smart grids, nonlinear network optimiza-tion, and signal processing for graphs. Dr. Marques has served the IEEE ina number of posts (currently, he is an Associate Editor of the IEEE SignalProcess. Letters and a member of the Signal Process. Theory and MethodsTechnical Committee), and his work has been awarded in several conferences.

IEEE TRANSACTIONS ON SMART GRID (TO APPEAR) 11

Georgios B. Giannakis (F’97) received his Diplomain Electrical Engr. from the Ntl. Tech. Univ. ofAthens, Greece, 1981. From 1982 to 1986 he waswith the Univ. of Southern California (USC), wherehe received his MSc. in Electrical Engineering,1983, MSc. in Mathematics, 1986, and Ph.D. inElectrical Engr., 1986. He was with the Univ. of Vir-ginia from 1987 to 1998, and since 1999 he has beena professor with the Univ. of Minnesota, where heholds an Endowed Chair in Wireless Telecommuni-cations, a Univ. of Minnesota McKnight Presidential

Chair in ECE, and serves as director of the Digital Technology Center.His general interests span the areas of communications, networking and

statistical signal processing - subjects on which he has published more than

400 journal papers, 680 conference papers, 25 book chapters, two editedbooks and two research monographs (h-index 119). Current research focuseson learning from Big Data, wireless cognitive radios, and network sciencewith applications to social, brain, and power networks with renewables. He isthe (co-) inventor of 28 patents issued, and the (co-) recipient of 8 best paperawards from the IEEE Signal Processing (SP) and Communications Societies,including the G. Marconi Prize Paper Award in Wireless Communications.He also received Technical Achievement Awards from the SP Society (2000),from EURASIP (2005), a Young Faculty Teaching Award, the G. W. TaylorAward for Distinguished Research from the University of Minnesota, and theIEEE Fourier Technical Field Award (2015). He is a Fellow of EURASIP, andhas served the IEEE in a number of posts, including that of a DistinguishedLecturer for the IEEE-SP Society.