992 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

Distributed MPC for Efficient Coordination ofStorage and Renewable Energy Sources

Across Control AreasKyri Baker, Member, IEEE, Junyao Guo, Student Member, IEEE,

Gabriela Hug, Senior Member, IEEE, and Xin Li, Senior Member, IEEE

Abstract—In electric power systems, multiple entities areresponsible for ensuring an economic and reliable way of deliv-ering power from producers to consumers. With the increase ofvariable renewable generation it is becoming increasingly impor-tant to take advantage of the individual entities’ (and their areas’)capabilities for balancing variability. Hence, in this paper, weemploy and extend the approximate Newton directions method tooptimally coordinate control areas leveraging storage available inone area to balance variable resources in another area with onlyminimal information exchange among the areas. The problem tobe decomposed is a model predictive control problem includinggeneration constraints, energy storage constraints, and AC powerflow constraints. Singularity issues encountered when formulat-ing the respective Newton–Raphson steps due to intertemporalconstraints are addressed and extensions to the original decom-position method are proposed to improve the convergence rateand required communication of the method.

Index Terms—Distributed optimization, model predictive con-trol, storage, AC optimal power flow, approximate Newtondirection method.

NOMENCLATURE

N optimization horizonNB number of buses in the systemPGi active power output of generator at bus iai, bi, ci cost parameters of generator at bus iPWi active power output of wind generator at bus iPLi active power load at bus iPIi power injected into storage at bus iPOi power drawn from storage at bus iPij active power flowing on line ij�i set of buses connected to bus i

Manuscript received June 23, 2014; revised April 13, 2015 andDecember 7, 2015; accepted December 23, 2015. Date of publicationJanuary 7, 2016; date of current version February 17, 2016. This workwas supported in part by the National Science Foundation under GrantECCS 1027576, and in part by the SYSU-CMU Joint Institute of Engineering.Paper no. TSG-00644-2014.

K. Baker is with the National Renewable Energy Laboratory, Golden,CO 80401 USA (e-mail: kyri.baker@nrel.gov).

J. Guo, G. Hug, and X. Li are with the Department of Electricaland Computer Engineering, Carnegie Mellon University, Pittsburgh,PA 15213 USA (e-mail: junyaog@andrew.cmu.edu; ghug@ece.cmu.edu;xinli@ece.cmu.edu).

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

Digital Object Identifier 10.1109/TSG.2015.2512503

QGi reactive power output of generator at bus iQLi reactive power load at bus iQij reactive power flowing on line ijEi energy level in storage at bus iT time between two consecutive timestepsηci charging efficiency of storage at bus iηdi discharging efficiency of storage at bus iVi maximum voltage magnitude at bus iVi minimum voltage magnitude at bus iPGi maximum power output of generator at bus i�PGi maximum ramp rate of generator at bus iEi lower energy limit of storage at bus iEi energy capacity of storage at bus iPEi maximum dis/charging rate of storage at bus i

I. INTRODUCTION

AS ELECTRIC power systems span entire continents, thecontrol responsibility for these large systems is shared

among multiple entities. Each of these entities is responsiblefor a specific geographic area called control area. The couplingof the control areas via tie lines allows for exchanging poweracross their boundaries but also leads to the need to coordinatethe actions in the areas. Traditionally, this is being done byagreeing on a tie line flow, e.g., based on market mechanisms,and then optimize the schedule of generation within the areasto balance supply and demand. This leads to an overall subop-timal usage of the available resources because the optimizationis limited to the localized areas.

As long as resources are distributed throughout the systemin a fairly homogenous way in terms of their capabilities andcosts, the suboptimality may be acceptable. However, oncethe resources in one area have considerably different charac-teristics compared to the resources in the neighboring area,substantial improvements in terms of providing reliable andcost effective electric power supply may be achieved. In thispaper, we particularly consider the situation in which onearea has significant amounts of variable renewable genera-tion resources and the other area has significant amounts ofstorage. In that case, it is beneficial for both areas to improvethe coordination such that the storage is being used to balancethe variability of the renewable resources.

Generally, this means that the control areas should be opti-mized jointly in a single centralized optimization problem

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BAKER et al.: DISTRIBUTED MPC FOR EFFICIENT COORDINATION OF STORAGE AND RENEWABLE ENERGY SOURCES 993

requiring that the areas share their system information eitherwith the other entities or a centralized entity overseeing allareas. Another option is to use decomposition techniques todecompose the centralized problem into subproblems eachassociated with a particular control area. The result is an iter-ative process where each area solves the problem assignedto it and then provides some information about the solutionat the buses located at the boundary of the area to the otherareas. Specifically, the voltage magnitude and angle as well asthe Lagrange multiplier corresponding to the power balanceequation at the boundary buses are exchanged between areas.Based on the information received the areas update their solu-tion and keep exchanging until convergence has been achieved.The final solution should be equal to the solution obtained ifthe problem would be solved by a centralized entity acrossall areas.

Given that the intention is to optimize the usage of storageand that how a storage can be used in the future timestepsheavily depends on how it is used at the current timestep,we formulate a multi-step AC Optimal Power Flow prob-lem and implement a receding horizon. Consequently, theresulting problem is a Model Predictive Control (MPC) prob-lem [1]. MPC, or look-ahead optimization, has been shown tobenefit the operation of power systems significantly [2]. Wedecompose this problem using the Optimality Decompositionand Approximate Newton Directions method [3] where eachsubproblem is associated with a particular control areas. Inorder to improve the convergence rate of the algorithm, twoapproaches are proposed: the first is derived from the Jacobimethod for solving a linear system of equations and the sec-ond adds an additional term to the update which better reflectsthe impact of one area on the other. The consequence is thatthe number of times that the areas need to communicate toconverge toward the overall optimum is reduced.

Hence, the outline of the paper is as follows: Section IIprovides an overview of some related work. In Section III,the MPC problem formulation for the centralized problem isgiven. This problem is then decomposed in Section IV wherethe proposed modifications to the decomposition algorithm arepresented as well. Cases when the Jacobian matrix becomessingular due to the intertemporal constraints from storage arediscussed and resolved in Section V. Simulation results areshown for all three distributed methods in Section VI, andSection VII concludes the paper.

II. RELATED WORK

The distributed AC Optimal Power Flow problem has beenaddressed in the literature by a variety of approaches, most ofthem derived from the Augmented Lagrangian method [4]–[6].The Approximate Newton Directions method is used todecompose the AC OPF problem in [7], and is also usedin [8] for decentralized control of power flow devices acrossoverlapping control areas. Fully decentralized optimization onthe nodal level for the AC Optimal Power Flow problem isdiscussed in [9] on a 4 and 6-bus network.

Here, the distributed problem is extended across optimiza-tion timesteps to optimize over a prediction horizon using

Model Predictive Control in order to determine the optimaluse of the storage device and optimal generation settings.Geographical regions are decomposed and a small amountof information is communicated between areas without theneed for a centralized controller, coordinating storage in onearea with renewable energy in an adjacent area. MPC haspreviously been applied to power systems for energy storagecontrol. In [10]–[13], centralized MPC is used on relativelysmall scale systems to optimally control a battery to reducethe effect of fluctuations in the power supply due to renewablegeneration. Rolling horizon control has been applied to theunit commitment problem in [14]–[16]. A Model PredictiveControl AC OPF problem was solved in a distributed man-ner in [17] using AND on the IEEE-14 bus system. In [18],an MPC problem including DC power flow constraints issolved in a distributed way using a proximal message passingmethod. Distributed MPC is implemented for another purposein [19] for Automatic Generation Control, and in [20] for themitigation of cascading failures in a power system.

In this paper, distributed MPC is used to coordinate renew-able generation in one area with storage in another area. Twoextensions are derived for the original AND method whichfor a variety of cases significantly improve the rate of con-vergence of the optimization. A proof of concept is givenfor the IEEE-57 and IEEE-118 bus test systems. Generally,solving the AC OPF problem for each step in an entire pre-diction horizon results in a very large nonlinear optimizationproblem; however, the straightforward decomposition of theproblems using AND makes the optimization of the individ-ual subproblems easily parallelizable. When this method isapplied to the multi-area OPF problem as in [7] and [17], thevariables exchanged between the areas corresponds to the volt-age magnitudes and angles at buses connected across areas, aswell as the Lagrange multipliers at these connected buses andlines. There is no need for areas to share information withother areas that is not related to their physically connectedbuses, and the problem converges to the centralized solutionprovided the convergence criteria is met [3].

III. PROBLEM FORMULATION

The problem that we address in this paper is a ModelPredictive Control problem to minimize the overall cost of sup-plying the load by optimally using the available energy storage.Hence, it is a multi-step optimal power flow problem whichincludes inter-temporal constraints on energy storages and theAC power flow constraints. The overall problem formulationis therefore given by

minPG

N∑

k=1

(NB∑

i=1

aiP2Gi

+ biPGi + ci

)(1)

s.t. PGi(k) + PWi(k) − PLi(k) (2)

− PIi(k) + POi(k) −∑

j∈�i

Pij(k) = 0, (3)

QGi(k) − QLi(k) −∑

j∈�i

Qij(k) = 0, (4)

994 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

Ei(k + 1) = Ei(k) + ηcTPIi(k) − T

ηdPOi(k), (5)

Ei ≤ Ei(k + 1) ≤ Ei, (6)

0 ≤ PIi(k) ≤ PEi , (7)

0 ≤ POi(k) ≤ PEi , (8)

0 ≤ PGi(k) ≤ PGi , (9)

Vi ≤ Vi(k) ≤ Vi, (10)

|PGi(k + 1) − PGi(k)| ≤ �PGi , (11)

for k = {0, . . . , N − 1} and i = {1, . . . , NB}.Whenever there is no generator connected to bus i, it is

assumed that equations (2) – (4) are reduced to not includethe generation output variable. The same holds for the windgenerator output, the loads and the storage variables. Equalitiesand inequalities (5) – (9) are only included if there is a gen-erator or a storage connected to bus i. Equations (2) – (4)represent the power flow equations in the system where theflows Pij and Qij are functions of the voltage magnitudes andangles at the ends of the lines and the line parameters. Forthe generator buses, the voltage magnitude at the respectivebuses are set to fixed values and for the slack bus additionallythe voltage angle is set to zero. Equation (10) represents aconstraint on the upper and lower level of the voltage magni-tude at each bus. If there is a generator at bus i, equation (11)represents the ramping limit of that generator.

As Model Predictive Control is used here, once the problem(1) – (11) is solved for timestep t, the solution for the first stepk = 0 is applied. Then, the optimization horizon is shifted byT and the problem is solved for the new time horizon.

IV. DISTRIBUTED MPC

The resulting optimization problem (1) – (9) correspondsto the centralized problem including multiple control areas.Decomposing the problem allows each control area to solve theoptimization problem associated with its own part of the sys-tem while optimally coordinating with its neighboring areas.It is assumed that these control areas are separate and do notcoordinate in any other means other than exchanging the tie-line variables. In the situation considered here, energy storageis located in one of these areas and renewable generation inthe other.

The focus of this paper is on vertically integrated utili-ties where the goal is to minimize overall generation costto supply the load. Even if the optimization takes place forone timestep, or over a horizon, or over a rolling horizon,the approach can be used to optimally coordinate neighboringcontrol areas. The original method has been presented for thepurpose of optimizing for a single timestep [7]. Expandingit to multiple timesteps also allows for an optimal integra-tion of storage devices. Hence, the main focus in this paper ison enabling optimal coordination across areas and particularlythe coordination of variable renewable generation with storage.However, decomposition of the considered problem generallyallows for parallelized computations and therefore for solvinglarge scale optimization problems which otherwise could notbe solved or not solved within a reasonable amount of time.As the prediction horizon N increases, the considered MPC

problem may lead to such a large scale problem and despitethe potentially existing centralized coordinator could requirea distributed solution process.

In this section, we first show how the Unlimited PointAlgorithm [21] is used to handle the inequality constraintsin the problem formulation, then we provide the decom-posed problem formulation using the Approximate NewtonDirection and finally describe the proposed modifications tothe algorithm which lead to an improved convergence speed.

A. Unlimited Point Method

There are various ways to handle inequality constraints inan optimization problem. In this paper, we use the UnlimitedPoint method [21], however, the derivations provided beyondthis subsection stay the same even if another method is usedto incorporate inequality constraints into a Newton-Raphsonupdate (such as Interior Point). In the Unlimited Point method,the inequality constraints in the general optimization problem

minx

f (x) (12)

s.t. g(x) = 0 (13)

h(x) ≤ 0 (14)

are transformed into equality constraints according to

hn(x) + s2n = 0 (15)

for inequality n and where sn is a slack variable. Squaringthe slack variable ensures that the original inequality con-straint is fulfilled. The first order optimality conditions arethen formulated as

∂f

∂x+ λT · ∂g

∂x+ μ2T · ∂h

∂x= 0 (16)

g(x) = 0 (17)

h(x) + s2 = 0 (18)

diag(μ) · s = 0 (19)

Hence, similar to the slack variables also the LagrangeMultipliers are replaced with squared variables to ensure thatthese Lagrange Multipliers take values which are greater thanzero without having to explicitly include such non-negativityconstraints.

The Unlimited Point formulation is applied to (1) – (11)in this paper. The next step now is to apply the ApproximateNewton Direction method [3] to the resulting first order opti-mality conditions in order to be able to solve these in adistributed way.

B. Application of Approximate Newton Directions Method

Generally, an optimization problem can be solved by apply-ing the Newton-Raphson method to the first order optimalityconditions of the problem. Such an update is given by

x(p+1) = x(p) + α · �x(p) = x(p) − α ·(

J(p)tot

)−1 · d(p), (20)

where p is the iteration counter. In our case, the righthand side vector d(p) includes the first order optimalityconditions (16) – (19) for problem (1) – (11) ordered according

BAKER et al.: DISTRIBUTED MPC FOR EFFICIENT COORDINATION OF STORAGE AND RENEWABLE ENERGY SOURCES 995

to the subproblems and evaluated at x(p), and the update vectoris given by �x(p). The parameter α is used to control the stepsize to avoid divergence due to overshooting. The Jacobianmatrix J(p)

tot is also evaluated at x(p) and is given by

J(p)tot =

⎛

⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

J(p)

1,1 J(p)

1,2 · · · · · · J(p)

1,M

J(p)

2,1. . .

. . ....

...

.... . .

. . . J(p)

q−1,q

· · · J(p)

q,q−1 J(p)q,q J(p)

q,q+1 · · ·J(p)

q+1,q. . .

. . ....

......

. . .. . . J(p)

M−1,M

J(p)

M,1 · · · · · · J(p)

M,M−1 J(p)M,M

⎞

⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

.

(21)

The block element J(p)

l,j corresponds to the Jacobian matrix ofthe first order optimality conditions associated with the con-straints in area l with respect to the variables associated witharea j.

In the Approximate Newton Directions method, the decom-position into M subproblems is achieved by setting theoff-diagonal block matrices J(p)

l,j , l �= j, equal to zero. Theseoff-diagonal matrices are generally sparse because the onlynon-zero elements arise from coupling constraints, i.e., con-straints which couple the variables of area l with the variablesof area j. The resulting Newton-Raphson update can then besolved in a distributed way, i.e.,

x(p+1)q = x(p)

q + α · �x(p)q = x(p)

q − α ·(

J(p)q,q

)−1 · d(p)q , (22)

for q = 1, . . . , M. Hence,

d(p) =[d(p)

1 , . . . , d(p)M

]T, (23)

x(p) =[x(p)

1 , . . . , x(p)M

]T(24)

�x(p) =[�x(p)

1 , . . . ,�x(p)M

]T. (25)

In the considered problem, the optimization problem is decom-posed according to geographical areas, i.e., the variables inx(p)

q correspond to the variables associated with buses inarea q. As the considered problem spans multiple timesteps,this variable vector includes copies of all the variableswithin that area for all timesteps in the optimization horizon,i.e., PGi(0), . . . , PGi(N − 1).

The advantage of this method is that instead of solvingeach subproblem to optimality before exchanging informa-tion with the other subproblems, data can be exchanged aftereach Newton-Raphson iteration. And unlike other Lagrangian-based decomposition methods such as Lagrangian Relaxationand Augmented Lagrangian, there is no need for a centralizedentity or tuning of parameters to update the Lagrange multi-pliers; subproblems simply exchange data directly with theirneighbors and the updates for the multipliers come directlyfrom the other subproblems.

C. Modifications of the AND Method

In this paper, we consider two adjustments to theApproximate Newton method, both with the intention to

reduce the gap between the distributed variable update andthe centralized update, thus improving the convergence rate insome cases.

1) Jacobi Update: The first modification is derived fromthe Jacobi method for solving a linear system of equa-tions [22]. Instead of setting the off-diagonal block matrices inthe Jacobian matrix to zero, the information from the previousiteration p − 1 is used to update the variables at iteration p.The variable update for each subproblem p = 1, . . . , M is nowequal to

J(p)q,q · �x(p)

p = −d(p)q −

M∑

m=1,m�=q

(J(p−1)m,q · �x(p−1)

m ). (26)

Even with these additional terms in the update it is not nec-essary to exchange the full update vectors �x(p−1)

m among theareas. Area m can, without additional information exchange,evaluate J(p−1)

m,q at iteration p−1 and then compute the multipli-cation with the update vector �x(p−1)

m . As J(p−1)m,q is very sparse,

the multiplication with �x(p−1)m results in a sparse vector and

only the non-sparse elements need to be shared.The issue with this update is that it basically builds upon

the assumption that �x(p−1)m and �x(p)

m will be similar, whichdoes not necessarily have to be the case and may thereforeonly result in improved performance in certain cases.

2) Additional Term in Right Hand Vector: The secondapproach is a bit more involved in its derivation. Hence, we usea two area example to present the main idea. The centralizedupdate is given by

[J(p)

11 J(p)

12

J(p)

21 J(p)

22

]·[

�x(p)

1

�x(p)

2

]= −

[d(p)

1

d(p)

2

](27)

By reordering the terms in the rows, the following formulasfor the updates result,

�x(p)

1 = −J(p)−1

11 · d(p)

1 − J(p)−1

11 J(p)

12 · �x(p)

2 , (28)

�x(p)

2 = −J(p)−1

22 · d(p)

2 − J(p)−1

22 J(p)

21 · �x(p)

1 . (29)

By substituting (29) into (28) and vice versa, the updates canbe written as a function of the matrices and the right handside vectors, i.e., (for simplification, we do not indicate theiteration counter in these equations)

�x1 =(

J11 − J12J−122 J21

)−1 ·(−d1 + J12J−1

22 · d2

), (30)

�x2 =(

J22 − J21J−111 J12

)−1 ·(−d2 + J21J−1

11 · d1

). (31)

This update corresponds to the exact update, i.e., the updatethat is obtained if Newton Raphson is applied to the first orderoptimality conditions of the centralized problem. As can beseen, even for two areas, a fairly complicated update resultsif this is to be done in a distributed way. Consequently, wepropose to simplify this update to

�x(p)

1 = J(p)−1

11 ·(−d(p)

1 + J(p)

12 J(p)−1

22 · d(p)

2

)

= J(p)−1

11 ·(−d(p)

1 + d(p)

12

)(32)

�x(p)

2 = J(p)−1

22 ·(−d(p)

2 + J(p)

21 J(p)−1

11 · d(p)

1

)

= J(p)−1

22 ·(−d(p)

2 + d(p)

21

)(33)

996 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

This update is significantly less computationally intense than(30) – (31). Area 1 can compute d(p)

21 without additionalknowledge from area 2 and then provide area 2 with the non-zero entries in this vector. The computation of d(p)

21 involvesthe inverse of J(p)

11 . However, that inverse is needed for theupdate of �x(p)

1 anyway. Consequently, area 1 can reuse theinverse for the computation of d(p)

21 . Generally, there shouldonly be very few terms in the vectors d(p)

1 and d(p)

2 whichare non zero, namely the ones which correspond to firstorder optimality constraints which include variables from bothsubproblems. Hence, only a limited amount of additional infor-mation needs to be exchanged among the subproblems (not theentire additional vector) to carry out (32) and (33).

We now generalize the update for the case with multipleareas. Hence, we propose the following update

�x(p)q = J(p)−1

q,q ·⎛

⎝−d(p)q +

M∑

m=1,m�=q

J(p)q,mJ(p)−1

m,m · d(p)m

⎞

⎠

= J(p)−1

q,q ·⎛

⎝−d(p)q +

M∑

m=1,m�=q

d(p)q,m

⎞

⎠ (34)

With the communication of these few extra terms, the updatescarried out locally for the areas is closer to the centralizedupdate. Hence, it can be expected that the number of itera-tions until convergence is reached is reduced compared to theoriginal method or the method with the modification based onthe Jacobi update.

V. SINGULARITY ISSUES

One major challenge encountered when including inter-temporal constraints such as the constraint on energy stor-age (5) in combination with the inequality constraints (6) – (8)on the variables of the storage into a multi-timestep optimiza-tion problem is that in some cases, the Jacobian matrix ofthe first order optimality conditions may become singular atthe optimal solution. This may cause the Jacobian to be ill-conditioned as it approaches the optimal solution, increasingthe number of iterations to optimality or in some cases caus-ing the optimization to diverge. In this section, we discusswhen such singularity occurs by analyzing the structure of theJacobian and a solution to the problem is presented.

A. Causes of Singularities

The formulation given in (1) – (11) will result in a singularJacobian when the gradients of storage constraints (5) – (8)are simultaneously binding. An inequality constraint is called“binding” or “active” at the optimal solution if its corre-sponding slack variable is zero (all equality constraints areconsidered active). This is due to the Linear IndependenceConstraint Qualification (LICQ), which states that at the opti-mal solution, the gradients of all binding constraints must belinearly independent or there exists no unique solution for theLagrange Multipliers [23].

The following analysis of the structure of the Jacobianmatrix is shown for the Jacobian of the first order opti-mality conditions without the Unlimited Point modification

because the issue is encountered independent of if UnlimitedPoint, Interior Point, or another method which uses theNewton-Raphson Jacobian is used. This is because the sin-gularity is due to the linear dependence of the gradients ofbinding constraints and has nothing to do with the way howinequality constraints are handled.

The Jacobian has the following structure:⎡

⎢⎢⎣

∇2xxL(x, z, λ, μ) ∇g(x)T ∇h(x)T 0

∇g(x) 0 0 0∇h(x) 0 0 I

0 0 diag{s} diag{μ}

⎤

⎥⎥⎦ (35)

The rows of the Jacobian which become singular when LICQis not satisfied are:

⎡

⎣∇g(x) 0 0 0∇h(x) 0 0 I

0 0 diag{s} diag{μ}

⎤

⎦. (36)

When an inequality constraint hn(x) is binding, sn = 0 andμn �= 0. If a set of binding constraints �g(x) and �h(x) arelinearly dependent, (36) will have dependent rows and thusthe entire Jacobian (35) will be singular.

B. Singularities From Storage Constraints

Although the aforementioned Jacobian singularity can occurwith many various sets of constraints, it may particularly occurin multi-step optimization problem formulations which includeinter-temporal constraints such as generator ramp limits andconstraints on storage devices [24]. With respect to storage,the singularity occurs when it is optimal for the energy storagesystem to keep its energy level at a minimum or maximumfor multiple timesteps. This can be seen by considering thefollowing variable vector:

x = [Ei(k) PIi(k) POi(k) Ei(k + 1)], (37)

If the optimal solution for these variables is x∗ = [Ei 0 0 Ei],i.e., the storage is empty for two consecutive timesteps (andthus there is no charging/discharging during these times), thematrix of the gradients of the binding constraints (5) – (8) isgiven by

⎡

⎢⎢⎢⎢⎣

0 · · · 0 −1 −TηcTηd

1 0 · · · 00 · · · 0 −1 0 0 0 0 · · · 00 · · · 0 0 0 0 − 1 0 · · · 00 · · · 0 0 −1 0 0 0 · · · 00 · · · 0 0 0 −1 0 0 · · · 0

⎤

⎥⎥⎥⎥⎦(38)

which, upon inspection, has linearly dependent rows. Hence,the LICQ is not fulfilled, and the Jacobian matrix is singular.Similar effects can be observed for ramping constraints but asdiscussed in [24], situations in which singularity is caused bythe ramping constraints are rather rare. The reader is thereforereferred to [24] for a more extensive discussion on singularitycaused by ramping constraints.

BAKER et al.: DISTRIBUTED MPC FOR EFFICIENT COORDINATION OF STORAGE AND RENEWABLE ENERGY SOURCES 997

C. Modification of Storage Model

Multiple solution techniques are presented in [24] to avoidthe singularity problem. The singularity of the matrix at theoptimal solution indicates that there are multiple solutions thatsatisfy the first order optimality conditions and result in thesame objective function value. Consequently, a Moore-PenrosePseudoinverse may be used to solve the underdetermined setof equations. While the pseudoinverse will be able to solve thesystem, it requires the extra step of decomposing the Jacobianmatrix using Singular Value Decomposition. The time requiredto perform this operation for every Newton-Raphson iterationcan be costly.

Another method that is proposed in [24] is to detect whenthe storage constraints are close to being binding, and then toremove these constraints from the set of constraints. However,this method runs the risk of prematurely removing constraintsand variables before they are actually at the optimal solution,resulting in the original first order optimality conditions notbeing satisfied. If the constraints are removed too late, theJacobian may already be too ill-conditioned for the iterationsto continue.

The method that is adopted here is to incorporate storagestandby losses into the model. A constant, small ε can beincluded to represent a loss in the current energy level overtime. For example, this could represent charge leakage froma battery or inertia losses from a flywheel. This does notonly prevent the Jacobian matrix to become singular but alsomore accurately models the behavior of a storage device. Thisvalue ε is subtracted from the energy balance equation at eachtimestep, hence modifying (5) to

Ei(k + 1) = Ei(k) + ηcTPIi(k) − T

ηdPOi(k) − ε. (39)

This loss prevents all of the storage constraints from beingsimultaneously binding. If the storage is at a minimum at timek, it must store energy (have a nonzero PIi(k)) to avoid dippingbelow the minimum value at time k + 1. If the storage is fullat time k, it cannot be full at k + 1 unless POi(k) is nonzero.Thus, the standby loss prevents all of the storage variablesfrom simultaneously being at their limits, and the Jacobiansingularity is avoided. This of course only makes sense if thelower limit Ei is not equal to zero because an empty storagecannot lose any energy. However, it is reasonable to set Ei �= 0to avoid very deep discharging of the storage.

VI. SIMULATION RESULTS

In this section, simulation results are provided as a proofof concept for the IEEE-57 and IEEE-118 bus system forvarying horizons. Results of the distributed MPC and compar-isons between the conventional and modified AND methodsare shown.

A. IEEE-57 Bus Simulation Setup

The IEEE-57 bus system is decomposed into two geograph-ical regions as shown in Figure 1. Two wind generators areplaced at buses 17 and 43, and a storage device with roundtrip

Fig. 1. IEEE-57 bus system decomposed into two areas.

Fig. 2. 24-Hour input data with 10-minute intervals.

efficiency of ηc · ηd = 0.95%, standby loss of 0.005p.u.·10-minutes and maximum capacity of 0.5p.u.·10-minutes isplaced at bus 7. Simulations were run for a 24-hour period withprediction horizons of N = 1, 3, 6 and 9 where T = 10min,hence, the horizon length correspond to no horizon, 30-minute, 60-minute, and 90-minute horizons. Historical datafor the wind and load was used from the Bonneville PowerAdministration and for the particular simulation presented herethe load and wind curve as given in Fig. 2 are used. Thereis a 27% level of wind energy penetration in the system byenergy. The cost functions and maximum output limits forthe generators were obtained from the IEEE-57 bus specifica-tions in MATPOWER [25]. Here, the storage is operated at the10-minute scale to balance out the intra-hourly fluctuations inthe power supply caused by variations in the wind and load.Storage could also be used with this method on an hourlyscale for longer term load shifting applications. For this testsystem, we neglect generator ramping and voltage constraintsbut later include them for the simulations in the IEEE-118 bustest system.

The results were compared with the solution of the central-ized problem achieved by the SNOPT solver in the commercialoptimization package TOMLAB and found to be within 1e−3

of the solution. The algorithm is considered to have converged

998 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

Fig. 3. Optimal power input (positive) and output (negative) from storage.

Fig. 4. Optimal state of charge of storage device.

once the maximum absolute value over all elements in the vec-tor of the first order optimality conditions d is less than 1e−4.It is important to note that in these simulations, because thisis a proof of concept simulation, the predictions for the windare assumed to be perfect; i.e., there is no forecast error, and alonger horizon always results in a lowered objective functionvalue. This may not necessarily be the case if prediction errorsare considered. With prediction errors the results depend onthe level of the prediction error.

B. Effect of Optimization Horizon

Figure 3 shows the power charged/discharged from the stor-age device over the 24-hour period, and Figure 4 shows thestate of charge of the storage over the 24-hour period. Thelonger the horizon, the better is the utilization of the energystorage as longer term variations in net load can be predictedand be accounted for. The effect is that less ramping is needfrom the generators as can be seen in Fig. 5, where the totalgeneration output from dispatchable generators is shown forthe horizon N = 9. It should be noted that as AC power flowis used, the overproduction in generation is mostly due to ACpower flow losses.

The total amount of generator ramping summed over allindividual generators was measured for each horizon N = 1(no MPC), 3, 6, and 9 and the total generation costs were cal-culated. As indicated by the results shown in Table I, with theuse of MPC, the overall required amount of generator rampingdecreases. Without the use of storage, generators must adjusttheir output more frequently to account for the fluctuationsin the power supply introduced by the wind. The reductionin overall generation cost for the considered day and com-pared for the different horizon lengths is quite low. However,that measure heavily depends on the particular load and wind

Fig. 5. Optimal generation levels for N = 9.

TABLE IREQUIRED GENERATOR RAMPING AND TOTAL GENERATION COST

curves for the considered day as well as the composition of thegenerators, i.e., which generators become the marginal gener-ators and which reach their capacity limit. Also note that thisdoes not say anything about the difference in cost if coordina-tion is used and if it is not. The comparison solely is focusedon the different lengths in horizon. As this paper focuses onthe method of how to coordinate the areas, a full economicanalysis is beyond the scope of this paper. However, it canbe expected that the greater the differences in cost parametersand the higher the fluctuations in net load are, i.e., the higherthe penetration of variable renewable generation, the higherthe benefit of longer horizons.

C. Comparison of Convergence Rates

1) Requirement for Convergence: In order for the ANDmethod to converge to the optimal solution x∗ of the problemdescribed in (1) – (9), the following must hold true at theoptimal solution [3]:

ρ(

I − Jdec · Jtot

)< 1 (40)

where ρ indicates the spectral radius. Matrix Jtot is theJacobian matrix (21) evaluated at the optimal solution and Jdec

is the Jacobian matrix with off-diagonal elements set to zeroagain at the optimal solution. If the condition on the spectralradius is not fulfilled, the optimization may be unable to con-verge to the optimal solution. In these cases, a preconditionedconjugate gradient method such as the generalized minimalresidual method (GMRES) [22] may be used to improve theconvergence. In the system decomposition used in this paper,the spectral radius was calculated to be around 0.88, fulfillingthe convergence criteria. This value was not found to changedramatically depending on the horizon length or point in thesimulation for the considered case.

2) Comparison of Distributed Methods: In Table II, theminimum, maximum, and median number of iterations to con-vergence for each method for horizons N = 1, 3, 6 and 9 isshown. Figure 6 shows the rate of convergence at simulation

BAKER et al.: DISTRIBUTED MPC FOR EFFICIENT COORDINATION OF STORAGE AND RENEWABLE ENERGY SOURCES 999

TABLE IINUMBER OF ITERATIONS TO CONVERGENCE

Fig. 6. Convergence rates for N=9.

timestep t = 2 for each of the three distributed optimizationmethods for N = 9. To ensure better convergence properties atthe cost of a higher number of iterations, the damping param-eter α on the Newton-Raphson step was initially chosen tobe 0.25. In the rare cases where the method still continued todiverge, a higher damping of 0.1 was chosen for the iterationswhich leads to a few outliers in terms of iteration numbers.To reduce the number of iterations, an adaptive approach forsetting the damping parameter could be used. For the sakeof comparison, the same damping factors have been usedthroughout the iteration process for a particular timestep.

As the results indicate, the Jacobi update method only leadsto significant improvements for N = 1, whereas the methodwith the additional term leads to significant reduction for hori-zons of N = 1 and N = 9 and stays roughly the same for theother horizons. The conclusion that can be drawn is that ananalysis should be done for the particular considered prob-lem to determine whether or not it is useful to add in theadditional term.

D. IEEE-118 Bus Test System

To demonstrate the effect of scaling these methods to alarger system, the IEEE-118 bus test case was decomposedinto two areas as shown in Figure 7. For this test case, we nowdo include generator ramping and bus voltage constraints. Thecost function and maximum output capacities for the genera-tors were taken from the IEEE-118 bus case in MATPOWER.The minimum and maximum voltage magnitude Vi and Vi

are set to 0.9p.u. and 1.1p.u., respectively. Generator ramp-ing limits �PGi are set to PGi . The wind and load data were

Fig. 7. IEEE-118 bus system decomposed into two areas.

Fig. 8. Optimal state of charge of storage device.

obtained from the Bonneville Power Administration, and thestorage parameters were chosen to be the same as in the 57-buscase. Wind generation is located in area 1 at bus 19, and thestorage is located in area 2 at bus 70.

Similarly to the IEEE-57 bus system, the storage is uti-lized more during simulations with longer horizons, as seenin Figure 8. This is again due to the fact that the longer thehorizon, the more the long-term variations in net load can beaccounted for.

However, in contrast to the 57-bus system, theNewton-Raphson damping parameter α was only steppeddown to 0.5 in these simulations, resulting in fewer iterationsoverall. As distributed algorithms usually perform better insystems where the coupling among areas is reduced, it can beexpected that larger systems generally benefit from a strongerinternal coupling and weaker inter-area coupling, i.e., thenumber of buses per area increases which overall shouldbenefit the performance of the algorithm, as seen in Table III.The original AND method failed to converge for timestep36 for N=3 and timestep 122 for N=6, and these timestepsare not included in the calculations. However, the other twoextensions of the AND method were able to converge for alltimesteps.

1000 IEEE TRANSACTIONS ON SMART GRID, VOL. 7, NO. 2, MARCH 2016

TABLE IIINUMBER OF ITERATIONS TO CONVERGENCE

Fig. 9. Convergence rates for N=9 at timestep 2.

The results from Table III indicate that in general, the Jacobimethod resulted in the least number of iterations for this sys-tem. And overall, the iterations from all methods indicatean improvement over the 57-bus system, perhaps due to thedecrease in coupling between subproblems. A comparison ofthe convergence rates for each of the three methods for N = 9at timestep 2 is shown in Figure 9.

E. Simulation With Four Areas

In order to compare convergence rates for a greater numberof subproblems, the 118-bus test system was decomposed intofour areas as seen in Figure 10. Simulations were performedfor the original AND method as well as the Additional Termmethod and shown in Table IV. The Additional Term methodshows a significant improvement over the original method inthe four-area case and performs more robustly in terms of themaximum number of iterations, when compared with the two-area results seen in Table III. In general, it can be seen thatthe number of iterations increases as the number of areas isincreased. This is due to the fact that the variable exchangerequired in the four-area case is much greater, requiring morevariables to iterate to optimality between areas rather thanwithin a single area.

Further case studies can be found in [26] for non-MPCpower flow simulations where AND and the Additional Termmethod are used including line constraints on a synthetic sys-tem with 12 118-bus systems. More specifically, if the tie linesget congested, the convergence speed will decrease with theabove two methods, which can be resolved by devising a newpartition of the system. However, if the inner lines withinareas are congested, the distributed methods will hardly beaffected.

Fig. 10. IEEE-118 bus system decomposed into four areas.

TABLE IVNUMBER OF ITERATIONS TO CONVERGENCE FOR FOUR AREAS

VII. CONCLUSION

In this paper, a distributed Model Predictive Control prob-lem was solved to coordinate resources across areas byonly exchanging the variables corresponding to the tie-linesbetween control areas. Two methods were developed thatextended the Approximate Newton Directions method for non-linear optimization decomposition; one method was basedon the Jacobi method, and the other method was deriveddirectly from AND and utilized a few additional terms inthe variable update. With all of these distributed methods,areas only have to exchange a limited amount of informationto achieve the same solution as the centralized optimizationproblem, maximizing the amount of social welfare in thesystem.

The convergence rates of each of the distributed approachesis shown in the simulation results, indicating that it dependson the horizon and most likely also the considered problemif the extensions of the AND method improve upon the rateof convergence of the original method. For the 57-bus system,a considerable improvement can be seen for the longer timehorizon and the second approach in which the off diagonalelements are approximated by additionally exchanged infor-mation. However, the results from the 118-bus system indicatethat the Jacobi-based method is the most beneficial.

The optimal generation values and state of charge of thestorage device is shown for various optimization horizons

BAKER et al.: DISTRIBUTED MPC FOR EFFICIENT COORDINATION OF STORAGE AND RENEWABLE ENERGY SOURCES 1001

in the MPC problem, showing a utilization of the storagefor the purpose of reducing the fluctuations in the powersupply introduced by an increase in renewable energy penetra-tion. Overall, the results look promising for predictive controlOptimal Power Flow problems across areas that do not fullycommunicate their system data, achieving the centralized solu-tion in a more efficient, distributed way, and allowing for amore effective utilization of energy resources across areas.

ACKNOWLEDGMENT

The authors are very grateful for the financial assistance thatmade this project possible.

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Kyri Baker (S’08–M’15) received the B.S., M.S.,and Ph.D. degrees in electrical and computerengineering from Carnegie Mellon University, in2009, 2010, and 2014, respectively. She is cur-rently a Post-Doctoral Researcher with the NationalRenewable Energy Laboratory, Golden, CO. Herresearch interests include power system optimiza-tion and planning, smart grid technologies, andrenewable energy integration.

Junyao Guo (S’13) received the B.S. degree inelectronic engineering from Tsinghua University, in2013. She is currently pursuing the Ph.D. degreewith Carnegie Mellon University. Her research inter-ests include distributed optimization and communi-cations in smart grids.

Gabriela Hug (S’05–M’08–SM’14) was born inBaden, Switzerland. She received the M.Sc. degreein electrical engineering and the Ph.D. degree fromthe Swiss Federal Institute of Technology, Zürich,Switzerland, in 2004 and 2008, respectively. Sheworked with the Special Studies Group of HydroOne in Toronto, Canada. Since 2009, she hasbeen an Assistant Professor with Carnegie MellonUniversity, Pittsburgh, USA. Her research is dedi-cated to control and optimization of electric powersystems.

Xin Li (S’01–M’06–SM’10) received the Ph.D.degree in electrical and computer engineering fromCarnegie Mellon University, in 2005. He is cur-rently an Associate Professor with the Electricaland Computer Engineering Department, CarnegieMellon. From 2009 to 2012, he was the AssistantDirector for FCRP Focus Research Center forCircuit and System Solutions. He is currently theAssistant Director for the Center for Silicon SystemImplementation. His research interests include inte-grated circuits and signal processing.