LNCS 4416 - Price-Based Optimal Control of Power Flow in

Price-Based Optimal Control of Power Flow inElectrical Energy Transmission Networks

A. Jokic, M. Lazar, and P.P.J. van den Bosch

Dept. of Electrical Eng., Eindhoven Univ. of Technology,P.O. Box 513, 5600 MB Eindhoven, The Netherlands

{a.jokic,m.lazar,p.p.j.v.d.bosch}@tue.nl

Abstract. This article presents a novel control scheme for achieving op-timal power balancing and congestion control in electrical energy trans-mission networks via nodal prices. We develop an explicit controller thatguarantees economically optimal steady-state operation while respectingall line flow constraints in steady-state. Due to these constraints, theresulting optimal control law has a piecewise affine structure. To opti-mize the dynamic response of the system and to satisfy line overloadconstraints during the transient period, the explicit optimal controller iscomplemented with a hybrid MPC controller. An example illustrates theeffectiveness of the proposed hybrid control scheme.

1 Introduction

During the past decade there has been a tremendous amount of research de-voted to a market-oriented approach for the electrical power system sector [1].Electrical power systems have some unique properties, which make this a chal-lenging task. For example, electrical energy cannot be efficiently stored in largeamounts, which implies that production has to meet rapidly changing demandsin real-time, making electricity a commodity with fast changing production costs.Furthermore, unlike other transportation systems, which assume a free choiceamong alternative paths between source and destination, the flow of power inelectrical energy transmission networks is governed by physical laws and, forsome fixed pattern of power injections, it can be influenced only to a certaindegree. Therefore, physical and security limits on the maximal power flow inthe lines of electrical energy transmission networks represent crucial system con-straints, which cannot be neglected [2]. Due to the fast changing variable produc-tion costs, there is a general tendency in power markets towards increasing thespeed with which the market price is updated. The usage of price as a real-timefeedback signal for power balance control has been investigated in [3], however,with no line congestion limits considered. Analysis of real-time market dynamicswas performed in [4], where the effects of congestion have been modeled by astatic equality constraint representing the power flow in a congested line. A de-tailed overview of problems and current schemes for price-based control of powersystems is presented in [2] and the references therein. The common characteristic

A. Bemporad, A. Bicchi, and G. Buttazzo (Eds.): HSCC 2007, LNCS 4416, pp. 315–328, 2007.c© Springer-Verlag Berlin Heidelberg 2007

316 A. Jokic, M. Lazar, and P.P.J. van den Bosch

of virtually all existing approaches is that the congestion management problemis treated in a static manner, while for real-time control of power flow in thelines the automatic generation control (AGC) scheme [5] is utilized.

One of the main contributions of this paper is the development of an explicit,price-based, dynamic controller for real-time optimal power balancing and con-gestion management. The proposed explicit controller guarantees that, followingany admissible change in the load, the power system will settle in the corre-sponding economically optimal steady-state point with all line flow constraintssatisfied. Due to the inequality constraints representing the line flow limits, theoptimal controller has a piecewise affine structure. Although steady-state opti-mal, the explicit controller does not guarantee that during the transients follow-ing load changes some of the power lines will not become overloaded to suchan extent that it will threaten the safety of the system’s operation. To solvethis issue, we complement the explicit optimal controller with a hybrid modelpredictive controller (MPC), i.e. a MPC controller that uses a piecewise affinemodel for predictions. The MPC control action amends the explicit optimal con-trol law so that constraints are met during the transients following load changes,and it converges to zero in steady-state. As a result, the response of the systemis optimized and the constraints are satisfied in the transient period as well,while the advantageous steady-state properties of the explicit optimal controllerare preserved. Consequently, the proposed hybrid control strategy achieves bothoptimal congestion management (in a dynamic manner) and price-based AGC.

The results of this paper add the optimal power flow problem with conges-tion constraints to a list of previously considered problems in electrical energytransmission systems, where the benefits of utilizing hybrid control schemes werealready illustrated. The interested reader is referred to [6], [7], where hybrid MPCwas utilized for efficient emergency voltage control, and to [8], where hybrid MPCwas used for optimal control of co-generation power plants. Recently, in [9], hy-brid MPC was employed for solving the problem of multi-period investments formaintenance and upgrade of electrical energy distribution networks.

1.1 Nomenclature

The field of real numbers is denoted by R, while Rm×n denotes m by n matrices

with elements in R. For a matrix A ∈ Rm×n, [A]ij denotes the element in the i-th

row and j-th column of A. For a vector x ∈ Rn, [x]i denotes the i-th element of x.

kerA and imA denote the kernel and the image space of A, respectively. We useIn and 1n to denote an identity matrix of dimension n × n and a column vectorwith n elements all being equal to 1, respectively. The operator col(·, . . . , ·) stacksits operands into a column vector, and diag(·, . . . , ·) denotes a square matrixwith its operands on the main diagonal and zeros elsewhere. All inequalities areinterpreted elementwise. We will use graph-theoretic terminology to representpower networks. With a slight abuse of notation we will often use the samesymbol to denote a signal, i.e. a function of time, as well as possible values thatthe signal may take at any time instant.

Price-Based Optimal Control of Power Flow 317

2 Steady-State Optimal Controller Design

Consider a connected undirected graph G = (V, E, A) as an abstraction of anelectrical power network. V = {v1, . . . , vn} is the set of nodes, E ⊆ V × V isthe set of undirected edges, and A is a weighted adjacency matrix. Undirectededges are denoted as eij = (vi, vj), and the adjacency matrix A ∈ R

n×n satisfies[A]ij �= 0 ⇔ eij ∈ E and [A]ij = 0 ⇔ eij �∈ E. No self-connecting edges areallowed, i.e. eii �∈ E. We associate the edges with the power lines of the electricalnetwork and, for convenience, we set the weights in the adjacency matrix asfollows: [A]ij = − 1

zij= −bij, where zij is the inductive reactance of a line, i.e.

the imaginary part of the line impedance, and bij is the line susceptance, see [2]for details. Note that the matrix A has zeros on its main diagonal and A = A�.The set of neighbors of a node vi is defined as Ni � {vj ∈ V | (vi, vj) ∈ E}.Often we will use the index i to refer the node vi. Define I(Ni) as the set ofindices corresponding to the neighbors of node i, i.e. I(Ni) � {j | vj ∈ Ni}. Weassociate the nodes with the buses in the electrical energy transmission network.

2.1 Steady-State Optimal Control Problem

To define the steady-state optimization problem, with each node vi we associatea singlet pi and a quadruplet (pi, pi

, pi, Ji) , where pi, pi, pi, pi ∈ R, p

i< pi and

Ji : R → R is a strictly convex, continuously differentiable function. The valuespi and pi denote the reference values for node power injections into the network.Positive values correspond to a flow of power into the network (production),while negative values denote power extracted from the network (consumption).Both pi and pi can take positive as well as negative values, and the only differenceis that, in contrast to pi, the value pi has an associated objective function Ji anda constraint p

i≤ pi ≤ pi. In the case of a positive pi, the function Ji represents

the variable costs of production, while for negative values of pi, it denotes thenegated benefit function of a consumer. We will refer to pi as the power froma price-elastic producer/consumer (or simply, power from a price-elastic unit),and to pi as the power from a price-inelastic producer/consumer (price-inelasticunit).

We use a “dc power flow” model [2] to determine the power flows in thenetwork for given values of node power injections. This model is often used and,under certain reasonable assumptions, it is proven to be a relatively accurateapproximation of a complex “ac power flow” model. In particular, convexityof the dc power flow model is a crucial property that we will exploit. With δi

denoting a voltage phase angle at the node vi, the power flow in a line eij ∈ E isgiven by pij = bij(δi−δj) = −pji. If pij > 0, power in the line eij flows from nodevi to node vj . The power balance in a node yields pi + pi =

∑j∈I(Ni) pij . With

the abbreviations p = col(p1, . . . , pn), p = col(p1, . . . , pn), δ = col(δ1, . . . , δn) theoverall network balance condition is p+ p = Bδ, where the matrix B is given byB = A − diag(A1n). We define the optimal power flow problem as follows.


Problem 1. Optimal Power Flow (OPF) problem.For any constant value of p,

minp,δ

J(p) � minp,δ

n∑

i=1

Ji(pi) (1a)

subject top − Bδ + p = 0, (1b)p ≤ p ≤ p, (1c)

bij(δi − δj) ≤ pij , ∀(i, j ∈ I(Ni)), (1d)

where p = col(p1, . . . , p

n), p = col(p1, . . . , pn), and pij = pji is the maximal

allowed power flow in the line eij . �

We will refer to a vector p that solves the OPF problem as a vector of optimalpower injections. For an appropriately defined matrix L and a suitably definedvector of power line limits pL, the set of constraints in (1d) can be written in amore compact form as follows:

Lδ ≤ pL. (2)

In a liberalized, market-oriented power system, different units are owned byseparate parties and each of them acts autonomously to maximize its own benefit.In other words, when a price-elastic unit at node i receives the current pricefor electrical power, i.e. λi, it adjusts its production level pi to be equal topi, where pi = argminpi

{Ji(pi) − λipi subject to p

i≤ pi ≤ pi

}. Since Ji is a

strictly convex, continuously differentiable function, this relation defines a uniquemapping from λi to pi for any λi ∈ R. For convenience, we denote this mappingwith Υi : λi → pi, i.e.

pi = Υi(λi) � arg minpi∈[p

i, pi]

Ji(pi) − λipi, (3)

and define Υ (λ) � col(Υ1(λ1), . . . , Υn(λn)). The operational goal in a liberalizedpower system is to determine the nodal price λi for each node i in the network, insuch a way that the total benefit of the system is maximized, while all constraintsare fulfilled. Formally, we define the optimal nodal price problem as follows.

Problem 2. Optimal Nodal Prices (ONP) problem.For any constant value of p,

minλ,δ

n∑

i=1

Ji(Υi(λi)) subject to Υ (λ) − Bδ + p = 0, Lδ ≤ pL, (4)

where λ = col(λ1, . . . , λn) is a vector of nodal prices. �

We will refer to a vector λ that solves the ONP problem with the term vector ofoptimal nodal prices. The OPF and ONP problems are related through Lagrange


duality, as it will be shown later in this section. The ONP problem is employednext to define the optimal steady-state control problem.

Consider a power network where each price-elastic unit is a dynamical system,and assign to each such unit an appropriate model Gi of its dynamics. We assumefor simplicity that each model Gi is an LTI system with respect to the inputpi = Υi(λi), which is specified by its state-space realization, i.e.

Gi :

{xi = Aixi + Bipi = Aixi + BiΥi(λi)pA

i = Cixi

, ∀i, (5)

where the power reference signal pi is the input, and the actual node powerinjection pA

i is the output. We denote the actual power injection of a price-inelastic unit with pA

i . Note that (1b) is always fulfilled when p and p are replacedwith pA = col(pA

1 , . . . , pAn ) and pA = col(pA

1 , . . . , pAn ), since in that case (1b)

represents the conservation law, i.e. pA − Bδ + pA = 0. Achieving balance inreference values (1b), i.e. balance of the desired production and consumption,is a control problem. The production/consumption of price-inelastic units is anexogenous signal to the system and pA = p, which yields:

pA − Bδ + p = 0. (6)

The desired production/consumption of price-elastic units is a function of currentnodal prices. Therefore, nodal prices can be effectively used as a feedback signalfor power balance control. Each system Gi receives a price signal and, based onits benefit maximization objective (3), it maps the signal into a reference pi. Wewill assume this mapping to be instantaneous, although the model can easily beextended with dynamics, time delays, threshold based rules, etc. The completedynamical model of the power system is described with the set of differentialalgebraic equations (5)-(6). For a detailed presentation of power system modelingfor real-time power balance control problems we refer to [5], Chapter 11, or [10],Chapter 12.

Note that the mapping Υi : λi → pi is linear only if Ji is a quadratic functionand p

i= −∞, pi = ∞. In practice, however, it will always be a nonlinear

mapping and therefore, the model (5)-(6) with the nodal prices λi as inputs, isnonlinear. Since for any strictly convex and continuously differentiable Ji themapping Υi : λi → pi can be arbitrarily well approximated with a continuouspiecewise affine mapping, one can always obtain a piecewise affine model thatapproximates (5) arbitrarily well. We will assume for the remainder of the articlethat each Υi is a piecewise affine function and thus, the overall system model (5)-(6) is piecewise affine with respect to λi as inputs.

To control the system, a measure of imbalance in (1b) has to be available.The network frequency serves that purpose. The system is in balance, in thesense of equality (1b), if the frequency is equal to its reference value, e.g. 50Hzin Europe. A change in any reference for power value causes the frequency tochange. In steady-state, the frequency is equal for all nodes in the network and,if it is above its reference value, the total production in a network exceeds thetotal consumption. Finally, we are able to define the control problem.


Problem 3. Optimal steady-state control problem.Design a feedback controller that has the network frequency and the line powerflows as input, and the nodal prices as output (see Figure 1), such that thefollowing objective is met: for any constant value of p such that the ONP problemis feasible, the state of the closed-loop system converges to a steady-state wherethe nodal prices are the optimal nodal prices as defined in Problem 2. �

2.2 Explicit Dynamic Optimal Controller

In this subsection we employ the relation between the solutions of the OPF andthe ONP problems to obtain a solution to Problem 3. We start by presentingthe following basic result from power system economics.

Proposition 1. The optimal dual variable (Lagrange multiplier) associated withthe power balance constraint (1b) in the Lagrange dual problem of OPF, is thevector of optimal nodal prices for the corresponding ONP problem.

Proof. Consider some constant value p such that the OPF and ONP problems arefeasible. The OPF problem is a convex problem which satisfies Slater’s constraintqualification. This implies that strong duality holds and that first-order Karush-Kuhn-Tucker (KKT) conditions are necessary and sufficient conditions for opti-mality. For the OPF problem, the Lagrangian is given by L(p, δ, ν+, ν−, λ, μ) =J(p) − λ�(p − Bδ + p) + (ν−)�(p − p) + (ν+)�(p − p) + μ�(Lδ − pL) and theKKT conditions are given by:

p − Bδ + p = 0, (7a)

p − p ≤ 0, ν+ ≥ 0, (ν+)�(p − p) = 0, (7b)

− p + p ≤ 0, ν− ≥ 0, (ν−)�(−p + p) = 0, (7c)

Lδ − pL ≤ 0, μ ≥ 0, μ�(Lδ − pL) = 0, (7d)

∇pJ(p) − λ + ν+ − ν− = 0, (7e)

Bλ + L�μ = 0, (7f)

where λ, ν+, ν− and μ are (vector) Lagrange multipliers. The multiplier λ isassociated with the equality constraint (7a) and is therefore not sign restricted.At the optimum, the value of the objective function and the vector of optimalpower injections in the OPF problem are equal to the value of the objectivefunction and the vector of power injections in the ONP problem, i.e. p = Υ (λ).Assume that the optimal Lagrange multiplier λ from the OPF dual problem istaken to be the vector of nodal prices. In this case the conditions (7b),(7c) and(7e) correspond to the KKT conditions for the constrained minimization problemin (3) for all price-elastic units in the network. Therefore, for this particular λ,the vector of power injections p = Υ (λ) in the ONP problem corresponds tothe vector of optimal power injections in the OPF problem, which concludes theproof. �


Remark 1. B is a singular matrix with rank deficiency one and with the kernelspace spanned by the vector 1n. Physically, this reflects the fact that only therelative voltage phase angles determine the power flow. Note also that 1n /∈ im B,since B = B� implies 1�

n B = 0, i.e. 1n is orthogonal to each column of B.Finally, analyzing the structure of L (see (1d) and (2)) one can easily observethat 1�

n L� = 0 and therefore, 1n /∈ im L�. These properties of B and L will beused later in this section to prove Proposition 2.

Consider the OPF problem solution for some constant value p such that theproblem is feasible. We denote the minimizers of OPF with p, δ, and with λthe value of the corresponding Lagrange multiplier. Strict convexity of each Ji

implies that at the optimum p is unique. On the other hand, due to singularity ofB, if δ is a minimizer so is δ+1nc where c ∈ R is an arbitrary constant. However,note that for all minimizers the set of active constraints is uniquely determined.Furthermore, we denote with μ and μ the Lagrange multipliers correspondingto inactive and active line power flow constraints, respectively. Analogously, wedefine ν+, ν+ and ν−, ν−. For inactive constraints col(μ, ν+, ν−) = 0. Theequality (7f) yields Bλ = −L�μ. This condition implies that λ ∈ kerB if nolines are congested. This further implies λ = 1nλ�, λ� ∈ R, i.e. at the optimum,there is one price in the network for all nodes. In case at least one line in thesystem is congested, it follows that the optimal nodal prices will in general bedifferent for each node in the system.

Next, we present the explicit dynamic controller that solves Problem 3. Wedefine fi = δi

2π as the network frequency [Hz] at node i, the abbreviation f �col(f1, . . . , fn), and we use fref (fref ∈ R) to denote the frequency reference value,e.g. fref = 50Hz. Let Kλ, Kf and Kp be diagonal matrices with positive elementson the diagonal, such that Kf = αKλ, α ∈ R, α > 0, and let Γ denote a diagonalmatrix of the same size as Kp. Consider the following explicit piecewise affinedynamic controller:

(xλ

xμ

)

=(

−KλB −KλL�

0 0

) (xλ

xμ

)

+(

−Kf 00 Γ

) (ΔfΔpL

)

, (8a)

λ =(In 0

)(

xλ

xμ

)

, (8b)⎧⎪⎨

⎪⎩

[Γ ]ii = [Kp]ii if [xμ]i ≥ 0 & [ΔpL]i ≥ 0[Γ ]ii = [Kp]ii if [xμ]i > 0 & [ΔpL]i < 0[Γ ]ii = 0 if [xμ]i = 0 & [ΔpL]i < 0,

(8c)

xμ(0) ≥ 0, (8d)

where xλ and xμ denote the controller states and the matrices Kλ, Kf and Kp

represent the controller gains. In (8), the inputs ΔpL = Lδ − pL and Δf = f −1nfref denote the line power overflow and the frequency deviation, respectively,while the output λ denotes the vector of nodal prices. Note that, due to theinitialization constraint (8d), it holds that xμ(t) ≥ 0 for all t ≥ 0.

Assumption 1 The closed-loop system resulting from the interconnection ofthe explicit dynamic controller (8) with the overall network model given by the


differential algebraic equations (5)-(6) is globally asymptotically stable for anyconstant value of p (i.e. with respect to the corresponding steady-state) suchthat the ONP problem is feasible. �

Proposition 2. Suppose that Assumption 1 holds. Then the explicit dynamiccontroller (8) solves the optimal steady-state control problem, as defined in Prob-lem 3.

Proof. To prove Proposition 2, it suffices to show that in steady-state, the vectorof nodal prices λ in (8) coincides with the Lagrange multiplier λ in (7), andtherefore, by Proposition 1, is a vector of optimal nodal prices. In steady-state,the frequency is equal for all nodes, i.e. Δf = 1nΔf�, Δf� ∈ R. Since in thepower system dynamics (5)-(6) there is no direct feedthrough from the input λ tothe outputs Δf and ΔpL, in steady-state the following conditions are satisfied:(i) for μ := xμ and from (8a) and (8b) it follows that Bλ + L�μ + 1nαΔf� = 0,which, together with 1n /∈ im

(B L�)

(see Remark 1), implies that Δf� = 0and Bλ+L�μ = 0, i.e. that the power balance constraint (7a) and the optimalitycondition (7f) are satisfied; (ii) from (8a) and (8c) it follows that (7d) is satisfiedfor μ := xμ; (iii) (7b), (7c), (7e) are satisfied since they correspond to the KKTconditions for the optimization problem in (3), which concludes the proof. �

Remark 2. The property 1n /∈ im(B L�)

, which ensures zero frequency devi-ation in steady-state (Δf� = 0), is not robust with respect to possible errors inthe controller implementation, i.e. to perturbations of matrices B and L in (8a).To overcome this drawback and to robustly ensure Δf� = 0 in steady-state, thecontroller (8) can easily be modified to explicitly include integral action on thenetwork frequency deviation. This modification is presented in Appendix A.

Remark 3. Due to the steady-state related complementarity conditions in (7d),there does not exist an LTI controller that solves Problem 3 and a hybrid con-troller, such as the piecewise affine controller (8), is a necessity. For fixed valuesof the gain matrices Kλ, Kf and Kp in (8), one can a posteriori check asymptoticstability of the resulting closed-loop system by searching for quadratic Lyapunovfunctions, and therefore, validate Assumption 1.

3 Hybrid MPC Control Scheme

Following a large disturbance acting on the system, for example, a relativelylarge, sudden change in p, it is desirable that the closed-loop system responseis such that the power lines overload ΔpL and the frequency deviations Δf arelimited during the transient period, i.e.

ΔpL ≤ ΔpmaxL , −Δfmax ≤ Δf ≤ Δfmax, (9)

at all times, where ΔpmaxL > 0, Δfmax > 0 are some predefined values. Since

the lines can be overloaded only for a short period of time, satisfying the limit


Fig. 1. Hybrid control scheme

constraints imposed on ΔpL and Δf , as well as fast convergence to the newsteady-state, are crucial.

To guarantee the fulfillment of the constraints (9) during the transient period,following a change in p, we design a hybrid MPC algorithm whose purposeis to complement the output λEX of the explicit optimal controller (8) suchthat inequalities (9) are satisfied at all times. The hybrid control scheme isdepicted in Figure 1. Let G denote the continuous-time model of the overallnetwork (5)-(6) in closed-loop with the explicit dynamic controller (8), as shownin Figure 1. To define the MPC algorithm, we first consider the following discrete-time approximation of G in closed-loop with the MPC controller:

GD :

{xk+1 = Ajxk + Bjλ

MPCk + aj(pk)

yk = Cjxk = col(ΔpLk, Δfk, −Δfk)if Hjxk ≤ hj , k ≥ 0, (10)

where Aj , Bj , aj(pk), Cj , Hj , hj are matrices and vectors of appropriate dimen-sions for all j ∈ S = {1, . . . , s}, with S a finite set of indices. The state vector xincorporates now both the states of the model describing the transmission net-work and the states of the explicit controller. As G is a piecewise affine system,an equivalent discrete-time piecewise affine counterpart cannot be obtained ingeneral. However, one can obtain a discrete-time piecewise affine approximationof G by discretizing each continuous-time affine sub-system of G and takinginto account physical insights when deriving the switching hyperplanes of thediscrete-time model, see, for example, [6], [7].

The outputs of system (10) are given by ΔpLk - the power flow deviation inthe lines of the transmission network and Δfk - the frequency deviation in thenodes of the transmission network (−Δfk is considered an output just to easilyspecify the constraints (9) in the MPC algorithm).

The MPC control action λMPCk acts additively on the output of the explicit

controller (8) present in the discrete-time dynamics GD such that the constraints(9) are fulfilled at all times k ≥ 0. In steady-state, λMPC

k converges to zero so


that the network is controlled only by the output of the explicit controller, whichthen guarantees optimality and constraint satisfaction.

Problem 4. MPC optimization problem. Let N ≥ 1 be given and let xk =:x0|k denote the measured state at time k ≥ 0. Let ck := (c0|k, . . . , cN |k) denote asequence of optimization variables and let Ψ be a positive definite and symmetricmatrix. Minimize the cost:

J(ck) �N−1∑

i=0

c�i|kΨci|k, (11)

subject to the constraints{

xi+1|k = Ajxi|k + Bjci|k + aj(pi|k)yi|k = Cjxi|k = col(Δpi|k, Δfi|k, −Δfi|k)

if Hjxi|k ≤ hj , ∀i = 0, . . . , N,

yN+1|k = CjxN+1|k = col(ΔpN+1|k, ΔfN+1|k, −ΔfN+1|k) if HjxN+1|k ≤ hj ,(12a)

yi|k ≤ col(ΔpLmax, Δfmax, Δfmax), ∀i = 1, . . . , N + 1, (12b)

cN |k = 0. (12c)

�

Let c∗k := (c∗0|k, . . . , c∗N−1|k, 0) denote an optimal sequence of variables obtainedby solving Problem 4. Then, the MPC control law is defined as follows:

λMPCk := c∗0|k; k ≥ 0. (13)

The value of pk, which is employed in (12a), can be estimated from the measuredvalues of the system state. With the current estimated value pk available, toobtain the future outputs in (12a), we assume pi|k = pk for all i = 0, . . . , N .

Assumption 2 For any constant value of pk, the prediction horizon N is longenough such that the predicted state xN |k lies in a subset of the state-space thatis invariant for the closed-loop system (10) with λMPC

k = 0 for all k ≥ 0, andwhere the constraints (12b) are satisfied. Furthermore, the global optimum isattained in Problem 4 for any xk and all k ≥ 0. �

Proposition 3. (i) Suppose that Assumption 2 holds. If Problem 4 is feasibleat time k ≥ 0 for the measured state xk = x0|k, then Problem 4 remains feasibleat time k + 1 for state xk+1 = Ajxk + Bjλ

MPCk + aj(pk) if Hjxk ≤ hj.

(ii) Suppose that Assumption 1 holds for GD with λMPCk = 0 for all k ≥ 0 and

for any constant value of pk such that the ONP problem is feasible. Furthermore,suppose that Assumption 2 holds. Then the discrete-time closed-loop system (10)with the input λMPC

k defined as in (13) is asymptotically stable for any constantvalue of pk such that the ONP problem is feasible.

Fulfilment of the first part of Assumption 2 can be a priori guaranteed by addinga terminal1 equality or inequality constraint to Problem 4, see, for example, [11].1 By this we mean a constraint on the predicted state xN|k.


4 Illustrative Example

The proposed hybrid control scheme was implemented for a three nodes triangu-lar network with a synchronous generator at each node. Each generator is mod-eled by a third order model, which is standardly used in AGC studies [5], [10].The generator model is taken from [10], pp. 543-545. We have used the follow-ing parameter values: τg ∈ {0.2, 0.25, 0.2}, τT ∈ {0.5, 0.45, 0.5}, H ∈ {5, 5, 5},D ∈ {0.7, 0.7, 0.8}, R ∈ { 1

22 , 124 , 1

20}, where the first element in each set denotesthe parameter value for a generator at the first node, etc., and the symbols are theones from [10]. We use quadratic functions to represent the variable productioncosts Ji, with quadratic terms {1.2, 1.24, 1.4} and linear terms {35.9, 36.1, 36}, re-spectively. For simplicity, no saturation limits were considered for the generators,i.e. p

i= −∞, pi = ∞, ∀i. Furthermore, we use the following reactance values for

each line: z12 = 0.1, z13 = 0.09, z23 = 0.13. Appropriate, although not optimal,values for the gains of the explicit controller were chosen as Kλ = diag(2, 2, 2),Kf = diag(7, 7, 7). For the line connecting nodes 1 and 2, the steady-state powerflow limit was set equal to 1.2, i.e. p12 = 1.2, and the maximal allowed violationof this constraint in a transient period was set to 0.1, i.e. the corresponding ele-ment in Δpmax

L from (9) is ΔpmaxL12

= 0.1. For simplicity, no line flow limits wereconsidered in the remaining lines, and the explicit controller is implemented toact only on Δp12 with gain Kp = 2. No frequency deviation constraints wereimposed during transients.

The resulting closed-loop system G is a continuous-time piecewise affine modelwith 15 states and 3 affine sub-systems. The (robust) asymptotic stability ofG for any values of p in the interval indicated below was established via aquadratic Lyapunov function, which validates Assumption 1. We obtained adiscrete-time model GD by discretizing each affine sub-system of G with a sam-pling period of 0.1s and keeping the same switching hyperplanes as the ones ofthe continuous-time model. The simulations showed that GD is a good (i.e.for control purposes) approximation. The asymptotic stability test was suc-cessfully carried out for the discrete-time model GD with λMPC

k = 0 for allk ≥ 0 as well, which validates Assumption 1 for GD. For the MPC controller,the following tuning parameters were used: N = 12, Ψ = diag(0.1, 0.1, 0.1).For simplicity, we have used the following estimate for the value of aj(pk):aj(pk) = aj(pk−1) = xk − (Ajxk−1 + Bjλ

MPCk−1 ) if Hjxk−1 ≤ hj. A price-inelastic

load was connected to each node. In the simulations, in nodes 1 and 3, the corre-sponding loads were set equal to constant values: p1(t) = 0, p3(t) = 2. The loadat node 2 was taken to be a time varying signal given by p2(t) = 3 for t < 5s andp2(t) = 4 for t ≥ 5s. The system’s response to the change in the load is pre-sented in Figures 2 - 5. At time instant t = 0 the system is in a steady-state. InFigure 2, solid lines represent simulated trajectories of power flow in line e12. Onetrajectory (labeled “Explicit”) corresponds to the system in closed-loop with thethe explicit optimal controller alone, while the other trajectory (labeled “Explicit+ MPC”) corresponds to the MPC closed-loop system described in Figure 1. InFigure 2, dashed lines represent the steady-state line flow limit p12 = 1.2 andthe transient line flow limit p12 + Δpmax

L12= 1.3, respectively. Figure 3 presents


0 5 10 15 20 251.05

1.1

1.15

1.2

1.25

1.3

1.35

1.4

1.45

1.5

1.55

time [s]

p 12

Explicit

Explicit+MPC

Fig. 2. Line power flow p12. Dashed lines represent power flow restriction: 1.2 in steady-state; 1.3 in transient.

0 5 10 15 20 25

36.4

36.6

36.8

37

37.2

37.4

37.6

time [s]

λ λ1

λ3

λ2

λ1=λ

2=λ

3

Fig. 3. Nodal prices λ as sum of λEX and λMPC. Dashed lines represent steady-stateoptimal nodal prices.

the values of the nodal prices λ as a function of time (solid lines). These trajec-tories correspond to the summation of the outputs from the explicit controllerand the MPC controller. In the same figure, dashed lines represent the off-linecalculated values for the steady-state optimal nodal prices, which depend on thevalue of p2(t). The MPC control actions are presented in Figure 4, and the sim-ulated trajectories of the node frequency deviations are presented in Figure 5.Note that, before the change in the load p2, the line is not congested and allnodal prices are equal for the steady-state optimal operating point. After thestep increase in the load, in the optimal steady-state operating point the line iscongested and, as a consequence, the optimal nodal prices have different values.The simulation results clearly illustrate the efficiency of the proposed hybridcontrol scheme for both steady-state and transient behavior of the closed-loopsystem.


0 5 10 15 20 25

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

time [s]

λMPC

λ2MPC

λ1MPC λ

3MPC

Fig. 4. The MPC control actions

0 5 10 15 20 25−0.04

−0.03

−0.02

−0.01

0

0.01

time [s]

Δ f / f

ref

Δ f3

Δ f1

Δ f2

Fig. 5. Relative frequency deviations Δffref

5 Conclusion

This paper presented a novel hybrid control scheme for achieving optimal powerbalancing and congestion control in electrical energy transmission networks. Wedeveloped an explicit dynamic controller that guarantees economically optimalsteady-state operation while respecting all line flow constraints in steady-state.Due to these constraints, the resulting optimal control law has a piecewise affinestructure. To improve the response of the system and to effectively and efficientlyhandle constraints during the transient period, the explicit optimal controllerwas complemented with a hybrid MPC controller. An example illustrated theproposed hybrid control scheme.

A scalable approach for stability analysis and distributed implementation ofthe proposed hybrid control scheme will be considered in future work.


Acknowledgments. This work is part of the research program “Intelligent PowerSystems” that is supported financially by SenterNovem. SenterNovem is anagency of the Dutch Ministry of Economic Affairs.

References

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2. Christie, R.D., Wollenberg, B.F., Wangensteen, I.: Transmission management inthe deregulated enviroment. Proceedings of the IEEE 88(2) (2000) 170–195

3. Alvarado, F.L., Meng, J., DeMarco, C.L., Mota, W.S.: Stability analysis of inter-connected power systems coupled with market dynamics. IEEE Transactions onPower Systems 16(4) (2001) 695–701

4. Alvarado, F.: The stability of power system markets. IEEE Transactions on PowerSystems 14(2) (1999) 505–511

5. Kundur, P.: Power System Stability and Control. McGraw-Hill (1994)6. Geyer, T., Larsson, M., Morari, M.: Hybrid emergency voltage control in power

systems. In: European Control Conference, Cambridge, UK (2003)7. Beccuti, A.G., Geyer, T., Morari, M.: A hybrid system approach to power systems

voltage control. In: IEEE Conference on Decision and Control, Seville, Spain (2005)8. Ferrari-Trecate, G., Gallestey, E., Letizia, P., Spedicato, M., Morari, M., Antoine,

M.: Modeling and control of co-generation power plants: A hybrid system approach.IEEE Transactions on Control Systems Technology 12(5) (2004) 694–705

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PhD thesis, Eindhoven University of Technology, The Netherlands (2006)

A Appendix

This appendix concerns Remark 2 in Section 2. The controller modification in-cludes replacing (8a) and (8b) with (14a) and (14b), respectively, where:

⎛

⎝xλ0

xΔλ

xμ

⎞

⎠ =

⎛

⎝0 0 00 −KΔBΔ −KΔL�

Δ

0 0 0

⎞

⎠

⎛

⎝xλ0

xΔλ

xμ

⎞

⎠ +

⎛

⎝−kf1�

n 00 00 Γ

⎞

⎠(

ΔfΔpL

)

, (14a)

λ =((

11n−1

) (0

In−1

) (00

))

col(xλ0 , xΔλ, xμ). (14b)

The diagonal and positive definite matrix KΔ (KΔ ∈ R(n−1)×(n−1)) and the

positive scalar kf represent controller gains. BΔ is a submatrix of B obtained byremoving the first column and the first row of B, and LΔ is a submatrix of L,obtained by removing the first column of L. The dynamics of the scalar valuedcontroller state xλ0 acts as an integral control action on frequency deviationsΔf , while the dynamics of the controller state xΔλ acts as an integral controlaction on violations of the optimality condition (7f).

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