Modern Optimization Models and Techniques for … Optimization Models and Techniques for Electric Power Systems Operation Andy Sun and Dzung T. Phan Abstract This article introduces

Modern Optimization Models and Techniques for

Electric Power Systems Operation

Andy Sun and Dzung T. Phan

Abstract This article introduces modern optimization models and solution methods for two fun-

damental decision making problems in electric power system operations, the optimal power flow

(OPF) problem and the unit commitment (UC) problem. The article surveys some of the most

recent advances, including global optimization techniques for exact solution of the OPF prob-

lem, adaptive robust optimization models for the UC problem under uncertainty of renewable

generation and demand-side response, contingency analysis for large-scale power systems, and

distributed optimization schemes for decentralized operation.

Key words: optimal power flow, unit commitment, security-constrained, contingency, Lagrangian

duality, semidefinite programming, branch and bound, adaptive robust optimization, Benders de-

composition, distributed algorithm

1 Introduction

It is not an exaggeration to say that modern electric power systems are built upon optimiza-

tion models and techniques. From long-term generation and transmission capacity planning to

Andy Sun

H. Milton Stewart School of Industrial and Systems Engineering, Georgia Institute of Technology, Atlanta, GA

30332, e-mail: [email protected]

Dzung Phan

Business Analytics and Mathematical Sciences Department, IBM T.J. Watson Research Center, Yorktown

Heights, NY 10598, e-mail: [email protected]

1

2 Andy Sun and Dzung Phan

medium-term maintenance scheduling and short-term daily and hourly operation, optimization

models and techniques are essential tools for decision making in power system operations.

In this article, we focus on two fundamental problems in the short-term operation of large-

scale electric power systems, namely, the day-ahead unit commitment (UC) problem and the

real-time economic dispatch problem based on optimal power flow (OPF). In the day-ahead UC

problem, the system operator needs to decide the commitment status of available generation units

to meet forecast electricity demand in the next day. Due to the discrete nature of the commitment

decisions, the UC problem is usually modeled as a mixed-integer optimization problem. After

the commitment decision is made, the system operator solves the real-time economic dispatch

problem, where the operator controls the dispatching of committed generators to meet varying

demands in hourly or smaller time intervals.

These two problems present unique challenges arising from the large spatial scale of modern

power systems, short solution time required by real-time or near real-time operations, and com-

plicated, nonconvex, discrete constraints. Uncertainty is another challenge emerging with the

increasing penetration of renewable energy resources, especially wind and solar power resources

which are intermittent in nature and difficult to forecast. We introduce fundamental models and

discuss some of the most recent advances in meeting these challenges.

The rest of the paper is organized as follows. Section 2 introduces the basic power flow equa-

tions and the optimal power flow models. Section 3 introduces global optimization methods to

exactly solve the OPF problems. Two lower-bounding techniques, the Lagrangian relaxation and

the semidefinite relaxation methods, are discussed in detail. Section 4 reviews the important

problem of security-constrained OPF problem and solution methods. Section 5 motivates decen-

tralized optimization schemes for solving large-scale OPF problems and reviews the literature.

The unit commitment and its optimization formulation are introduced in Section 6. The recent

development in the adaptive robust UC model is reviewed in Section 7.

2 Optimal Power Flow Problem

The system operator controls the dispatching of the committed generation units to meet varying

demand in hourly or smaller time intervals. The physics, described by the Kirchoff voltage and

current laws, dictates the relationship between nodal voltages and power flows on the transmis-

sion lines. In particular, this relationship can be modeled by a set of quadratic equations. The

Electric Power System Operations 3

associated OPF model is called the Alternating Current (AC) model. Due to nonlinearity and

especially nonconvexity of the AC OPF model, solving large-scale AC OPF problems is compu-

tationally challenging. A linearized model, usually referred to as the Direct Current (DC) model,

is widely used in practice. In the following, we introduce both models.

2.1 AC OPF models

Let N denote the set of buses (i.e. nodes) in the power system. Let D denote the subset of

demand buses, and let L denote the set of branches (e.g., transmission lines) in the power grid.

In an AC OPF model, the nodal voltage Vi can be modeled as a complex number in rectangular

coordinates

Vi = ei + j fi, ∀i ∈N ,

where j is the imaginary unit. The net active power Pi and reactive power Qi injections into bus

i ∈N are given by

Pi(e, f) = ∑j∈N

(ei(Gi je j−Bi j f j)+ fi(Gi j f j +Bi je j)

),

Qi(e, f) = ∑j∈N

(fi(Gi je j−Bi j f j)− ei(Gi j f j +Bi je j)

),

where e ∈ R|N | and f ∈ R|N | are vectors with voltage components, Gi j is the i j-th entry of the

bus conductance matrix G ∈ R|N |×|N |, and Bi j is the i j-th entry of the bus susceptance matrix

B ∈ R|N |×|N |. The active power flow Pi j from bus i to bus j is computed as

Pi j(e, f) = (e2i + f 2

i )Gii +(eie j + fi f j)Gi j− (ei f j− e j fi)Bi j,

where Gii is the self-conductance of branch admittance at bus i, Gi j is the mutual conductance,

and Bi j is the mutual susceptance of branch admittance from buses i to j.

The AC optimal power flow model that minimizes the total production cost can be written as

[15, 19, 38]:


mine,f,pg,qg

c(pg) = ∑i∈G

(ci0(pg

i )2 + ci1 pg

i + ci2)

(1a)

s.t. ∑j∈N

(ei(Gi je j−Bi j f j)+ fi(Gi j f j +Bi je j)

)= pg

i − pdi ,∀i ∈N (1b)

∑j∈N

(fi(Gi je j−Bi j f j)− ei(Gi j f j +Bi je j)

)= qg

i −qdi ,∀i ∈N (1c)

(e2i + f 2

i )Gii +(eie j + fi f j)Gi j− (ei f j− e j fi)Bi j ≤ pmaxi j ,∀(i, j) ∈L (1d)

pmini ≤ pg

i ≤ pmaxi ,∀i ∈ G (1e)

qmini ≤ qg

i ≤ qmaxi ,∀i ∈ G (1f)

(vmini )2 ≤ e2

i + f 2i ≤ (vmax

i )2,∀i ∈N , (1g)

where pgi and qg

i are the active and reactive powers of generation units at bus i, pdi and qd

i are

active and reactive powers of demands at bus i, pmini ,qmin

i ,vmini and pmax

i ,qmaxi ,vmax

i are lower and

upper bounds on generation output and nodal voltage limits, respectively, pmaxi j is the branch flow

limit, and the cost of production is modeled as a quadratic function of active power outputs. For

different operational and planning requirements, other objective functions such as transmission

losses and reactive power output, as well as other functions, can be considered.

The nodal complex voltage can also be described in the polar representation as Vi = |Vi|exp(jθi),

where |Vi| and θi are the magnitude and angle of the voltage Vi, respectively. Using this represen-

tation, the power flow equations (1b) and (1c) can be replaced by the following equations

∑j∈N

(|Vi||Vj|(Gi j cos(θi−θ j)+Bi j sin(θi−θ j))) = pgi − pd

i , i ∈N (2a)

∑j∈N

(|Vi||Vj|(Gi j sin(θi−θ j)−Bi j cos(θi−θ j))) = qgi −qd

i , i ∈N . (2b)

2.2 DC OPF model

The AC OPF model involves quadratic or trigonometric constraints, which makes the problem

nonconvex. As an approximation to the AC model, the linearized DC model has been widely

used in practice. The DC model is obtained from the polar representation of power flows (2a)

and (2b) as follows. Firstly, since the line resistance is usually very small, we can assume Gi j =

0. Secondly, under normal operation conditions, the voltage angle difference between adjacent

nodes is small, i.e. (θi− θ j) ≈ 0, which makes sin(θi− θ j) ≈ (θi− θ j) and cos(θi− θ j) ≈ 1.


Thirdly, using a per-unit system [38], the magnitudes of nodal voltages are close to the unit 1.0,

so are their products. Combining the three approximations, the DC OPF model can be written as:

minpg,θ

∑i∈G

(ci2(pg

i )2 + ci1 pg

i + ci0)

s.t. ∑j∈N

Bi j (θi−θ j) = pgi − pd

i , ∀i ∈N

Bi j (θi−θ j)≤ pmaxi j , ∀(i, j) ∈L

pmini ≤ pg

i ≤ pmaxi , i ∈ G

Note that due to approximation, the DC model does not contain reactive power and voltage

magnitude.

3 Global Optimization Methods for AC OPF

In this section, we introduce exact methods based on branch and bound algorithms to solve the

optimization problem (1) in rectangular coordinates to global optimality. First, we discuss two

methods to obtain lower bounds on the global optimal solution, namely the Lagrangian relaxation

[35] and the semidefinite programming (SDP) relaxation [30].

3.1 Lagrangian relaxation

The Lagrangian dual problem of the primal AC OPF (1) is formed by dualizing all the compli-

cating quadratic constraints while retaining simple bounds such as box and sphere constraints in

the feasible set. The Lagrangian function becomes

L(e, f,pg,qg,λλλ ,βββ ,µµµ,γγγ,ηηη) = c(pg)+ ∑i∈N

(λi

(∑

j∈N(ei(Gi je j−Bi j f j)+ fi(Gi j f j +Bi je j))+

pdi

)+βi

(∑

j∈N( fi(Gi je j−Bi j f j)− ei(Gi j f j +Bi je j))+qd

i

))+ ∑

(i, j)∈Lµi j

((e2

i + f 2i )Gii

+(eie j + fi f j)Gi j− (ei f j− e j fi)Bi j

)+ ∑

i∈N(ηi− γi)(e2

i + f 2i )−∑

i∈G(λi p

gi +βiq

gi ).

Define the Lagrangian dual function g(λλλ ,βββ ,µµµ,γγγ,ηηη) as the optimal value of


mine,f,pg,pg L(e, f,pg,pg,λλλ ,βββ ,µµµ,γγγ,ηηη)−∑(i, j)∈L µi j pmaxi j +∑i∈N

(γi(vmin

i )2−ηi(vmaxi )2

)s.t. (1e),(1 f ),and ∑i∈N (vmin

i )2 ≤ ∑i∈N (e2i + f 2

i )≤ ∑i∈N (vmaxi )2.

(3)

It is known that for any value of the Lagrange multipliers (λλλ ,βββ ,µµµ,γγγ,ηηη) : µµµ ≥ 0,γγγ ≥ 0,ηηη ≥ 0,

the value of the dual objective function, g(λλλ ,βββ ,µµµ,γγγ,ηηη), provides a lower bound on the global

optimal objective function value of (1) (see, for example, [5, Theo. 6.2.1]). To obtain the best

lower bound for all possible multipliers, we need to solve the dual problem:

maxλλλ ,βββ ,µµµ,γγγ,ηηη g(λλλ ,βββ ,µµµ,γγγ,ηηη) s.t. µµµ ≥ 0,γγγ ≥ 0,ηηη ≥ 0. (4)

This is a concave maximization problem, so the optimal value can be obtained by convex

optimization methods. In general, the practical use of Lagrangian duality greatly depends on

how efficiently the inner nonconvex minimization, (3) in our case, can be solved. Fortunately,

problem (3) is tractable as shown below. The minimization problem (3) can be decomposed into

two subproblems

min pg,qg c(pg)−∑i∈G

(λi pgi +βiq

gi ) s.t. pmin

i ≤ pgi ≤ pmax

i ,qmini ≤ qg

i ≤ qmaxi , ∀i ∈ G (5)

and

min e,f xTA(λλλ ,βββ ,µµµ,γγγ,ηηη)x s.t. ∑i∈N

(vmini )2 ≤ xTx≤ ∑

i∈N(vmax

i )2. (6)

where x = [eT fT]T, and the matrix A(λλλ ,βββ ,µµµ,γγγ,ηηη) is given as

A(λλλ ,βββ ,µµµ,γγγ,ηηη) =

λ1G1:−β1B1:...λ|N |G|N |:−β|N |B|N |:

−β1G1:−λ1B1:...−β|N |G|N |:−λ|N |B|N |:

β1G1:+λ1B1:

...β|N |G|N |:+λ|N |B|N |:

λ1G1:−β1B1:...λ|N |G|N |:−β|N |B|N |:

+

µµµ G, −µµµ B

µµµ B, µµµ G

+diag([ηηη− γγγ;ηηη− γγγ]),

where Gi: and Bi: are the i-th rows of matrices G and B, respectively; diag(x) is a diagonal matrix

with x on the diagonal, and “” denotes componentwise product.

It is easy to recognize that (5) is a convex separable quadratic optimization problem, whereas

(6) is a nonconvex quadratic problem with two sphere constraints. The global optimal solution

of (6) has the following closed form


x =

v√

∑i∈N (vmaxi )2

‖v‖ , if λmin < 0

v√

∑i∈N (vmini )2

‖v‖ , otherwise,

where v is the eigenvector associated with the smallest eigenvalue λmin of the matrix A+AT.

The dual problem (4) is a convex nondifferentiable optimization problem, which can be solved

by a subgradient projection method because, for a given (λλλ ,βββ ,µµµ,γγγ,ηηη), one of the subgradients

for the objective function g is easy to evaluate. The problem of finding the optimal solution to

(6) amounts to computing the eigenvector corresponding to the smallest eigenvalue of A+AT,

which can be achieved by several efficient algorithms such as the power iteration method [20] and

the method based on inverse-free preconditioned Krylov subspace projection [21]. Furthermore,

the conductance G and susceptance B matrices are often sparse with graph density less than 0.1,

thus A+AT is also sparse. Therefore, sparse techniques can be applied which further reduces the

computation burden. In summary, the Lagrangian relaxation of the nonconvex AC OPF problem

can be solved efficiently, which gives a lower bound on the global optimal solution.

3.2 Semidefinite programming relaxation

In this subsection, we discuss another relaxation technique to solve the AC OPF problem, namely

the semidefinite programming relaxation, first proposed in [29, 30].

The nodal power injections, power flows, and current flows on branches can be fully charac-

terized by the nodal voltages Vi and the admittance matrix Y = G+ jB. Therefore, the AC OPF

problem can be reformulated in the voltage variables Vi’s as follows.

Let ei denote the i-th unit vector in R|N |, and let Mi denote a square matrix of size 2|N | with

a 1 in the i-th and (i+ |N |)-th diagonal entries and 0’s elsewhere. Define

Yi =12

ReYi +YTi ImYT

i −Yi

ImYi−YTi ReYi +YT

i

, Yi =−12

ImYi +YTi ReYi−YT

i

ReYTi −Yi ImYi +YT

i

, Yi = eieTi Y,

Yi j =12

ReYi j +YTi j ImYT

i j −Yi j

ImYi j−YTi j ReYi j +YT

i j

, Yi j = (Yi j + Yi j)eieTi − Yi jeie

Tj ,

and Yi j denotes the shunt admittance at bus i associated with the line (i, j), and Yi j = Gi j + jBi j.

Following [30], the net nodal active and reactive powers Pi(e, f) and Qi(e, f) are given as


Pi(e, f) = ReVi I∗i = ReV∗YiV

= xT

ReYi −ImYi

ImYi ReYi

x

=12

xT

ReYi +YTi ImYT

i −Yi

ImYi−YTi ReYi +YT

i

x

= xTYix = trace

YixxT.

Note that x = [ReVT ImVT]T. Similarly, we have

Qi(e, f) = ImVi I∗i = trace

YixxT,

e2i + f 2

i = xTMix = trace

MixxT,

Pi j(e, f) = trace

Yi jxxT.

It is well known that a matrix W can be represented as xxT if and only if W is positive

semidefinite, i.e., W 0, and has rank 1. Dropping the rank condition on W yields the following

relaxation of the AC OPF problem:

minW ∑

i∈G

(ci0(traceYiW+ pd

i )2 + ci1(traceYiW+ pd

i )+ ci2

)(7a)

s.t. pmini − pd

i ≤ traceYiW ≤ pmaxi − pd

i ,∀i ∈N (7b)

qmini −qd

i ≤ trace

YiW≤ qmax

i −qdi ,∀i ∈N (7c)

trace

Yi jW≤ pmax

i j ,∀(i, j) ∈L (7d)

(vmini )2 ≤ traceMiW ≤ (vmax

i )2,∀i ∈N (7e)

W 0. (7f)

By Schur’s complement condition, the constraint

ci0(traceYiW+ pdi )

2 + ci1(traceYiW+ pdi )+ ci2 ≤ αi

for a given αi is equivalent toci1(traceYiW+ pdi )+ ci2−αi

√ci0(traceYiW+ pd

i )

√ci0(traceYiW+ pd

i ) −1

0,∀i ∈ G . (8)

With this equivalence, (7) is equivalent to the following semidefinite program


minW,α

∑i∈G

αi (9a)

s.t. (7b),(7c),(7d),(7e),(7 f ),and (8). (9b)

Problem (9), as a semidefinite program, can be solved to optimality in polynomial time. If

the matrix W has rank 1, then (9) is equivalent to the original AC OPF. In particular, an optimal

solution for OPF can be recovered from eigen-decomposition of the matrix W . It has been shown

that if (9) has a rank-one solution, then it must has an infinite number of rank-two solutions [30].

As a result, it is numerically difficult to verify the existence of a rank-one solution. It leads to

consider the dual of (9).

We introduce dual variables λ i,γ i,µ

ifor the lower bound inequalities and λi, γi, µi for upper

bound inequalities in (7b), (7c), (7e), variable λi j for constraint (7d), and matrix variable[ 1 ri1

ri1 ri2

]for constraint (8). The Lagrangian dual problem for (9) is

min ∑i∈N

λ i p

mini − λi pmax

i + λi pdi + γ

iqmin

i − γiqmaxi + γiqd

i +µi(vmin

i )2− µi(vmaxi )2

+ ∑i∈G

(ci2− ri2)− ∑(i, j)∈B

λi j pmaxi j (10a)

s.t. ∑i∈N

λiYi + γiYi + µiMi

+ ∑

(i, j)∈Bλi jYi j 0 (10b)

1 ri1

ri1 ri2

0, ∀i ∈ G , (10c)

where aggregate dual variables are defined as follows:

λidef=

−λ i + λi + ci1 +2√

ci0ri1 if i ∈ G

−λ i + λi otherwise,

γidef=−λ i + λi, µi

def=−µ

i+ µi.

Problem (10) has several advantages over problem (9). The number of decision variables is es-

sentially linear with respect to the number of buses |N |, as apposed to the quadratic growth for

(9). In addition, problem (10) solves the OPF problem if matrix ∑i∈N

λiYi + γiYi + µiMi+

∑(i, j)∈L λi jYi j has a zero eigenvalue of multiplicity at most 2. This sufficient condition is often

found to hold in practice [30].


3.3 Global optimization algorithms

Note that zero duality gap property is satisfied for a wide range of power grid test instances

including all IEEE benchmark systems with 14, 30, 57, 118, and 300 buses. However, there are

also instances that these above relaxations lead to strictly positive duality gaps. In these cases,

global optimization algorithms based on the branch-and-bound principle have been proposed to

compute the globally optimal solution of the AC OPF problem [35]. The lower bound at a node in

the branch-and-bound search tree is obtained by solving Lagrangian or SDP relaxation problems,

while the branching procedure is based on the subdivision of bounds on the decision variables.

The search is restricted in the set

B0 = (pg,qg,e, f) : pmin ≤ pg ≤ pmax,qmin ≤ qg ≤ qmax,(vmini )2 ≤ e2

i + f 2i ≤ (vmax

i )2,∀i ∈N .

The search space B0 can be partitioned either by a rectangular bisection on upper and lower

bounds on active and reactive powers [35], or by a radial bisection on the voltage magnitudes

[22]. Let Ω denote the feasible set of problem (1). For any set B with given bounds on power

outputs and voltage magnitudes, we define `(B) as the lower bound of c(pq) over B ∩Ω ob-

tained from the optimal value of (4) or (10). In the following procedure, uk is the best upper

bound value at the k-th iteration, and let Sk be the set containing live nodes at the k-th iteration.

BRANCH-AND-BOUND ALGORITHM

1. Set S0 = B0, evaluate `(B0) and apply a local algorithm to compute u0 and a feasible

point z0.

2. For k = 0,1,2, . . .

(a)If Sk = /0, then the optimal solution of OPF (1) has been found.

(b)Choose Bk ∈Sk such that `(Bk) = min`(B) : B ∈Sk. Subdivide Bk into two subsets

Bk1 and Bk2.

(c)For i = 1,2, if Bki∩Ω = /0, set `(Bki) = ∞; otherwise, evaluate `(Bki).

(d)Let zk+1 denote a feasible point associated with the lowest function value that has been

generated so far. If c(zk+1)< uk, then define uk+1 = c(zk+1);

otherwise, set uk+1 = uk.

(e)Set Sk+1 = B ∈Sk ∪Bk1∪Bk2 : `(B)< uk+1,B 6= Bk.


Because the diameter of Bk tends to zero as k tends to infinity, the bounding scheme is con-

sistent. In addition, because of the selection of the live node in Step 2b, the smallest lower bound

is increasing, thus the algorithm is convergent [24].

4 Security-Constrained Optimal Power Flow

Contingency analysis is routinely performed in the economic dispatch practice. The goal is to

ensure that demand can be satisfied in the normal case and in the case of contingencies, where

any one or more components in the power system, such as generators, transmission lines, trans-

formers, or other equipments, experience unexpected failure. The OPF problem with contingency

constraints is often referred to as the Security-Constrained OPF (SCOPF). There are two major

types of SCOPF models: the preventive model [2] and the corrective model [33].

The preventive model finds a minimum cost normal case dispatch solution that is also feasible

for all pre-specified contingency conditions. A general formulation is given below:

minx0,...,xC ,u0

c(x0,u0)

s.t. gk(xk,u0) = 0, k = 0, . . . ,C,

hk(xk,u0)≤ 0, k = 0, . . . ,C,

where x0 and xk represent the state variables, such as the complex voltages, in the normal case

and in the k-th contingency, respectively; u0 represents the control variable, such as the active and

reactive powers, in the normal case. As stated above, the preventive model requires the normal

case control variable u0 to be feasible for all constraints involved in total C contingencies.

The second type of SCOPF model is the so-called corrective model, which allows the system

operator to re-adjust control variables after the contingency occurs. The rationale is that the power

system components, such as transmission lines and transformers, can usually sustain a short

period of overloading without being damaged [33]. This capability gives the system operator a

window to adjust control variables to eliminate any violations caused by the contingency. The

problem can be formulated as follows:


minx0,...,xC ,u0,...,uC

c(x0,u0)

s.t. gk(xk,uk) = 0, k = 0, . . . ,C,

hk(xk,uk)≤ 0, k = 0, . . . ,C,

|uk−u0| ≤ umaxk , k = 1, . . . ,C.

Here, the system operator has the flexibility to choose a control variable uk for each contingency

k. The last constraint is imposed to limit the maximum deviation between the normal case control

and the post-contingency control to be within umaxk for each k. Note that these constraints should

be understood componentwise, i.e., |uk,i−u0,i| ≤ umaxk,i for every generator i ∈ G .

Sometimes, system operators are only able to handle a limited number of corrective actions.

This can be modeled by imposing the constraints on the number of corrective actions [32]:

|uk−u0| ≤ sk umaxk ,

∑i∈G

sk,i ≤ Nk, sk,i ∈ 0,1,

where Nk is the maximum number of corrective actions allowed.

One of the major challenges of SCOPF is its huge dimensionality. For C contingency sce-

narios, the problem size of the SCOPF problem is roughly C + 1 times larger than that of the

normal OPF problem. For large-scale power systems involving numerous contingencies, central-

ized solution algorithms may encounter prohibitive memory usage and long execution times. In

the literature, solution approaches include iterative contingency selection schemes [1, 13, 14, 18],

decomposition methods [31, 33, 36], and network compression [26]. It is well-known that many

post-contingency constraints are redundant, contingency filtering techniques identify and include

only those potentially binding contingencies into the formulation. Benders methods decompose

the SCOPF into a master problem and subproblems, where subproblems check the solution fea-

sibility and possibly generate a linear cut for the master problem. A network compression of

post-contingency states technique aims to reduce the size of the problem based on the observa-

tion that the impact of an outage is, in general, related to a localized area of the power grid.


5 Distributed Algorithms for AC OPF

Previous sections introduce models and solution methods for AC OPF problems, which require

a central control entity gathering system wide information and implementing the solution algo-

rithms. However, in some situations, decentralized decision making is more appealing or even

necessary. For instance, in a large interconnected power system involving multiple Independent

System Operators (ISO) or utilities, information pooling and sharing may be limited by institu-

tional arrangements. The central controller in this case has no access to detailed information to

implement the centralized optimization algorithms. To achieve the system wide optimal dispatch

decision, a decentralized algorithm is needed, where subsystems solve localized problems and

exchange limited amount of information with neighboring subsystems. A distributed algorithm

is also appealing from reliability perspective. In the event of power system disruption such as

communication failure between regions, centralized control would be disrupted, while a decen-

tralized scheme can handle the situation in a more flexible way.

Baldick and his colleagues in a series of papers first demonstrated the viability of distributed

algorithms for optimal power flow problems [27, 4, 28]. Since then, several papers have con-

tributed to the topic. In the following, we categorize existing approaches based on the type of

OPF models, decomposition methods, and algorithmic frameworks.

1. OPF models: Both AC and DC OPF problems have been studied. In particular, [27, 4, 28, 17,

34, 11] have proposed distributed algorithms for AC OPF problems, while [16, 3, 10, 12] have

proposed for DC OPF problems. In [16], the authors use a nonlinear DC model to account for

network losses. Distributed algorithms for security-constrained DC OPF problem have also

been proposed [12].

2. Decomposition methods: A large power system is usually divided to sub-control regions,

such as ISOs and utilities, interconnected by tie-lines. The coupling between subsystems

arises from constraints such as power flow balance equations and thermal limits on trans-

mission lines. In order to apply a distributed algorithm, the coupling between subsystems

need to be broken down. A common idea behind all the decomposition methods proposed

in the literature is to introduce redundant variables in the overlapping regions between sub-

systems. These redundant variables can be power injections and voltages on fictitious buses

introduced on tie-lines [27, 4, 28, 16, 17, 34]; or, redundant power flow variables on tie-lines

[3, 10, 12, 11].


3. Algorithmic framework: After redundant variables are introduced, the centralized OPF

model is decomposed into subproblems. A common approach is to relax coupling constraints

using Lagrangian relaxation or some of its variants. In particular, the existing proposals can

be divided into two groups. The first group uses the exact framework of Lagrangian or aug-

mented Lagrangian relaxation, where the coupling constraints are relaxed by introducing La-

grange multipliers, the resulting Lagrangian problem is solved by a distributed algorithm, and

the Lagrange multipliers (the shadow prices) are updated by a subgradient scheme. For exam-

ple, the auxiliary problem principle (APP) is used to solve the Lagrangian problem in [27, 4].

The alternating direction method and the predictor-corrector proximal method are applied and

compared to the APP method in [28]. These three methods have comparable performance in

terms of number of iterations and computation time. The other group of methods uses a variant

of Lagrangian relaxation framework, where the KKT system of the subproblem is solved ap-

proximately in each iteration [17, 34, 3, 10, 12]. The key observation is that stacking together

the KKT conditions of subsystems gives the KKT condition of the entire system. Therefore,

when the distributed algorithm converges, the solution satisfies the overall KKT condition and

thus is at least a stationary point of the centralized problem. It is worth mentioning that the

distributed algorithm frameworks such as the auxiliary problem principle or the alternating di-

rection methods are originally proposed for convex optimization problems. The convergence

is not guaranteed for general non-convex problems like the AC OPF problems.

6 Security-Constrained Unit Commitment Problem

In a centralized control system, the system operator has access to a wide range of detailed eco-

nomic and operational data, including production costs and physical constraints of the generation

units, system load forecast, system reserve requirement, and network parameters. Based on these

data, the system operator need to decide a commitment schedule of generation units for the next

24 hours or a longer period of time, under which the forecast demand can be met with the least

total commitment and dispatch cost, and very importantly, the security constraints induced by the

N−1 criterion are satisfied. Therefore, the unit commitment model used in the current practice

is a deterministic scheduling model with many complicating constraints.

It is also important that the system operator accounts for uncertainties in the load forecast and

unexpected outages of generators and transmission lines when making the unit commitment de-


cision. The traditional and still widely used approach is to reserve a certain amount of generation

resources for the purpose of fast response to emergencies such as loss of generators and trans-

mission lines or sudden load peak. This amount of stand-by capacity is called reserves, which

are further classified into different categories based on the speed of response, such as ten-minute

spinning reserve (TMSR), ten-minute nonspinning reserve (TMNSR), and thirty-minute oper-

ating reserve (TMOR). Other types of reserves exist. For example, regulation reserves are the

generation capacity controlled by automatic systems that respond to frequency changes in the

power system every few seconds.

The unit commitment model then includes decision variables for the amount of reserves pro-

vided by each generator, and also includes constraints that the total reserves in the power system

should meet certain reserve requirements. Such reserve requirements are usually pre-determined

by the system operator as a part of the operating rules. In the following, the security-constrained

unit commitment model is presented.

The decision variables are: binary variables xti ,u

ti,v

ti for commitment decisions and continuous

variables pti and qt

i,a for generation and reserve decisions, where xti = 1 if generator i is producing

electricity, i.e. in the on state, at time t, and xti = 0 otherwise; ut

i = 1 if generator i is turned on from

the off state at time t; vti = 1 if generator i is turned off at time t; pt

i is the amount of electricity

produced by generator i at time t; and qti,a is the amount of reserve provided by generator i for

reserve type a at time t.

The basic optimization model is given as below [23, 37]:


minxxx,uuu,vvv,ppp,qqq ∑

t∈T∑i∈G

(f ti (x

ti ,u

ti,v

ti)+ ct

i(pti))

(11)

s.t. xt−1i − xt

i +uti ≥ 0, ∀i ∈ G , t ∈T , (12)

xti− xt−1

i + vti ≥ 0, ∀i ∈ G , t ∈T , (13)

xti− xt−1

i ≤ xτi , ∀τ ∈ [t +1,mint +MinUpi−1,T], t ∈ [2,T ], (14)

xt−1i − xt

i ≤ 1− xτi , ∀τ ∈ [t +1,mint +MinDwi−1,T], t ∈ [2,T ], (15)

∑i∈G

pti = ∑

j∈Ddt

j, ∀t ∈T , (16)

−RDi ≤ pti− pt−1

i ≤ RU i, ∀i ∈ G , t ∈T , (17)

− f maxl ≤ aaa

T

l (pppt −dddt)≤ f maxl , ∀t ∈T , l ∈L , (18)

− f maxl,i ≤ aaa

T

l,i(pppt −dddt)≤ f maxl,i , ∀t ∈T , l ∈ Ci, i ∈L , (19)

pmini ≤ pt

i + ∑a∈A

qti,a ≤ pmax

i , ∀i ∈ G , t ∈T , (20)

pmini xt

i ≤ pti ≤ pmax

i xti , ∀i ∈ G , t ∈T , (21)

∑i∈G ,a∈Ak

qti,a ≥ qt

k, ∀t ∈T ,k ∈Nr, (22)

∑a∈Ak

qti,a ≤ qt

i,k, ∀i ∈ G , t ∈T ,k ∈Nr. (23)

The parameters in the above model are explained as follows. The function f ti (x

ti ,u

ti,v

ti) is the

total fixed cost of generator i at time t in the horizon T , including start-up cost, shut-down cost,

and other fixed costs; cti(pt

i) is the operating cost, which is usually approximated as a convex

piece-wise linear function of the electricity output pti; dt

j is the forecast nodal load at bus j, time

t; RUi and RDi are respectively the ramp-up and ramp-down rates of generator i; aaal ∈R|N | is the

vector of shift factors of branch l, whose component al,i is defined as the change of power flow

on branch l if a unit of power is injected at bus i and taken out at the reference bus; similarly,

aaal,i ∈ R|N | is the vector of shift factors of branch l in the contingency that line i is offline; qtk

is the reserve requirement for reserve type k at time t; Nr is the set of all reserve types; qti,k is

the maximum amount of reserve type k that generator i can provide at time t; Ci is the set of

remaining lines when line i is tripped; A is the set of reserve types such as TMSR, TMNSR, and

TMOR. For more details on the reserve requirement, interested readers can refer to [8].

Constraints (12) and (13) represent logic relations between on and off status and the turn-on

and turn-off actions. Constraints (14) and (15) restrict the minimum up and down times for each

generator, i.e. if a generator is turned on at time t, then it must remain on at least for the next


(MinUpi−1) periods, and similar for the shutdown constraint. Constraint (16) is the energy bal-

ance for each time period. Constraint (17) is the ramping constraint, which couples consecutive

time periods. Constraint (18) is the linearized power flow equations using shift factors for the

base case, where all transmission lines are functioning. Constraint (19) is the power flow limits

under the i-th contingency. Constraint (20) limits the total production and reserve for each gener-

ator. Constraint (21) limits the production level of each generator. Constraint (22) is the system

reserve requirement for reserve category k at time t. Constraint (23) provides an upper bound on

generator i’s reserve capacity.

7 Adaptive Robust Optimization for SCUC under Uncertainty

The rapid growth of renewable energy resources such as wind and solar power leads to signif-

icantly increased uncertainty in both supply and demand in the power systems. The traditional

deterministic approach introduced in the previous section is challenged in both its economic ef-

ficiency and operational effectiveness. In this section, we introduce recent developments in unit

commitment models using the methodology of adaptive robust optimization [6, 7, 9].

7.1 Adaptive robust UC model

The primitive of the adaptive robust optimization model is the so-called uncertainty set, which is

a deterministic set representation of the uncertain net nodal load (load minus supply). In partic-

ular, consider the following uncertainty set of net load at each time period t:

D t(dddt, ddd

t,∆ t) :=

dddt ∈ RNd : ∑

i∈Nd

|dti − dt

i |dt

i≤ ∆

t ,dti ∈[dt

i − dti , d

ti + dt

i],∀i ∈Nd

, (24)

where Nd is the set of nodes that have uncertain injections, Nd is the number of such nodes,

dddt = (dti , i ∈Nd) is the vector of uncertain net injections at time t, dt

i is the nominal value of the

net injection of node i at time t, dti is the deviation from the nominal net injection value of node i

at time t, the interval[dt

i − dti , d

ti + dt

i]

is the range of the uncertain dti , and the inequality in (24)

controls the total deviation of all injections from their nominal values at time t. The parameter ∆ t

is the “budget of uncertainty”, taking values between 0 and Nd . When ∆ t = 0, the uncertainty set

D t = dddt is a singleton, corresponding to the nominal deterministic case. As ∆ t increases, the


size of the uncertainty set D t enlarges. This means that larger total deviation from the expected

net injection is considered, so that the resulting robust UC solutions are more conservative and

the system is protected against a higher degree of uncertainty. When ∆ t = Nd , D t equals to the

entire hypercube defined by the intervals for each dti for i ∈Nd .

Now we formulate the adaptive robust UC model as follows,

minxxx

(cccT xxx+max

ddd∈Dmin

yyy∈Ω(xxx,ddd)bbbT yyy

)(25)

s.t. FFFxxx≤ fff , xxx binary,

where xxx is the vector of commitment related decisions, yyy is the vector of dispatch related vari-

ables, D is the Cartesian product of nodal load uncertainty sets over time, Ω(xxx,ddd) = yyy : HHHyyy ≤

hhh,AAAxxx+BBByyy ≤ ggg, IIIuyyy = ddd is the set of feasible dispatch solutions for a fixed commitment deci-

sion xxx and nodal net injection realization ddd, the constraints HHHyyy ≤ hhh,AAAxxx+BBByyy ≤ ggg, IIIuyyy = ddd are a

matrix representation of constraints (16)-(23), and constraints FFFxxx≤ fff is a matrix representation

of commitment constraints (12)-(15).

From this formulation, we can see that the commitment decision xxx takes into account all pos-

sible future net load represented in the uncertainty set D . Such a commitment solution remains

feasible, thus robust, to any realization of the uncertain net load. In comparison, the traditional

UC solution only guarantees feasibility for a single nominal net load, whereas the stochastic op-

timization UC solution only considers a finite set of preselected scenarios of the uncertain net

load. Furthermore, in this formulation the optimal second-stage decision yyy is a function of the

uncertain net load ddd, where the functional form yyy(ddd) is implicitly determined by the economic

dispatch problem minyyy∈Ω(xxx,ddd) bbbT yyy. The dispatch solution is thus fully adaptive to any realization

of the uncertainty.

7.2 Solution method to solve adaptive robust model

This subsection outlines algorithms to solve the adaptive robust UC model (25) [8, 25, 39].

Observe that the adaptive robust UC model (25) is a two-stage problem, where the first stage

finds the optimal commitment decision and the second stage finds the worst-case dispatch cost

under a fixed commitment solution. The proposed solution method is a two-level algorithm. The

outer level uses a Benders decomposition type method which uses cuts generated from the inner


level algorithm. The inner level solves the second-stage problem maxddd∈D minyyy∈Ω(xxx,ddd) bbbT yyy, which

can be reformulated either as a bilinear optimization problem by dualizing the inner dispatch

problem minyyy∈Ω(xxx,ddd) bbbT yyy [8], or as a mixed-integer linear optimization problem by introducing

binary variables and big-M constants to linearize the bilinear terms [25, 39]. In [8], the bilinear

optimization problem is solved by an outer approximation algorithm. In [25, 39], the MILP

problem is solved by commercial MIP solvers. The overall framework of the two-level algorithm

is as follows: At the k-th iteration:

Step 1: Solve the following outer level master problem:

minxxx,α

cccT xxx+α

s.t. α ≥ λλλTl (AAAxxx−ggg)−ϕϕϕ

Tl hhh+ηηη

Tl dddl , ∀l ≤ k, (26)

FFFxxx≤ fff ,xxx binary.

Let (xxxk,αk) be the optimum. Set LBD = cccT xxxk +αk. The inequality (26) is the cut generated

from the second-stage problem.

Step 2: For the fixed commitment decision xxxk, solve the second-stage problem either as a

bilinear program [8] or an MILP [25, 39]. Denote its optimal objective value as R(xxxk). Set

UBD = cccT xxxk +R(xxxk).

Step 3: If UBD−LBD < ε , stop and return xxxk. Otherwise, let k = k+1, and go to step 1.

Extensive computational experiments have conducted on some large scale power systems,

such as the ISO New England’s power system [8]. Comparing to the reserve adjustment approach

used in the current practice, the adaptive robust UC model demonstrates better economic effi-

ciency in terms of lower average commitment and dispatch costs, significantly reduced volatility

of the dispatch cost, and robustness against different uncertainty distributions. Interested readers

can refer to [8, 25, 39] for detailed numerical results.

8 Conclusions

In this article, basic optimization models and the most recent advances in modeling and solu-

tion methods are discussed for two fundamental decision making problems in electric power

system operations, the optimal power flow problem and the unit commitment problem. In par-

ticular, advanced global optimization techniques for solving the OPF problem are presented. An


adaptive robust optimization model and solution methods for the unit commitment problem are

introduced, which provide a new, effective framework for power system operations under uncer-

tainty. We have also discussed contingency analysis for large-scale power systems, as well as

distributed optimization schemes for decentralized control of the power systems. In conclusion,

modern optimization methods play a central role in power system operations. Further advances

are needed to meet new challenges arising in the fast evolving modern power systems.

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Modern Optimization Models and Techniques for … Optimization Models and Techniques for Electric Power Systems Operation Andy Sun and Dzung T. Phan Abstract This article introduces