SEMIDEFINITE RELAXATIONS FOR THE SCHEDULING

NUCLEAR OUTAGES PROBLEM

Agnes Gorge

, Abdel Lisser

and Riadh Zorgati

Universit´e Paris-Sud Orsay, LRI, 91405 Orsay, France

EDF R&D, OSIRIS, 92141 Clamart, France

Keywords:

Energy management, Combinatorial optimization, Semideﬁnite relaxation, Randomized rounding.

Abstract:

We investigate semideﬁnite relaxations for solving a MIQP (Mixed-Integer Quadratic Program) formulation of

the scheduling of nuclear power plants outages, which is extremely hard to solve with CPLEX. Based on our

numerical experiments, results obtained with semideﬁnite relaxations improve those obtained with continuous

relaxation: the gap between the optimal solution and the continuous relaxation is on average equal to 1.80%

whereas the semideﬁnite relaxation yields an average gap of 1.56%. These bounds are then used to obtain a

feasible solution with a randomized rounding procedure.

1 INTRODUCTION

The French electrical production facilities is charac-

terized by a high number of nuclear power plants,

which have to be shut down regurlarly in order to pro-

ceed to refueling and maintenance operations. Opti-

mizing the scheduling of these outages is therefore a

key factor for an efﬁcient economic performance.

This mid-term management problem consists in

determining, on the ﬁve years ahead i) the dates for

outages to refuel nuclear power plants, ii) the amount

of supplied fuel and iii) the nuclear power plants pro-

duction planning which satisfy the demandat minimal

cost, while respecting numerous technical constraints.

This real-life problem is far too difﬁcult to be

tackled exactly, due to its huge size, its non-linear

constraints, and because uncertainties affecting both

production and demand. Finally, modelling the on-

line/ofﬂine state of the plants requiresthe introduction

of binary variables, which make the problem combi-

natorial.

Many approaches for this problem have been in-

vestigated (Khemmoudj et al., 2006), (Porcheron

et al., 2009). In this paper, we deal with a determin-

istic version of the problem and emphasize its com-

binatorial nature in order to investigate efﬁciency of

semideﬁnite relaxations.

It is organized as follows. In Section 2, we derive

our model for the problem. In section 3, we outline

the semideﬁnite relaxations we use. We report some

numerical results in section 4 before concluding and

giving prospects for future work.

2 MODELING THE NUCLEAR

OUTAGES SCHEDULING

PROBLEM

We will consider in this paper a deterministic version

of the problem where only the most signiﬁcant tech-

nical constraints are taken into account.

2.1 Key Operating Features of Nuclear

Power Plants and Modeling

The nuclear park is composed of several sites, where

each site is a set of 2, 4 or 6 nuclear power plants. The

operation life of a nuclear power plant is decomposed

into cycles, each cycle being made up of a phase of

production, called production campaign, followed by

an outage.

Let’s introduce some convenient notations: in

what follows, x and y will denote respectively the bi-

nary and continuous variables. (i, j) refers to the j-th

cycle of the plant i. The index of the ﬁrst and last cy-

cle of each plant are respectively 1 and J

. {1, ··· , N

}

and {1, ··· , N

} are the set of sites and plants of the

nuclear park. If P is a set of nuclear plants, (i, j) ∈ P

denotes the whole cycles of the plants of this set. We

denote i ∈ k the fact that a plant belongs to the k-th

site and (i, j) ∈ k the cycles of the plants of this site.

Finally, a time step t corresponds to a week and the

horizon time is composed of N

weeks.

386

Lisser A., Gorge A. and Zorgati R..

SEMIDEFINITE RELAXATIONS FOR THE SCHEDULING NUCLEAR OUTAGES PROBLEM.

DOI: 10.5220/0003743203860391

In Proceedings of the 1st International Conference on Operations Research and Enterprise Systems (ICORES-2012), pages 386-391

ISBN: 978-989-8425-97-3

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

2.1.1 Phase I - Campaign

During this phase, the nuclear power plant produces

either in standard mode (at full power) or in mod-

ulation mode (at lower level). The standard mode,

that is producing at maximal power W

(expressed in

MW), is the best operating level for the plants. On the

contrary, when a nuclear plant doesn’t produce at full

power, it is said to ”modulate”. This mode of produc-

tion may alter the state of the plant, which requires

more maintenance afterwards. That is why the quan-

tity of modulation, which is measured as the amount

of ”non-produced” energy, is limited. Let y

i, j

be the

continuous variable that represents the modulation of

the cycle (i, j):

∀(i, j) ∈ N

, y

i, j

∈ [0, M

i, j

] (1)

2.1.2 Phase II - Outage

A nuclear power plant shall be stopped regularly for

refueling and maintenance operations. The duration

of the outage of the cycle (i, j) is denoted by the num-

ber of weeks δ

i, j

. The scheduling of the outages re-

quires to deﬁne a binary variable for each possible be-

ginning date of each outage: ∀t ∈ E

i, j

, x

i, j,t

∈ {0, 1}

where E

i, j

is the set of possible beginning dates for

the outage of the cycle (i, j). Among the variables

i, j,t

of the cycle (i, j), only the one for which t is the

actual beginning date of the outage shall be equal to

1. Consequently, we impose the so-called uniqueness

constraint:

∀(i, j < J

) ∈ N

∑

t∈E

i, j

i, j,t

= 1 (2)

With this modeling, the beginning date of the out-

age (i, j) can be easily computed using the formula

∑

t∈E

i, j

i, j,t

and the state of the plant i at week t, i.e.

1 if the plant is online, 0 otherwise, can be expressed

as follows: 1 −

∑

j=1

∑

′

=t−δ

i, j

i, j,t

′

. Note that for

the sake of simplicity, we sometimes drop the nota-

tion t ∈ E

i, j

for x

i, j,t

and consider it implicitly.

It comes that the maximal capacity of production

of the nuclear park y

at week t, a state variable, can

be computed as:

∀t = 1, ··· , N

∑

i∈N

(1−

∑

j=1

∑

′

=t−δ

i, j

i, j,t

′

)

(3)

Refueling and Final Stock of Energy. For safety

reasons, the stock of energy that remains in the reactor

of a nuclear plant at the beginning of an outage (i, j),

denoted y

i, j

, must lie within the interval [F

i, j

, F

i, j

except for the last cycle, for which only the lower

bound is required since the outage is not attained:

∀(i, j < J

) ∈ N

, F

i, j

≤ y

i, j

≤ F

i, j

∀i ∈ N

, F

i,J

≤ y

i,J

(4)

For the sake of concision, let’s just say that y

i, j

can

be computed as a afﬁne combination of the previous

ﬁnal stock y

i, j−1

, of the outages binary variables x

i, j,t

and of the variable y

i, j

and y

i, j

denoting the modula-

tion and the amount of the reload carried out during

outages respectively. Without detailing the particular

case of the ﬁrst and last cycles, we have the following

formula for the ﬁnal stock:

∀(i, 1 < j < J

) ∈ N

i, j

= W

i, j−1

+ β

i, j−1

+ y

i, j−1

+ y

i, j

−W

(

∑

t∈E

i, j

i, j,t

−

∑

t∈E

i, j−1

i, j−1,t

)

(5)

Besides, for each cycle (i, j < J

), the variable y

i, j

shall respect a maximal value

i, j

, corresponding to

the maximal capacity of the reactors:

∀(i, j < J

) ∈ N

, y

i, j

∈ [0,

i, j

] (6)

Managing Resourcesfor Outages. On a nuclear site,

in order to manage the limited resources required for

the refueling and maintenance operations, we impose

a maximal number of parallel outages at each time

step and a maximal lapping between outages.

Let N

par

be the maximum authorized number of

outages in parallel on site k. Then, the related con-

straint can be written:

∀k = 1, · · · , N

, ∀t = 1, ·· · , N

∑

(i, j)∈k

∑

′

=t−δ

i, j

i, j,t

′

≤ N

par

(7)

Let N

lap

be the maximum authorized lapping be-

tween the outages of site k. A negative value repre-

sents a minimum space. Then, for each concerned

pair (i, j), (i

′

, j

′

), there are two possibilities: either

(i, j) starts before (i

′

, j

′

), or it doesn’t. The computa-

tion of the lapping depends of the effective conﬁgura-

tion: let ∆

i, j,i

′

, j

′

denotes the space between beginning

of outages, the lapping might be:

i, j

+ ∆

i, j,i

′

, j

′

or δ

′

, j

′

− ∆

i, j,i

′

, j

′

(8)

This disjonction requires the introduction of new

binary variable: x

i, j,i

′

, j

′

that codes 0 in the ﬁrst case

and 1 otherwise. Let

M be a sufﬁciently large number.

Then both following constraints must be respected:

SEMIDEFINITE RELAXATIONS FOR THE SCHEDULING NUCLEAR OUTAGES PROBLEM

387

i, j

+∆

i, j,i

′

, j

′

−

i, j,i

′

, j

′

≤ N

lap

′

, j

′

−∆

i, j,i

′

, j

′

−

M(1− x

i, j,i

′

, j

′

) ≤ N

lap

(9)

2.2 Constraints Related to Demand

Satisfaction

In our problem, the production portfolio made up

of N

nuclear power plants and N

fossil-fuel power

plants has to satisfy the electrical demand on the two

following periods of each time step (e.g. a week):

• A peak period when the demand is high and can

not be satisﬁed by nuclear production (fossil-fuel

production is needed) ;

• An off-peak period when the demand is low (for

example, during the night) and can be satisﬁed by

nuclear production only.

2.2.1 Peak Demand

At peak time, the whole capacity of production of the

park y

∑

i∈N

i,t

, where U

i,t

is the capacity of pro-

duction of the fossil-fuel power plant i at time step t,

should satisfy the peak demand D

∀t = 1, ··· , N

∑

(i, j)∈N

≥ D

−

∑

i∈N

i,t

(10)

2.2.2 Off-Peak Demand

At off-peak time, the demand constraint comes to lim-

iting the whole modulation of the park at each time

step. Here, we make a reasonable simpliﬁcation con-

sisting in respecting the sum of these constraints on

the time horizon. This allows us to gather the modu-

lation throughout the cycles, so the constraint can be

written:

∑

(i, j)∈N

i, j

≤ D

−

(11)

2.3 The Objective Function

Our aim is to minimize the global cost of production

which is the sum of the nuclear production cost and

the fossil-fuel production cost. The ﬁrst one is pro-

portional to the amount of reloads and the fossil-fuel

production, which is computed as the difference be-

tween the peak demand and the nuclear production,

has a quadratic cost, so the global cost function is:

∑

(i, j)∈N

i, j

−

∑

i∈N

i,J

−1

i,J

∑

t=1

− y

]

(12)

3 RESOLUTION

Finally, gathering equations (1), (2), (3), (4), (5), (6),

(7), (9),(10), (11), (12) and introducing matrix formu-

lation leads to the compact form:

(P)











min

x,y

Qx+ p

x+ q

subject to Ax+ By ≤ c

y ≤ ¯y

x ∈ {0, 1}

, y ∈ R

(13)

Our problem is therefore a mixed quadratic opti-

mization problem with linear constraints, where the

quadratic terms of the objective function involve only

binary variables. This kind of problem is difﬁcult to

solve, even with a powerful commercial solver like

CPLEX. For this reason, we investigate semideﬁnite

relaxations in the view of obtaining better bounds of

the solution than we can obtain when using continu-

ous relaxations.

3.1 Semideﬁnite Relaxations

Semideﬁnite programming (SDP) is a subﬁeld of con-

vex optimization which deals with the optimization of

a linear function over an afﬁne subspace of the cone

of the semideﬁnite matrices. With A• B denoting the

Frobenius inner product, it has the following form:

(SDP)







min

X∈S

• X

subject to A

• X = b

, i = 1, ··· , m

X < 0

(14)

This area of mathematical programming has un-

dergone a rapid development in the last decades,

spurred by the development of efﬁcient resolution al-

gorithms (see (Helmberg et al., 1996), (Helmberg and

Rendl, 2000)) and by the discovery of widespread ap-

plications, in particular to relaxation of combinatorial

problems. For further reading on the subject, see for

example the surveys of Boyd and Vandenberghe(Van-

denberghe and Boyd, 1994) and Todd (Todd, 2001) or

the corresponding handbook (Wolkowicz et al., ). See

also the survey of Laurent ((Laurent et al., 2005)) on

the related relaxation of combinatorial problems.

Here we apply SDP to the relaxation of the pre-

viously described MIQP (13). For this, we introduce

the following symmetric matrix:

X =







x 1

∗ Diag(y)







(15)

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

388

where Diag(y) stands for the diagonal matrix made

up with vector y and ∗ means that any value can

be taken. Let’s note that the ﬁrst submatrix include

the vector x and the associated quadratic matrix xx

Then, by deﬁning the appropriate matrices C

, we can

express the objective quadratic function (because the

quadratic terms involve only x) and the linear con-

straints as C

• X.

About the binary constraints, we deﬁne the ma-

trices D

such that x

− x

= 0 ⇔ D

• X = 0. Fur-

thermore, a matrix E is used to impose that the last

component of the ﬁrst submatrix be equal to 1.

So, we have the following equivalent formulation for

our problem:

(Q)











min C

• X

s.t. C

• X ≤ 0, i = 1, · · · , M

• X = 0, i = 1, · · · , N

E • X = 1

i, j

= X

i,N

j,N

, i, j = 1, ··· , N

The last constraint comes to impose to X the speciﬁc

form described at (15). This constraint, which is nei-

ther linear or convex, is what makes the problem NP-

hard. Such as matrix is necessarily semideﬁnite pos-

itive, so the semideﬁnite relaxation is obtained by re-

placing this constraint by a constraint on its semidef-

initeness, which is convex. Consequently, the associ-

ated SDP is:

(SDP)











min C

• X

s.t. C

• X ≤ 0, i = 1, ··· , M

• X = 0, i = 1, · · · , N

E • X = 1

X < 0

This relaxation gives rise to a lower bound of

the optimal solution p

∗

of the initial problem, which

can be used either within an exact search, typically a

Branch & Bound procedure, or to compute an ap-

proximate solution of the problem, for example via a

randomized rounding scheme. It is the latter alterna-

tive we are using here. We will compare the approx-

imate solution obtained by applying this procedure

(described in the paragraph below) to the solution of

the Quadratic Program obtained by relaxing the in-

tegrality constraint (denoted here continuous relax-

ation). This programcan be solved with CPLEX since

the objective function is convex. In a second step, we

will try to improve the SDP bound by adding some

cuts based on the Sherali-Adams approach (Sherali

and Adams, 1990).

3.2 Randomized Rounding Procedure

Randomization has proved to be a powerful resource

to yield a feasible binary solution from a fractional

one. The basic idea is to interpret the fractional value

as the probability of the variable to take the value 1.

Then the values of the binary variables are drawn ac-

cording to this law and this process is iterated until

the solution satisﬁes the constraints.

Here, we slightly change this principle, in order

to ﬁnd more easily a feasible solution: instead of de-

ciding successively if a binary variable is 0 or 1, for

each cycle, we choose one date among the possible

beginning date for the associate outage, by using the

fractional valueas probabilily, since their sum is equal

to one from the uniqueness constraint. Then, the val-

ues of the lapping variables x

follow. About the con-

tinuous variables, for the modulation x

, we keep the

value of the relaxation and for the reload x

, we take

the minimal values that respects the ﬁnal stock con-

straint.

3.3 Tightening Semideﬁnite Relaxation

with Quadratic Cuts

Adding some valid appropriate equalities or inequal-

ities may improve the bound of the semideﬁnite re-

laxation. Here, we apply the Sherali-Adams (Sherali

and Adams, 1990) principle: let Ax = b be a set of

linear constraints and x

a binary variable, the con-

straints Axx

= bx

is valid. We apply this idea to the

uniqueness constraint (2), with all the variables x

that

appear in the constraint. By using x

= x

it comes:

∀(i, j < J

) ∈ N

, ∀t ∈ E

i, j

∑

′

∈E

i, j

, t

′

6=t

i, j,t

′

= 0

(16)

4 NUMERICAL EXPERIMENTS

Numerical experiments have been performed on a

three years time horizon (156 weeks), with one out-

age per year for each plant and two nuclear parks (re-

spectively 10 and 20 nuclear power plants for the data

set 1 to 12, and 13 to 24). Each park is declined into

two versions which differ from the maximum amount

of reload (

i, j

) and modulation (M

i, j

Finally, six instances have been tested for each

data set, varying by the size of the search spaces asso-

ciated to the outages dates variables (7 to 17 possibles

dates).

All the computations have been made on an Intel(R)

Core(TM) i7 processor with a clock speed of 2.13

GHz. In order to compare the solutions in the same

conditions, the CPLEX results are obtained without

activating the preprocessing. For each data set we

computed:

SEMIDEFINITE RELAXATIONS FOR THE SCHEDULING NUCLEAR OUTAGES PROBLEM

389

Table 1: Results of exact search, relaxations and randomized rounding.

Data Nb of Opt RelaxQP RelaxSDP RelaxSDP-Q

set bin. var. Obj Time Gap Time RR Gap Time RR Gap Time RR

D-1 215 3 343 1 0.73 0.02 2.35 0.54 12 2.35 0.26 12 0.70

D-2 278 3 254 21 0.80 0.00 3.88 0.64 19 1.49 0.46 21 1.70

D-3 341 3 174 183 0.94 0.02 4.86 0.82 31 2.43 0.65 36 3.25

D-4 406 3 110 1 286 1.10 0.02 4.23 0.97 44 5.04 0.83 54 5.14

D-5 469 3 051 7 200 1.18 0.02 11.70 1.08 63 3.72 0.96 79 4.04

D-6 530 2 994 5 780 1.17 0.03 14.56 1.09 81 3.35 1.00 108 4.73

D-7 215 3 297 2 1.24 0.02 3.31 1.03 5 2.82 0.68 6 0.82

D-8 278 3 223 8 1.89 0.03 10.28 1.72 8 7.15 1.38 11 3.35

D-9 341 3 176 39 2.94 0.08 11.31 2.81 15 9.95 2.49 64 2.11

D-10 406 3 133 169 3.91 0.13 14.69 3.80 26 11.94 3.52 98 8.98

D-11 469 3 070 76 3.87 0.18 13.56 3.78 38 13.81 3.53 147 11.79

D-12 530 3 024 232 4.25 0.20 14.47 4.17 53 17.98 3.95 236 16.20

D-13 539 12 580 7 200 0.85 0.05 3.16 0.77 154 3.28 0.61 171 2.08

D-14 698 12 431 7 200 0.95 0.10 3.47 0.89 252 3.76 0.76 286 4.06

D-15 852 12 290 7 200 1.13 0.14 5.78 1.08 373 4.58 0.99 436 4.83

D-16 1 011 12 156 7 200 1.14 0.14 6.16 1.09 578 5.19 1.02 750 5.29

D-17 1 170 12 034 7 200 1.15 0.22 5.72 1.12 791 5.77 1.08 1008 6.36

D-18 1 322 11 939 7 200 1.35 0.27 6.47 1.32 1030 5.67 1.30 1308 7.00

D-19 537 12 679 7 200 1.21 0.16 2.80 1.16 68 2.95 1.07 310 4.48

D-20 695 12 464 7 200 1.57 0.54 5.96 1.52 137 6.56 1.44 447 6.31

D-21 853 12 289 7 200 1.98 0.94 9.28 1.94 242 8.91 1.85 805 6.74

D-22 1 008 12 159 7 200 2.37 1.90 9.15 2.33 382 7.47 2.27 1113 8.80

D-23 1 165 12 034 7 200 2.65 2.95 7.87 2.62 628 7.70 2.58 2106 6.86

D-24 1 316 11 915 7 200 2.87 3.65 10.93 2.84 823 9.89 2.80 2231 8.52

Av. 651.83 7700.85 4224.91 1.80 0.49 7.75 1.71 243.88 6.41 1.56 493.46 5.59

• Opt: the best solution found within the time limit

(2 hours) by using CPLEX-Quadratic 12.1. The

time value7200 means that the time limit has been

reached, so the obtained integer solution is not op-

timal ;

• RelaxQP: the continuous relaxation computed

with CPLEX-Quadratic 12.1;

• RelaxSDP: the SDP relaxation computed with the

SDP solver CSDP 6.1.1 (cf (Borchers, 1999));

• RelaxSDP-Q: the SDP relaxation computed with

CSDP 6.1.1 with quadratic cuts (cf 3.3) ;

For each data set, the table 1 reports the number of

binary variables, the value of the objective function

(in currency unit), the computational time in second

and, for each kind of relaxation, the associated gap

(Gap) and the relative gap of the randomized rouding

(RR), whose formula are given below. The last line

(Av.) gives the average of the previous lines.

Gap =

opt

− p

relax

RR =

− p

opt

(17)

Analysis of the Results

First we observe that CPLEX reaches the limited time

for relatively small instances (e.g. 469 binary vari-

ables). This is in line with our expectations that this

kind of problem is very hard for CPLEX, despite a

quite small gap attained with continuous relaxation.

This may be related to the fact that, due to the

demand constraint, the variable part of the objective

function is very small w.r.t the absolute value of the

cost. In other words, the optimal value is high, even

with a ”perfect” outages scheduling. Let’s denote P

the best possible objective value for a given data set,

computed by considering the largest possible search

space, and let’s consider the variable part of the ob-

jective function, that is p − P, if p is the objective

value. Then, the gap would increase, as shown in the

following formula:

opt

− p

relax

− P

opt

− p

relax

(18)

This illustrates the importance of considering the

relative improvement of the gap achieved by semidef-

inite relaxation, rather that its absolute value.

For example, on the data set D-1, the gap is almost

divided by three. Unfortunately, this ratio decreases

as the number of binary variables raises, whereas the

gap increases. This can be explained by the fact that

the ”exact solution” provided here is not optimal, con-

sidered that the computational time of CPLEX is lim-

ited. Let’s denote p

′

opt

> p

opt

this value: then the ra-

tio computed with this value is greater than the ratio

computed with p:

ICORES 2012 - 1st International Conference on Operations Research and Enterprise Systems

390

opt

− p

relaxCPLEX

opt

− p

relaxSDP

′

opt

− p

relaxCPLEX

′

opt

− p

relaxSDP

(19)

On average, the gap improves from 1.80% to

1.71% with original SDP relaxation and 1.56% with

addition of valid equalitites. This latter improve-

ment is promising, even though it comes at high ad-

ditional computational cost, particularly on the larger

instances. This can be ascribed to the fact that SDP

solvers are only in their infancy, especially compared

to a commercial solver like CPLEX.

Finally, the randomized rounding yields satisfying

results: due to the random aspect of the procedure,

there are still some data set where the continuous re-

laxation gives better results than the semideﬁnite re-

laxation, but on average the loss of optimality reduces

from 7.75% to 6.41% and 5.59%, which is signiﬁcant

when considering the huge amount at stake.

5 CONCLUSIONS AND

PROSPECTS

We investigated in this paper, semideﬁnite relaxations

for a MIQP (Mixed-Integer Quadratic Program) ver-

sion of the scheduling of nuclear power plants out-

ages. Comparison of the results obtained on signiﬁ-

cant data sets shows the following main results. First,

our MIQP is extremely hard to solve with CPLEX.

Second, semideﬁnite relaxations provide a tighter

convex relaxation than the continuous relaxation. In

our experiments the gap between the optimal solu-

tion and the continuous relaxation is on average equal

to 1.80% whereas the semideﬁnite relaxation yields

an average gap of 1.56%. Third, the computational

time for computing these semideﬁnite relaxations is

reasonable. Exploiting those results in a randomized

rounding procedure instead of the result of the contin-

uous relaxation leads to a signiﬁcant improvement of

the feasible solution.

In the view of these preliminary results, additional

investigations will concern i) introduction of more

valid inequalities, ii) evaluation of others SDP resolu-

tion techniques, for instance Conic Bundle for facing

problems of huge size.

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gramming.

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