New Formulations for the Unit Commitment Problem

Optimal Control and Switching-Time Parameterization Approaches

ıs A. C. Roque

, Fernando A. C. C. Fontes

and Dalila B. M. M. Fontes

LIADD-INESC-TEC, DMA, Instituto Superior de Engenharia do Porto, 4200-072 Porto, Portugal

SYSTEC-ISR- Porto, Faculdade de Engenharia, Universidade do Porto, 4200-465 Porto, Portugal

LIADD-INESC-TEC, Faculdade de Economia, Universidade do Porto, 4200-464 Porto, Portugal

Keywords:

Unit Commitment, Switching Time Problem, Optimal Control, Electrical Power Systems.

Abstract:

The Unit Commitment Problem (UCP) is a well-known combinatorial optimization problem in power systems.

The main goal in the UCP is to schedule a subset of a given group of electrical power generating units and also

to determine their production output in order to meet energy demands at minimum cost. In addition, a set of

technological and operational constraints must be satisﬁed. A large variety of optimization methods addressing

the UCP is available in the literature. This panoply of methods includes exact methods (such as dynamic

programming, branch-and-bound) and heuristic methods (tabu search, simulated annealing, particle swarm,

genetic algorithms). This paper proposes two non-traditional formulations. First, the UCP is formulated as

a mixed-integer optimal control problem with both binary-valued control variables and real-valued control

variables. Then, the problem is formulated as a switching time dynamic optimization problem involving only

real-valued controls.

1 INTRODUCTION

The unit commitment problem (UCP) is well-studied

and practically relevant in the electrical power in-

dustry. This problem involves both the scheduling

of power units (i.e., the decisions when each unit is

turned on or turned off along a predeﬁned time hori-

zon) and the economic dispatch problem (the prob-

lem of deciding how much should produce each unit).

The scheduling of the units is typically seen as an

integer programming problem and the economic dis-

patch problem is a nonlinear (real-valued) program-

ming problem. The UCP is then a nonlinear, noncon-

vex, and mixed-integer optimization problem (Dang

and Li, 2007). The objective of the UCP is the mini-

mization of the total operating costs over the schedul-

ing horizon while satisfying demand, the spinning re-

serve requirements, and other generation and techno-

logical constraints such as capacity limits, ramp rate

limits, and minimum uptime/downtime. The objec-

tive function is expressed as the sum of the fuel, start-

up, and shutdown costs.

Several methods have been employed to ﬁnd solu-

tions for the UCP. The available approaches for solv-

ing unit commitment problem can usually be clas-

siﬁed into mathematical programming and heuris-

tic methods (Salam, 2007). In the past, the pro-

posed approaches were essentially based on exact

methods and include dynamic programming (Hobbs

et al., 1988; Patra et al., 2009), Lagrangian relax-

ation (Zhuang et al., 1992), Benders decomposi-

tion (Ma and Shahidehpour, 1998), and mixed inte-

ger programming (Frangioni et al., 2008; Frangioni

et al., 2009). More recently, several metaheuristic ap-

proaches have been used such as particle swarm op-

timization (Zhao et al., 2006), simulated annealing

(Simopoulos et al., 2006; Jenkins and Purushothama,

2003), tabu search (Rajan and Mohan, 2004) and ge-

netic algorithms (Dang and Li, 2007; Roque et al.,

2014; Roque et al., 2011). A bibliographical sur-

vey can be found in (Saravanan et al., 2013). Al-

though the UCP is a highly researched problem with

dynamical and multi-period characteristics, it appears

that it has not been addressed before by optimal con-

trol methods, except in (Fontes et al., 2012; Fontes

et al., 2014; Tuffaha and Gravdahl, 2016). In (Fontes

et al., 2012), the authors have formulated the UCP

as a discrete mixed-integer optimal control problem,

which has then been converted into one with only

real-valued controls through a variable-time transfor-

mation method. A mixed-integer state-space model

to solve both the UC and Economic Dispatch prob-

326

Roque, L., Fontes, F. and Fontes, D.

New Formulations for the Unit Commitment Problem - Optimal Control and Switching-Time Parameterization Approaches.

DOI: 10.5220/0006465603260331

In Proceedings of the 14th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2017) - Volume 1, pages 326-331

ISBN: 978-989-758-263-9

lems is reporteded in (Tuffaha and Gravdahl, 2016).

The output power generation, status of the generating

unit, and up and down time counters are considered

state variables.

Here, we discuss two formulations of the UCP as

an Optimal Control (OC) model. Initially, we formu-

late the UCP as a discrete-time mixed-integer optimal

control problem. Then, we propose a new optimal

control switching time approach. The model derived

is a continuous one and involves only real-valued de-

cision variables (controls).

The main contributions of the proposed modelling

approach are twofold. Firstly, since it allows deci-

sions to be taken at any time moment, and not only at

speciﬁc points in time (usually, hourly), it may ren-

der better solutions. It should be noticed that the pro-

posed approach allows for decisions on unit commit-

ment/decommitment and power production variation

at any moment in time. Secondly, it no longer forces

utilities to treat demand variations as instantaneous,

i.e., time steps. In addition, if one chooses to use

the approximated hourly data, as usual in the litera-

ture, the solution strategies (both regarding unit com-

mitment/decommitment and power production) of the

proposed model will approximate the discrete-time

solutions since actions are only required to be taken

hourly.

The remaining of this paper is organized as fol-

lows. The mixed-integer optimal control formulation

is given in section 2. Section 3 introduces a formula-

tion of the UCP as a switching time optimal control

problem including only real-valued controls, which is

proposed here for the ﬁrst time. Finally, in section 4

some conclusions are drawn.

2 UCP AS A DISCRETE-TIME

MIXED-INTEGER OPTIMAL

CONTROL MODEL

In this section a mixed–integer optimal control model

(OCM) is proposed to the UCP problem. The mixed–

integer optimal control model has two types of de-

cision/control variables: on the one hand, it consid-

ers the binary control variables β

(t) and γ

(t), which

are the start-up and shut down indicators at time pe-

riod t +1, respectively. β

(t)(γ

(t)) are either set to 1,

meaning that unit j is turned on-line (off-line) at time

period t + 1, or otherwise set to zero. On the other

hand, it also considers real-valued variables ∆

(t),

which enable to control the production, by increasing

or decreasing the power produced by unit j at time t.

The binary variables satisfy

(t − 1) = (1 − u

(t − 1))u

(t)

(t − 1) = u

(t) − u

(t − 1) + γ

(t − 1)

and

(t − 1) = (1 − u

(t))u

(t − 1)

(t − 1) = u

(t − 1) − u

(t) + β

(t − 1).

There are three types of state variables:

1. real variables y

(t), which represent the power

generated by unit j at time t; and y

(t), which

indicate the contribution for spinning reserve by

unit j at time t;

2. integer variables T

on/o f f

(t), which represent the

number of time periods for which unit j has been

continuously online/off-line until time t; and

3. binary variables u

(t), which are either set to 1,

meaning that unit j is committed at time t, or oth-

erwise set to zero.

The parameters used in the equations are deﬁned

as follows:

T: Number of time periods (hours) of the scheduling

time horizon

N: Number of generation units

R(t): System spinning reserve requirements at time

t, in [MW]

D(t): Load demand at time t, in [MW ]

Ymin

: Minimum generation limit of unit j, in [MW ]

Ymax

: Maximum generation limit of unit j, in

[MW ]

c,j

: Cold start time of unit j, in [hours]

on/off

min,j

: Minimum uptime/downtime of unit j, in

[hours]

0,j

: Initial state of unit j at time 0, time since the

last status switch off/on, in [hours]

off

0,j

: Initial state of unit j at time 0, time since the

last status switch on/off, in [hours]

∆

dn/up

: Maximum allowed output level de-

crease/increase in consecutive periods for unit j,

in [MW ].

For convenience, let us also deﬁne the index sets:

T := {1, ... ,T} and J := {1, 2,..., N}.

The UC problem can now be formulated as a mixed-

integer optimal control model.

Objective Function. The objective function has three

cost components: generation costs, start-up costs, and

shutdown costs. The generation costs, also known as

New Formulations for the Unit Commitment Problem - Optimal Control and Switching-Time Parameterization Approaches

327

the fuel costs, are conventionally given by the follow-

ing quadratic cost function:

(t)) = a

· (y

(t))

+ b

· y

(t) + c

, (1)

where a

are the cost coefﬁcients of unit j. The

start-up costs, that depend on the number of time pe-

riods during which the unit has been off, are repre-

sented by S

(t). The shutdown costs Sd

for each

unit, whenever considered in the literature, are con-

stant. Therefore, the cost incurred with an optimal

scheduling is given by the minimization of the total

costs for the whole planning period.

Minimize

∑

t=1

∑

j=1

(t))u

(t)+

(t) · β

(t − 1) + Sd

· γ

(t − 1)). (2)

The State Dynamics. The production of each unit,

at time t, depends on the amount produced in the pre-

vious time period and is limited by the maximum al-

lowed decrease and increase of the output that can oc-

cur during one time period:

(t) = y

(t − 1) + ∆

(t − 1) · (u

(t) − β

(t − 1))

+ ∆

· β

(t − 1) − ∆

· γ

(t − 1),

for all t ∈ T and j ∈ J .

Also

(t) − β

(t − 1) = u

(t − 1) − γ

(t − 1).

Thus,

(t) =y

(t − 1) + ∆

(t − 1) · (u

(t − 1) − γ

(t − 1))

+ ∆

· β

(t − 1) − ∆

· γ

(t − 1), (3)

for all t ∈ T and j ∈ J .

The term T

(t − 1) · γ

(t − 1) can be used to drive

the state variable T

(t) to zero whenever the unit j is

turned off-line at time period t. Then, the number of

time periods for which unit j has been continuously

online until time t is given by

(t) = T

(t − 1) + u

(t − 1) + β

(t − 1) − γ

(t − 1)

− T

(t − 1) · γ

(t − 1), (4)

for t ∈ T and j ∈ J .

The number of time periods for which unit j has

been continuously off-line until time t is given by

o f f

(t) = T

o f f

(t − 1) + 1 − u

(t − 1) − β

(t − 1)

+ γ

(t − 1) − T

o f f

(t − 1) · β

(t − 1), (5)

for all t ∈ T and j ∈ J , where the term T

o f f

(t − 1) ·

(t − 1) is used to drive the state variable T

o f f

(t)

to zero whenever the unit j is turned on-line at time

period t.

The dynamics concerning the unit status (binary)

indicator u

(t) is given by

(t) = u

(t − 1) + β

(t − 1) − γ

(t − 1), (6)

for t ∈ T and j ∈ J .

Pathwise Constraints. The pathwise constraints are

given by inequalities (7) to ( 27) and the ramp rate

constraints are handled by the equations (3) and con-

trol constraints.

∆

(t) ∈

−∆

,∆

, for t ∈ T and j ∈ J . (7)

In addition the control binary variables β

and γ

are

forced to take the 0 or 1 values using the inequations

(8-11).

0 ≤ β

(t − 1) ≤ 1 for t ∈ T , j ∈ J . (8)

(t − 1) · (1 − β

(t − 1)) ≤ 0 for t ∈ T , j ∈ J . (9)

0 ≤ γ

(t − 1) ≤ 1 for t ∈ T , j ∈ J . (10)

(t − 1) · (1 − γ

(t − 1)) ≤ 0 for t ∈ T , j ∈ J . (11)

In a similar way the auxiliary binary variables u

are

forced to take either the value 0 or the value 1.

0 ≤ u

(t − 1) ≤ 1 for t ∈ T , j ∈ J , (12)

(t − 1) · (1 − u

(t − 1)) ≤ 0 for t ∈ T , j ∈ J . (13)

The unit output range limits are expressed by

(t) −Y min

· u

(t) ≥ 0, ∀t ∈ T , ∀ j ∈ J (14)

(t) −Y max

· u

(t) ≤ 0, ∀t ∈ T , ∀ j ∈ J (15)

The limits for the number of time periods continu-

ously on-line are given by

(t) ≥ 0, (16)

(t) − K · T · u

(t) ≤ 0, (17)

(t − 1) − T

min, j

· γ

(t − 1) ≥ 0, (18)

for all t ∈ T and j ∈ J .

The limits for the number of time periods continu-

ously off-line are expressed by

o f f

(t − 1) ≥ 0, (19)

o f f

(t − 1) − K · T · (1 − u

(t − 1)) ≤ 0, (20)

o f f

(t − 1) − T

o f f

min, j

· β

(t − 1) ≥ 0, (21)

for all t ∈ T and j ∈ J .

The load requirements are modelled by

∑

j=1

(t) · u

(t) − D(t) ≥ 0, ∀t ∈ T . (22)

ICINCO 2017 - 14th International Conference on Informatics in Control, Automation and Robotics

328

The spinning reserve requirements are given by

∑

j=1

(t) · u

(t) − R(t) ≥ 0, ∀t ∈ T , (23)

and

(t) + y

(t) −Y min

· u

(t) ≥ 0, (24)

(t) + y

(t) −Y max

· u

(t) ≤ 0, (25)

(t) ≥ 0, (26)

(t) − ∆

· (u

(t) − β

(t − 1)) ≤ 0, (27)

for all t ∈ T and j ∈ J .

The Initial State Constraints. At the initial time t =

0 we have:

(0) = T

0, j

(given), (28)

o f f

(0) = T

o f f

0, j

(given), (29)

(0) =



0 if T

0, j

= 0

1 if T

0, j

> 0,

(30)

(0) =



0 if T

0, j

= 0

0, j

∈ [Y min

,Ymax

] if T

0, j

> 0,

(31)

3 UCP AS SWITCHING TIME

OPTIMIZATION PROBLEM

This section presents a continuous–time optimal con-

trol formulation, based on the concept of the Switch-

ing Time Optimization Problem, see e.g. (Sager,

2005; Kaya and Noakes, 2003). This model uses only

real-valued decision variables.

3.1 The Switching Time Optimization

Problem

Let us consider the set of time points Ψ when

the change of the unit status occurs. Ψ is ei-

ther Ψ

{

,...,

}

a ﬁnite set of possible

switching times. It should be noted for each k ∈

{

1,2,...,n

}

∈ ]

k−1

,T] and

= 0. Then, we

consider n

binary control functions

: [

k−1

] 7−→

{

0,1

}

deﬁned by

(t) =

(

0, if(w

(0) = 0 ∧ k odd)or(w

(0) = 1 ∧ k even)

1, if(w

(0) = 0 ∧ k even)or(w

(0) = 1 ∧ k odd)

t ∈ [

k−1

] (32)

Figure 1: Possible realisation.

with k = 1,2, ...,n

and 0 =

≤

≤ ... ≤

= T . If we assume a ﬁnite number of switches

, then the problem can be written in a multistage

formulation

min

∑

k=1

, p], (33)

where the h is the vector of stage lengths h

−

k−1

, subject to the dynamic model stages control

(k = 1,2,..., n

˙x

(t) = f

(t),z

(t),w

(t),u

(t), p) (34)

for t ∈ [

k−1

], and path constraints:

0 ≤ g

(t),z

(t),w

(t),u

(t), p), (35)

for t ∈ [

k−1

The maximum number of switching times is 2 ×

o f f

min

c where bXc represents the nearest integer

less than or equal to X . In Figure 1, an example of

one possible realisation with n

= 5 is given. For ex-

ample, we have the contraints concerning to the limits

for the time continuously on-line expressed by

−

k−1

≥ T

min

, if (

< T )∧

[(w

(0) = 0 ∧ k even) or (w

(0) = 1 ∧ k odd)],

and the limits for the time continuously off-line are

given by

−

k−1

≥ T

o f f

min

, if (

< T )∧

[(w

(0) = 0 ∧ k odd) or (w

(0) = 1 ∧ k even)] .

3.2 The Unit Commitment Problem

formulated as Switching Time

Optimization Problem

Let us consider

1, j

2, j

,...,

sw, j

, j

the possible

switching times for unit j. Then, we consider n

sw, j

binary control functions

k, j

: [

k−1, j

k, j

] 7−→

{

0,1

}

New Formulations for the Unit Commitment Problem - Optimal Control and Switching-Time Parameterization Approaches

329

deﬁned as in equation (32). The objective function to

be minimized is

min

,∆

k, j

∑

j=1

sw, j

∑

k=1

k, j

k−1, j

(t))·w

k, j

(t)·dt+

k, j

k−1, j

(t) · w

k, j

(t) · dt +

k, j

k−1, j



1 − w

k, j

(t)



· dt

− S

(t)w

0, j

(0) − Sd

(1 − w

0, j

(0)). (36)

The unit output range limits are expressed by

(t) −Y min

· w

k, j

(t) ≥ 0, (37)

(t) −Y max

· w

k, j

(t) ≤ 0, (38)

for all j ∈ J and t ∈ [

k−1, j

k, j

]. It should be noted

that the controls are all real-valued and satisfy

(t) ∈

−∆

,∆

In addition, the power production and, for conve-

nience, the unit status must also be deﬁned for each

time instant.

(t) =

(

0, if w

k, j

(t) = 0

(

k−1, j

) +

k−1, j

(s)ds, if w

k, j

(t) = 1

for j ∈ J and t ∈



k−1, j

k, j



. The limits for the time

continuously on-line of each unit j given by inequa-

tions (18) are now expressed by

k, j

−

k−1, j

≥ T

min, j

,if(

k, j

< T )∧

[(w

0, j

(0) = 0 ∧ k even)or(w

0, j

(0) = 1 ∧ k odd)],

while the limits for the time continuously off-line of

each unit j given in inequations (21), are now deﬁned

k, j

−

k−1, j

≥ T

o f f

min, j

, if (

k, j

< T )∧

[(w

0, j

(0) = 0 ∧ k odd) or (w

0, j

(0) = 1 ∧ k even)] .

The load requirements are given by

∑

j=1

(t) · w

k, j

(t) − D(t) ≥ 0,t ∈ [

k−1, j

k, j

]. (39)

The spinning reserve requirements are expressed by

∑

j=1

(t) · w

k, j

(t) − R(t) ≥ 0,t ∈ [

k−1, j

k, j

], (40)

and

(t) + y

(t) −Y min

· w

k, j

(t) ≥ 0, (41)

(t) + y

(t) −Y max

· w

k, j

(t) ≤ 0, (42)

(t) ≥ 0, (43)

for all j ∈ J ,t ∈ [

k−1, j

k, j

4 CONCLUSIONS

The UCP, an intensively researched problem in the lit-

erature, is addressed in this paper. The problem is

usually formulated using a mixed-integer nonlinear

programming model. Here, the formulation of this

problem using optimal control models is explored.

Previous works on an optimal control approach to the

UC problem, as far as we are aware of, are limited to

the works in (Fontes et al., 2012; Fontes et al., 2014;

Tuffaha and Gravdahl, 2016) that use a discrete-time

optimal control model.

We propose here two formulations of the UCP as

an Optimal Control model. First, we formulate it as a

discrete time mixed-integer optimal control problem.

Then, we propose a new optimal control switching

time approach using a continuous-time optimal con-

trol model. An interesting feature of the continuous-

time formulation is the fact that, contrary to the usual

mixed-integer programming models in the literature,

all decision variables are real-valued, which enables

the use of more efﬁcient optimization methods for its

solution.

Additional advantages of the continuous-time op-

timal control formulation are the possibility of dealing

more accurately with data provided with an irregular

or fast-sampled time intervals, or even continuous-

time varying (for instance, continuous-time varying

demand data). The schedule can be changed and

adapted according to the predictions and the ﬂuctu-

ations inherent to the renewable power supply. The

reduction in unbalance implies, simultaneously, a re-

duction on expensive spinning reserve. This can not

only account for signiﬁcant cost savings, but also al-

lows for better environmental characteristics.

Computational results will be reported in a future

work.

ACKNOWLEDGEMENTS

We acknowledge the support of COMPETE2020–

POCI/FEDER/Portugal2020/FCT funds through

grants PTDC/EEI–AUT/2933/2014–Toccatta,

POCI–01–0145–FEDER–006933–Systec, NORTE–

01–0145-FEDER–000020, and NORTE–45–2015–

02-Stride.

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