Constraint formulation

GMS model instances in the literature are typically formulated in terms of integer decision vari- ables or binary decision variables as exemplified in the discussion on scheduling objectives of the previous section. The adoption of each of these types of decision variables hold both advantages and disadvantages. In this section, both typical GMS integer decision variable constraint sets and typical binary decision variable constraint sets are described. In the case where integer de- cision variables are used, xu denotes the starting period for planned maintenance on PGU u ∈ U

during the scheduling window. In this case, an auxiliary variable yu,p is also employed which

takes the value 1 if planned maintenance is scheduled for PGU u ∈ U during planning period p ∈ P, or zero otherwise, as already introduced in §2.2.1. If, however, binary decision variables are adopted, the decision variable xu,p takes the value 1 if planned maintenance is scheduled for

PGU u ∈ U during planning period p ∈ P, or zero otherwise, and the same auxiliary variable yu,p is also adopted in this case.

As mentioned in §2.1.4, the demand satisfaction constraint is one of the most important con- straint sets in models of the GMS problem. The most basic way of imposing demand satisfaction involves requiring that

X

u∈U

Cu,p(1 − yu,p) ≥ Dp, p ∈ P (2.17)

in order to ensure that the demand Dp for energy in the power system during any planning

period is met. A slightly more complicated formulation of the demand satisfaction constraint may be found in [87] where the generating capacity Cu of PGU u ∈ U is allowed to vary over the

planning periods of the scheduling window. In this formulation, the parameter Cu in constraint

set (2.17) is replaced by the parameter Cu,p which denotes the generating capacity of PGU u

during planning period p ∈ P.

Another variation on the demand satisfaction constraint set (2.17) may be found in [66, 87], where the demand satisfaction constraint set takes into account the required reserve or safety margin in the system. This is achieved by adding a parameter Rp to the right-hand side of

constraint set (2.17) which denotes the reserve or safety margin required during planning period p ∈ P. In this case, the demand constraint is formulated as

X

u∈U

Cu,p(1 − yu,p) ≥ Dp+ Rp, p ∈ P (2.18)

instead of (2.17). In order to impose demand satisfaction in the case of using binary decision variables, two additional parameter sets are required. The first set U_p0 is the set of PGUs that are allowed to be in a state of planned maintenance during planning period p ∈ P. Hence, U0

p= {u | p ∈ Pu}, where Pu = {p ∈ P | eu ≤ p ≤ `u} denotes the set of planning periods during

planning period for planned maintenance on PGU u ∈ U and `u denoting the latest starting

planning period for planned maintenance on PGU u ∈ U . The second set S_u,p0 is the set of starting planning periods such that if planned maintenance were to start on PGU u ∈ U during such a planning period, then PGU u will be in a state of planned maintenance during planning period p ∈ P. Therefore, S_u,p0 = {j ∈ Pu | p − du+ 1 ≤ j ≤ p} [53]. The demand satisfaction

constraint set, when using binary decision variables, may then be formulated as X u∈U Cu,p− X u∈U0 p X j∈S0 u,p Cu,jxu,j ≥ Dp+ Rp, p ∈ P. (2.19)

Burke et al. [36], as well as Digalakis and Margaritis [57], took a slightly different approach in formulating the demand satisfaction constraint set. In their modelling approaches, the output level of a PGU is not considered a fixed value representing the capacity of the unit. A variable ou,p is rather employed (as introduced in the previous section) which denotes the output level

of PGU u ∈ U during planning period p ∈ P. Since this modelling approach takes into account variable output levels of the PGUs, the demand of the system is required to be equal to the output of the fleet of PGUs. The demand satisfaction constraint set of Burke et al. [36] and of Digalakis and Margaritis [57] is therefore

X

u∈U

ou,p = Dp, p ∈ P. (2.20)

An additional constraint set

0 ≤ ou,p≤ Cu,p(1 − yu,p), u ∈ U , p ∈ P (2.21)

is also enforced, which specifies output limits for each PGU. Constraint set (2.21) specifies that the output level of a PGU u ∈ U generating ou,p units of power should be between zero and the

generating capacity Cu,p of the unit during planning period p ∈ P. The value should, however,

be equal to zero if the PGU is in a state of planned maintenance.

Another basic class of constraints in GMS models are maintenance window constraints. This type of constraint set specifies earliest and latest starting times during which planned maintenance may commence on every PGU. If integer variables are employed in the model formulation, then the maintenance window constraints may be formulated as

eu≤ xu ≤ `u, u ∈ U (2.22)

[4, 133]. If, however, binary variables are employed in the formulation, then the more complex maintenance window constraint set

X

p∈Pu

xu,p = 1, u ∈ U (2.23)

is required, in addition to the constraint sets

xu,p = 0, u ∈ U , p /∈ Pu, (2.24)

yu,p = 0, u ∈ U , p < eu or p > `u+ du− 1. (2.25)

In (2.25), du denotes the duration of planned maintenance required for PGU u ∈ U , as before.

The duration of planned maintenance is furthermore required to be contiguous. In order to ensure that this requirement is met, the constraint set

yu,p =

(

1, if xu ≤ p ≤ xu+ du− 1,

is included in the model formulation when integer decision variables are used [57, 67, 68]. Once again, two more complicated sets of constraints are required in the case where binary variables are used in the formulation. The first is the maintenance duration constraint set and the second ensures contiguity of the planned maintenance period. The maintenance duration constraint set may be formulated as

X

p∈P

yu,p = du, u ∈ U (2.27)

[39, 185], while the constraint set ensuring contiguity of the planned maintenance period of duration du may be formulated as

yu,p− yu,p−1≤ xu,p, u ∈ U , p ∈ P\{1}, (2.28)

yu,1≤ xu,1, u ∈ U (2.29)

[39, 185]. There are, however, some cases in the literature [135, 152, 168] where the maintenance duration constraint set (2.27) and the maintenance contiguity constraint set (2.28) are combined into a single constraint set. This combined constraint set is

xu+du−1

X

p=xi

yu,p = du, u ∈ U . (2.30)

Instances may also be found in the literature where an undesirable nonlinear construct is em- ployed in the formulation of the maintenance duration and contiguity constraints [133]. In such cases, the duration constraint set is the same as (2.27), whereas the contiguity constraint set is formulated as

xu+du−1

Y

j=xu

yu,p = 1, u ∈ U . (2.31)

Resource constraints may also be included in GMS model formulations. These constraints specify the maximum number of resources available during a given planning period, of which some pre-specified minimum amount of resources are required to schedule planned maintenance for any PGU. These constraints ensure that the maximum amount of resources available during a planning period is not exceeded by the amount of resources required to carry out planned maintenance during that period. The resources available in GMS problem instances are typically the number of maintenance personnel available to perform planned maintenance. The most basic formulation of a resource constraint set of this kind was formulated for the case where integer decision variables are employed in the model formulation and assumes that during each period of planned maintenance, the same amount of resources is required for any given PGU [7, 68, 87]. Let fu,p denote the amount of resources required in order to perform planned maintenance on

PGU u ∈ U during planning period p ∈ P. Then the resource constraint set may be formulated as

X

u∈U

fu,pyu,p ≤ Mp, p ∈ P, (2.32)

where Mp denotes the maximum available amount of resources during planning period p ∈ P.

In the case where binary decision variables are employed in the formulation, a constraint set similar to (2.32) may be included in the model formulation [51, 53]. Following the same notation as in (2.19), the constraint set may be formulated as

X u∈U0 p X j∈S0 u,p fu,jxu,j ≤ Mp, p ∈ P. (2.33)

In more complex incarnations of the resource constraint set it is assumed that the resources required during each period of planned maintenance are not the same for some of the PGUs. This approach assumes that during the i-th planning period of planned maintenance, the PGU may require a specific amount of resources. Let f_ui denote the amount of resources required for planned maintenance on PGU u ∈ U during its i-th period of its planned maintenance and let fu,p,v denote the resources required for planned maintenance on PGU u ∈ U during planning

period p ∈ P if the maintenance were scheduled to start during planning period v ∈ P. The resources required may then be calculated as

fu,p,v =

(

fup−v+1, if p − v < du,

0, otherwise. (2.34)

From (2.34), a more general resource constraint set may be formulated as X

u∈U

X

v∈P

fu,p,vxu,v≤ Mp, p ∈ P. (2.35)

There are also formulations of the GMS problem in the literature that contain so-called exclusion constraints. These constraints specify sets of PGUs which are not allowed to be in a state of simultaneous planned maintenance. The reason for this type of constraint is that PGUs from the same power station are often not allowed to be in maintenance during the same period. It may, for example, also be required that certain PGUs of the same class (e.g. coal, nuclear, wind, sun, etc.) should not be in a state of simultaneous planned maintenance. In order to achieve this type of constraint, let J1, . . . , Jw be sets of PGUs which may not all be scheduled

simultaneously for planned maintenance and let Ii denote the maximum number of PGUs that

are allowed to be scheduled for planned maintenance simultaneously during any planning period within exclusion set Ji. Then the exclusion constraint set may be formulated as

X

u∈Ji

yu,p ≤ Ii, p ∈ P, i ∈ {1, . . . , w} (2.36)

[39, 51, 133]. In practice, it is also sometimes advantageous for a power utility to schedule planned maintenance of certain PGUs before certain other PGUs. This may be achieved by the incorporation of so-called precedence constraints which may either specify that some PGU u1

has to be scheduled for planned maintenance and return to full operation before a certain PGU u2 can be scheduled for planned maintenance, or that PGU u2 is only allowed to enter a state of

planned maintenance after planned maintenance has commenced on PGU u1. In the case where

integer decision variables are employed in the model formulation, the former type of precedence constraint may be formulated as

xu1+ du1 ≤ xu2. (2.37)

In the latter case, the precedence constraints may simply be formulated as

xu1 < xu2. (2.38)

In the case where binary decision variables are, however, employed in the model formulation, the former type of precedence constraint may be formulated as

X

v∈P

xi1,v− xi2,p≥ 0, p ∈ P (2.39)

[39], while the latter type of precedence constraint may be formulated as