If we assume the exogenous information is deterministic, the short-term economic dispatch problem can be formulated as a simple linear program, which can be solved using commercial packages such as GAMSR [80] and CPLEXR [81]. In this sec-
tion, we will describe the deterministic linear programming formulation in which random variables of exogenous information are replaced by their expected values
ωh = λh, βh
, 1≤h≤H, where
λh =E(λh) ;
βh =E(βh).
The objective of the deterministic linear program for the short-term economic dispatch problem (in which charging decisions are made by a system operator) is to minimize the costs of satisfying system demand over a 24-hour horizon, written as
min ghj, wh, qh, z+h, Y + h H X h=1 Chdisp(Sh, xh), (2.4)
subject to the following constraints J X j=1 ghj +wh+qh =Dh+Dh0+ L X l=1 CP ×z+ {h−l+1}>0, 1≤h≤H; (2.5) Y+ h = 0, h= 1; (2.6) Y+ h+1 =Y + h −z + h +λh, 1≤h≤H−L+ 1; (2.7)
z+ h =Y + h +λh, H−L+ 1≤h≤H; (2.8) 0≤ ghj ≤Gj, 1≤h≤H, 1≤j ≤J; (2.9) 0≤ wh ≤βh×W, 1≤h≤H; (2.10) qh, zh+, Yh+≥ 0, 1≤h≤H. (2.11)
Note that the PHEV backlog state variables Y+
h , which link different time periods together, are treated as decisions in the above formulation, since linear programming optimizes decisions at all time periods together.
Equation (2.5) is the power balance constraint. At any point of time, the total electricity supply should match the total system demand, which includes the elec- tricity demand associated with PHEV charging. We will explain in the following paragraphs how the electricity consumed for charging PHEVs at each hour is calcu- lated. There is a penalty measured by value of lost load (VOLL) [$/MWh] for any unsatisfied demand qh.
Our 24-hour daily cycle starts at 7 AM (h = 1). Let 1≤l ≤Lrepresent the hours within a complete PHEV charging cycle, e.g. L= 4 for charging a Chevy Volt using a Level II charger. Once it is started, the charging is assumed to continue forLhours till it is complete and the battery is fully charged. For example, if we start charging a PHEV at hour h = 21 (3 AM), the PHEV will remain being charged during hour 21, 22, 23, and 24 (from 3 AM to 6 AM).
Figure 2.1 illustrates a PHEV’s charging due time (by which its charging cycle needs to be completed), depending on when it is plugged in. For vehicles plugged in at and before hourh= 20, its charging due time is assumed to be the end of a day, i.e. 7 AM. This assumption makes sense since from a typical PHEV owner’s perspective, their vehicle needs to be charged overnight so that they can drive off in the morning. This gives opportunities for a system operator to strategically shift charging demand to increase system efficiency. The PHEVs plugged in at and after hour h = 21 are assumed to be charged immediately without any delay, and as a result the charging decisions for vehicles that arrive at home at and after 3 AM are fixed. The electricity
Fig. 2.1. Illustrating a PHEVs’ charging due time given its arrival time
Fig. 2.2. Computing the electricity consumed for charging PHEVs at each hour in a day
consumption associated with these vehicles, represented by D0
h in Equation (2.5), is known to the system at timeh= 1 and included as the initial state; that is, D0
h ∈S1.
We now explain the subscript ofz+
in Equation (2.5) with two examples. At hour
h= 22 (4 AM), highlighted in Figure 2.2, PHEVs being charged are those dispatched between 1 AM to 4 AM (hour 19, 20, 21, and 22). Vehicles dispatched at hour 22 are in the first hour of its charging cycle; while vehicles dispatched at hour 19 are in its last charging hour. Hence, the electricity consumed due to PHEV charging at hour 22 is equal to CP× z22+ +z + 21+z + 20+z + 19
, whereCP is battery charging power [kW], and z+
h is the number of batteries to charge at time h. Consider another hour
h = 1 (7 AM). PHEVs being charged are those dispatched between 4 AM to 7 AM (hour 22, 23, 24, and 1). As discussed earlier, the electricity consumption associated with vehicles dispatched at hour 22, 23, and 24 is treated as given data, and included in the initial state D0
h. Thus, the PHEV charging demand to be determined at hour 1 is equal to CP ×z1+, which only depends on the charging decision at hour 1, z
+ 1 .
Equations (2.6) and (2.7) are the transition functions for PHEV backlog Y+
h , as detailed in Section 2.1. Equation (2.8) enforces the charging due time for PHEVs. Equations (2.9) and (2.10) are capacity constraints for thermal units and wind energy, respectively. The power dispatched from a power plant at any time is constrained by its full nameplate capacity. The wind power production at each hour is confined by the total installed capacityW and availability factor for that particular hour. Finally, Equation (2.11) is the non-negativity restriction.