This section introduces the problem and formalises a deterministic version of it. The objective of the EMS is to minimise the cost of electricity and maximise occupant comfort through the control of household devices. To make the problem more concrete we have to make some assumptions, the first of which relates to the objectives.
Multi-objective problems can be interpreted and solved in a number of different ways [Collette and Siarry, 2013, Ehrgott and Gandibleux, 2000]. Our approach is to apply a linear scalarisation to the objectives, combining them into a single objective. This assumes that occupant discomfort can be weighted so that it is directly comparable to a monetary cost. Taking thermal comfort as an example, this assumes that it is possible for occupants to associate a price they are willing to pay to keep the temperature in their house within a certain range. To make this simple for occupants, we expect the EMS interface to present a “slider” which occupants can adjust between maximum comfort on one end and maximum frugality on the other. Behind the scenes this is interpreted as varying the weighting between the two objectives. Occupants can move the slider about until it matches their expectation through experience.
The next design choice we make is to discretise time, in effect taking a quasi-steady-state approximation of the control problem. This is appro- priate because we are focused on the longer time scale device actions that require planning in advance. Faster transients and power system frequency control are better handled by myopic fast acting controllers, for example, dynamic-demand devices [Angeli and Kountouriotis, 2012] which can work alongside the EMS. As long as time discretised into sufficiently small steps, the results will be close to what would be achievable with a fully continuous control signal.
For this discretisation, the index t represents the t-th time step, ∆τt is
the duration of thet-th time step in seconds andτtis the time at the end of
thet-th time step in seconds, whereτt> τt−1 and ∆τt=τt−τt−1 (see figure
3.1). This somewhat complicated formulation allows us to have variable step sizes over the forward scheduling horizon which, as we will discuss later, will allow us to focus computational resources where they are most needed.
3.3. DETERMINISTIC EMS PROBLEM 29
1 2
T
τ0 τ1
time
τ2 τT-1 τT
Figure 3.1: The timings over a forward horizon withT time steps. The time steps (represented by the indext) range from 1 toT. The timesτ are used to mark the times at which the steps begin and end, as the steps might have varying lengths.
3.3.1 Devices
At a high level, a device has a vector of variables xt ∈ RM and a vec-
tor of parameters1 rt ∈ RL for each time step t. The variables repres-
ent device actions or states and the parameters represent external factors which impact the operation of the device (e.g., occupant usage requests or ambient temperatures). Devices have an operation cost function f :
RM ×RL 7→ R and a power function h :RM ×RL 7→ R which take these
vectors as inputs. The operation cost represents any comfort, fuel, deteri- oration or other cost associated with the operation of the device, and the power function returns the power that the device either consumes (+ve) or produces (−ve). Finally, devices have a vector-valued constraint function
g:RM×RL×RM×RL7→RN which applies to the variables and parameters
in consecutive time steps and which is satisfied when the component-wise in- equalityg(xt, rt, xt−1, rt−1)≤0 holds. This generic constraint function can
be used to do anything from placing bounds on variables to constraining state transitions.
3.3.2 Optimisation Problem
A house is simply a set of devicesD, together with bounds ¯
P and ¯P on the instantaneous amount of power that the house can transfer to or from the grid. The EMS controls the decision variables for each of these devices in order to minimise the costs for the overall house.
We make use of a deterministic formulation of the EMS optimisation problem as a building block for the stochastic formulations. In this formula- tion we have a forward horizon of T time steps, over which the values of all parameters are known. The objective is to choose device actions to reduce the total cost over this future horizon. Inputs include the device initial states
xd,0, electricity priceλt, house background power usagePtb (the aggregation
of uncontrollable electrical consumption, e.g., lighting, entertainment and cooking), and device parameters rd,t. The variables at each time step in-
clude the device variables and the total house power consumptionPt.
1
The deterministic optimisation problem is as follows: min xd,t,Pt T X t=1 ∆τtλtPt+ X d∈D T X t=1 fd(xd,t, rd,t) (3.6) s.t. Pt∈[ ¯ P,P¯] ∀t∈ {1, . . . , T} (3.7) Pt=Ptb+ X d∈D hd(xd,t, rd,t) ∀t∈ {1, . . . , T} (3.8) gd(xd,t, rd,t, xd,t−1, rd,t−1)≤0 ∀d∈ D, t∈ {1, . . . , T} (3.9)
The appropriate solver for this optimisation problem depends on the form of the device functions. For example, if all functions are linear, then an linear programming (LP) solver can be used, or if they are convex an interior point convex solver. The device models presented in the next section result in a mixed-integer linear program (MILP), which in our experiments is solved with Gurobi [Gurobi Optimization, Inc., 2014].