Fitting an Emulator

Section 2.3 gave an overview for the existing literature on transmission expansion plan-ning. It was noted that a large amount of the literature gave little thought to uncer-tainty in the power system background when identifying optimal expansion decisions of the power system’s transmission system. Further, the literature which did consider uncertainty did so by considering a small number of possible states of the power sys-tem. This thesis will overcome this by treating the variables which define the power system as continuous variables, which would allow the variables to take any potential value over a given range (e.g. we could model the peak demand level to be anywhere between 62100 MW and 63600 MW) instead of a small number of possible values.

As mentioned, in the problem presented there is uncertainty in the values of the input variables, X. However, the function f_T(X) (the simulator of total costs), is far too ex-pensive to evaluate at every set of input parameters desired. Therefore, it is necessary to evaluate f_T a small number of times for particular values of X, and approximate f_T everywhere else as carefully as possible by some alternative function, ˜fT. ˜fT should be much less computationally demanding to evaluate in order to allow for efficient estima-tion for input values not simulated. An estimate of the uncertainty that exists in the response when approximating f_T by ˜f_T is also required. The process of approximating fT by ˜fT is known as emulation [74, 45, 89].

Suppose there are N_v variables with uncertainty of interest, labelled v₁, ..., v_Nv and N_d decision variables, d₁, ..., d_Nd. The goal is to fit a model to the output of f_T (the sum

5.2. Emulation 113

of mean constraint costs plus reinforcement costs) based on these N_v + N_d variables.

As this process assumes all other variables are considered fixed as if known, ˜fT can be treated as if it is a function of v₁, ..., v_Nv and d₁, ..., d_Nd only.

A small set of training runs of the simulator are used to construct the function ˜f_T which approximates f_T. These training runs vary the values of v₁, ..., v_Nv, and d₁, ..., d_Nd over a given range. Details on how training runs are selected to give a dense coverage of the space defined by v₁, ..., v_Nv and d₁, ..., d_Nd are given in Section 5.2.4.

Suppose y_T represents the response of the simulator (f_T) for the training data. For simplicity in the following notation, also suppose x = (v, d) = (v₁, ..., v_Nv, d₁, ..., d_Nd).

The emulator, ˜f_T, is usually constructed such that ˜f_T is the sum of a parametric model plus a non-parametric smoother applied to the residuals of the parametric model [45, 89, 84, 92], i.e.

f˜_T(x) = g(x)^Tβ + e(x) (5.2.1) where g(x)^Tβ is the parametric portion of the emulator and e(x) is the non-parametric smoother applied to the residuals of g(x)^Tβ. The emulator used in this thesis will use a polynomial regression model as the parametric portion (i.e. g(x)^Tβ from Equa-tion 5.2.1) of the emulator model. This can be thought of as

y_T(v₁, ...v_Nv, d₁, ..., d_Nd) = y_T(x) = β₀+

p

X

t=1 Nv+Nd

X

i=1

β_t,ix^t_i+

p

X

t=1 Nv+Nd

X

i=1

X

j6=i

β_t,i,jx^t_ix^t_j+...+ε(x) (5.2.2) i.e. the dot product of a coefficient vector with the polynomial form of the predictor variables with error term ε(v₁, ..., v_Nv, d₁, ..., d_Nd). This thesis only considers interac-tions between equal powers of variables (e.g. the polynomial may include an interaction between v₁ and d₂ but not between v₁ and d²₂). Values of β are estimated via least squares regression, as will be detailed in Section 5.2.2.

Whilst this methodology applies very generally, the terms included in the polynomial regression model should be carefully selected based on the specifics of the example considered. This is considered in Sections 5.3.3 and 6.1.3, where both the R² value of the polynomial regression model and a measure of the resulting emulator’s predictive power for a second set of training data which was not used to fit the model are used

to identify terms to be included in the model.

If the residual of the polynomial regression model, ε(x^∗), is positive for a given set of input values, x^∗, this indicates that the polynomial regression model under-estimates the response of the simulator, y_T(x), at x^∗. As y_T(x) is modelled as a continuous variable, ε(x) is also modelled as a continuous variable. By the continuity of y_T(x), the polynomial model will also under-estimate the response for values of x close to x^∗, i.e. values of ε(x) are not independent but rather contain information about the response of the simulator near x.

This means that the fit of the model can be improved further by using a Gaussian process model to smooth the residuals, ε(v₁, ..., v_Nv, d₁, ..., d_Nd). This is important as the polynomial model will aim to give a good fit over the entire range of the variables considered, whereas the Gaussian process model can model local behaviour much more accurately. The idea of a Gaussian process is to smooth the residuals in order for the model to agree with the training data and provide an accurate estimate of response (and variance of the estimated response) for values where training data is unavailable.

In this sense, the Gaussian process model can be thought of as a smooth interpolator of the residuals [74]. In simple terms, the idea of the Gaussian process is if a response is to be estimated at point x_p, training data closer to x_p carries more relevant information and should be given more weight when making an estimation. Further details on Gaussian process models are given in Section 5.2.3.

In this thesis, the polynomial portion of the emulator model will be denoted by ˜fT1 and the Gaussian process model fitted to the resulting residuals of the polynomial model will be denoted by ˜fT2. Thus, the emulator model for given inputs is the sum of these two, denoted ˜f_T, such that

f˜T(v₁, ..., vNv, d1, ..., dN_d) = ˜fT1(v₁, ..., vNv, d1, ..., dN_d) + ˜fT2(v₁, ..., vNv, d1, ..., dN_d) (5.2.3) That is to say ˜fT is our emulator which approximates the simulator, and the estimated response of the emulator is acquired by evaluating the sum of the estimated responses of the fitted polynomial regression model and the corresponding Gaussian process model for given input values.

After the emulator has been fitted to a set of training data, a second set of training

5.2. Emulation 115

data will then be simulated. The emulator model fitted to the first set of training data, f˜T, will then be used to estimate a response for the output of the simulator at the second set of training data. By comparing the estimated response of the emulator to the value simulated the predictive power of the emulator can be assessed, as will be demonstrated in Section 5.3.3.