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92
10_Introduction.tex
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@ -1,5 +1,7 @@
\section{Introduction} \section{Introduction}
\subsection{Motivation}
Buildings are a major consumer of energy, with more than 25\% of the total Buildings are a major consumer of energy, with more than 25\% of the total
energy consumed in the EU coming from residential energy consumed in the EU coming from residential
buildings~\cite{tsemekiditzeiranakiAnalysisEUResidential2019}. Combined with a buildings~\cite{tsemekiditzeiranakiAnalysisEUResidential2019}. Combined with a
@ -18,10 +20,88 @@ Gaussian Processes have been previously used to model building dynamics, but
they are usually limited by a fixed computational budget. This limits the they are usually limited by a fixed computational budget. This limits the
approaches that can be taken for identification and update of said models. approaches that can be taken for identification and update of said models.
Learning \acrshort{gp} models have also been previously used in the context of Learning \acrshort{gp} models have also been previously used in the context of
autonomous racing cars, but there the Sparse \acrshort{gp} model was built on autonomous racing cars \cite{kabzanLearningBasedModelPredictive2019}, but there
top of a white-box model and only responsible for fitting the unmodeled the Sparse \acrshort{gp} model was built on top of a white-box model and only
dynamics. responsible for fitting the unmodeled dynamics.
This project means to provide a further expansion of the use of black-box \subsection{Previous Research}
\acrlong{gp} Models in the context of building control, through online learning With the increase in computational power and availability of data over time,
of building dynamics at new operating points as more data gets collected. the accessibility of data-driven methods for system identification and control
has also risen significantly.
The idea of using Gaussian Processes as regression models for control of dynamic
systems is not new, and has already been explored a number of times. A general
description of their use, along with the necessary theory and some example
implementations is given in~\cite{kocijanModellingControlDynamic2016}.
In~\cite{pleweSupervisoryModelPredictive2020}, a \acrlong{gp} Model with a
\acrlong{rq} Kernel is used for temperature set point optimization.
Gaussian Processes for building control have also been studied before in the
context of Demand Response~\cite{nghiemDatadrivenDemandResponse2017,
jainLearningControlUsing2018}, where the buildings are used for their heat
capacity in order to reduce the stress on energy providers during peak load
times.
There are, however, multiple limitations with these approaches.
In~\cite{nghiemDatadrivenDemandResponse2017} the model is only identified once,
ignoring changes in weather or plant parameters, which could lead to different
dynamics. This is addressed in \cite{jainLearningControlUsing2018} by
re-identifying the model every two weeks using new information. Another
limitation is that of the scalability of the \acrshort{gp}s, which become
prohibitively expensive from a computational point of view when too much data is
added.
Outside of the context of building control, Sparse \acrlong{gp}es have been used
in autonomous racing in order to complement the physics-based model by fitting
the unmodeled dynamics of the
system~\cite{kabzanLearningBasedModelPredictive2019}.
\subsection{Contribution}
The ability to learn the plant's behaviour in new regions is very helpful in
maintaining model performance over time, as its behaviour starts deviating and
the original identified model goes further and further into the extrapolated
regions.
This project tries to combine the use of online learning control schemes with
\acrlong{gp} Models through implementing \acrlong{svgp} Models. \acrshort{svgp}s
provide means of extending the use of \acrshort{gp}s to larger datasets, thus
enabling the periodic re-training of the model to include all the historically
available data.
\subsection{Project Outline}
The main body of work can be divided in two parts: the development of a computer
model of the \pdome\ building and the synthesis, validation and comparison of
multiple control schemes using both classical \acrshort{gp}s, as well as
\acrshort{svgp}s.
Section~\ref{sec:gaussian_processes} provides the mathematical background for
understanding \acrshort{gp}s, as well as the definition in very broad strokes of
\acrshort{svgp}s and their differences from the classical implementation of
\acrlong{gp}es. This information is later used for comparing their performances
and outlining their respective pros and cons.
Section~\ref{sec:CARNOT} goes into the details of the implementation of the
\pdome\ computer model. The structure of the CARNOT model is described, and all
the parameters required for the simulation are either directly sourced from
existing literature, computed from available values or estimated using publicly
available data from \textit{Google Maps}~\cite{GoogleMaps}.
Following that, Section~\ref{sec:hyperparameters} discusses the choice of the
hyperparameters for the \acrshort{gp} and \acrshort{svgp} models respectively in
the context of their multi-step ahead simulation performance.
The \acrlong{ocp} implemented in the \acrshort{mpc} controller is described in
Section~\ref{sec:mpc_problem}, followed by a description of the complete
controller implementation in Python in Section~\ref{sec:implementation}.
Section~\ref{sec:results} presents and analyzes the results of full-year
simulations for both \acrshort{gp} and \acrshort{svgp} models. A few variations
on the initial \acrshort{svgp} model are further analyzed in order to identify
the most important parameteres for long time operation.
Finally, Section~\ref{sec:further_research} discusses the possible improvements
to both the current \acrshort{gp} and \acrshort{svgp} implementations.
\clearpage

44
20_Previous_Research.tex
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@ -1,44 +0,0 @@
\section{Previous Research}
With the increase in computational power and availability of data over time,
the accessibility of data-driven methods for System Identification and Control
has also risen significantly.
The idea of using Gaussian Processes as regression models for control of dynamic
systems is not new, and has already been explored a number of times. A general
description of their use, along with the necessary theory and some example
implementations is given in~\cite{kocijanModellingControlDynamic2016}.
In~\cite{pleweSupervisoryModelPredictive2020} a \acrlong{gp} Model with a
\acrlong{rq} Kernel is used for temperature set point optimization.
Gaussian Processes for building control have also been studied before in the
context of Demand Response~\cite{nghiemDatadrivenDemandResponse2017,
jainLearningControlUsing2018}, where the buildings are used for their heat
capacity in order to reduce the stress on energy providers during peak load
times.
There are, however, multiple limitations with these approaches.
In~\cite{nghiemDatadrivenDemandResponse2017} the model is only identified once,
ignoring changes in weather or plant parameters, which could lead to different
dynamics. This is addressed in \cite{jainLearningControlUsing2018} by
re-identifying the model every two weeks using new information. Another
limitation is that of the scalability of the \acrshort{gp}s, which become
prohibitively expensive from a computational point of view when too much data is
added.
Outside of the context of building control, Sparse \acrlong{gp}es have been used
in autonomous racing in order to complement the physics-based model by fitting
the unmodeled dynamics of the
system~\cite{kabzanLearningBasedModelPredictive2019}.
The ability to learn the plant's behaviour in new regions is very helpful in
maintaining model performance over time, as its behaviour starts deviating and
the original identified model goes further and further into the extrapolated
regions.
This project will therefore try to combine the use of online learning schemes
with \acrlong{gp}es by implementing \acrlong{svgp}es, which provide means of
employing \acrshort{gp} Models on larger datasets, and re-training the models
every day at midnight to include all the historically available data.
\clearpage

29
30_Gaussian_Processes_Background.tex
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@ -36,7 +36,7 @@ to the given prior:
\begin{bmatrix} \begin{bmatrix}
\mathbf{f} \\ \mathbf{f} \\
\mathbf{f_*} \\ \mathbf{f_*} \\
\end{bmatrix} = \end{bmatrix} \sim
\mathcal{N}\left( \mathcal{N}\left(
\mathbf{0}, \mathbf{0},
\begin{bmatrix} \begin{bmatrix}
@ -53,7 +53,7 @@ In the case of noisy observations, assuming $y = f + \epsilon$ with $\epsilon
\begin{bmatrix} \begin{bmatrix}
\mathbf{y} \\ \mathbf{y} \\
\mathbf{f_*} \\ \mathbf{f_*} \\
\end{bmatrix} = \end{bmatrix} \sim
\mathcal{N}\left( \mathcal{N}\left(
\mathbf{0}, \mathbf{0},
\begin{bmatrix} \begin{bmatrix}
@ -69,7 +69,7 @@ which, for the rest of the section, will be used in the abbreviated form:
\begin{bmatrix} \begin{bmatrix}
\mathbf{y} \\ \mathbf{y} \\
\mathbf{f_*} \\ \mathbf{f_*} \\
\end{bmatrix} = \end{bmatrix} \sim
\mathcal{N}\left( \mathcal{N}\left(
\mathbf{0}, \mathbf{0},
\begin{bmatrix} \begin{bmatrix}
@ -96,7 +96,7 @@ value that gets maximized is the log of Equation~\ref{eq:gp_likelihood}, the log
marginal likelihood: marginal likelihood:
\begin{equation}\label{eq:gp_log_likelihood} \begin{equation}\label{eq:gp_log_likelihood}
log(p(y)) = - \frac{1}{2}\log{\left( \log(p(y)) = - \frac{1}{2}\log{\left(
\det{\left( \det{\left(
K + \sigma_n^2I K + \sigma_n^2I
\right)} \right)}
@ -172,7 +172,7 @@ $\mathbf{x'}$'s dimensions:
where $w_d = \frac{1}{l_d^2}; d = 1 ,\dots, D$, with D being the dimension of the where $w_d = \frac{1}{l_d^2}; d = 1 ,\dots, D$, with D being the dimension of the
data. data.
The special case of $\Lambda^{-1} = \text{diag}{\left([l_1^{-2},\dots,l_D^{-2}]\right)}$ This special case of $\Lambda^{-1} = \text{diag}{\left([l_1^{-2},\dots,l_D^{-2}]\right)}$
is equivalent to implementing different lengthscales on different regressors. is equivalent to implementing different lengthscales on different regressors.
This can be used to asses the relative importance of each regressor through the This can be used to asses the relative importance of each regressor through the
value of the hyperparameters. This is the \acrfull{ard} property. value of the hyperparameters. This is the \acrfull{ard} property.
@ -207,7 +207,7 @@ without inquiring the penalty of inverting the covariance matrix. An overview
and comparison of multiple methods is given and comparison of multiple methods is given
at~\cite{liuUnderstandingComparingScalable2019}. at~\cite{liuUnderstandingComparingScalable2019}.
For the scope of this project the choice of using the \acrfull{svgp} models has For the scope of this project, the choice of using the \acrfull{svgp} models has
been made, since it provides a very good balance of scalability, capability, been made, since it provides a very good balance of scalability, capability,
robustness and controllability~\cite{liuUnderstandingComparingScalable2019}. robustness and controllability~\cite{liuUnderstandingComparingScalable2019}.
@ -230,9 +230,9 @@ $f(X_s)$, usually denoted as $f_s$ is introduced, with the requirement that this
new dataset has size $n_s$ smaller than the size $n$ of the original dataset. new dataset has size $n_s$ smaller than the size $n$ of the original dataset.
The $X_s$ are called \textit{inducing locations}, and $f_s$ --- \textit{inducing The $X_s$ are called \textit{inducing locations}, and $f_s$ --- \textit{inducing
random variables}. They are said to summarize the data in the sense that a model random variables}. They summarize the data in the sense that a model trained on
trained on this new dataset should be able to generate the original dataset with this new dataset should be able to generate the original dataset with a high
a high probability. probability.
The multivariate Gaussian distribution is used to establish the relationship The multivariate Gaussian distribution is used to establish the relationship
between $f_s$ and $f$, which will serve the role of the prior, now called the between $f_s$ and $f$, which will serve the role of the prior, now called the
@ -242,7 +242,7 @@ sparse prior:
\begin{bmatrix} \begin{bmatrix}
\mathbf{f}(X) \\ \mathbf{f}(X) \\
\mathbf{f}(X_s) \\ \mathbf{f}(X_s) \\
\end{bmatrix} = \end{bmatrix} \sim
\mathcal{N}\left( \mathcal{N}\left(
\mathbf{0}, \mathbf{0},
\begin{bmatrix} \begin{bmatrix}
@ -267,8 +267,8 @@ computationally tractable on larger sets of data.
The following derivation of the \acrshort{elbo} is based on the one presented The following derivation of the \acrshort{elbo} is based on the one presented
in~\cite{yangUnderstandingVariationalLower}. in~\cite{yangUnderstandingVariationalLower}.
Assume $X$ to be the observations, and $Z$ the set of hidden (latent) Assume $X$ to be the observations, and $Z$ the set parameters of the
variables --- the parameters of the \acrshort{gp} model. The posterior \acrshort{gp} model, also known as the latent variables. The posterior
distribution of the hidden variables can be written as follows: distribution of the hidden variables can be written as follows:
\begin{equation} \begin{equation}
@ -337,7 +337,10 @@ The \acrfull{noe} uses the past predictions $\hat{y}$ for future predictions:
f(w(k-1),\dots,w(k-l_w),\hat{y}(k-1),\dots,\hat{y}(k-l_y),u(k-1),\dots,u(k-l_u)) f(w(k-1),\dots,w(k-l_w),\hat{y}(k-1),\dots,\hat{y}(k-l_y),u(k-1),\dots,u(k-l_u))
\end{equation} \end{equation}
The \acrshort{noe} structure is therefore a \textit{simulation model}. Due to its use for multi-step ahead simulation of system behaviour, as opposed
to only predicting one state ahead using current information, the \acrshort{noe}
structure can be considered a \textit{simulation model}.
In order to get the best simulation results from a \acrshort{gp} model, the In order to get the best simulation results from a \acrshort{gp} model, the
\acrshort{noe} structure would have to be employed. Due to the high algorithmic \acrshort{noe} structure would have to be employed. Due to the high algorithmic

52
40_CARNOT_model.tex
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@ -94,7 +94,7 @@ Table~\ref{tab:GIMP_measurements}:
\hline \hline
Object & Size [px] & Size[mm] & Size[m]\\ Object & Size [px] & Size[mm] & Size[m]\\
\hline \hline \hline \hline
acrshort{hvac} height & 70 & 2100 & 2.1 \\ \acrshort{hvac} height & 70 & 2100 & 2.1 \\
Building height & 230 & 6900 & 6.9 \\ Building height & 230 & 6900 & 6.9 \\
Stem wall & 45 & 1350 & 1.35 \\ Stem wall & 45 & 1350 & 1.35 \\
Dome height & 185 & 5550 & 5.55 \\ Dome height & 185 & 5550 & 5.55 \\
@ -174,13 +174,13 @@ The stem of the \pdome\ is approximated to a cube of edge 25m, and its volume ca
therefore be calculated as: therefore be calculated as:
\begin{equation} \begin{equation}
V_s = l_s^2 * h_s V_s = l_s^2 \times h_s
\end{equation} \end{equation}
The total volume of the building is then given as: The total volume of the building is then given as:
\begin{equation} \begin{equation}
V = V_d + V_s = \frac{1}{6} \pi h (3r^2 + h^2) + l_s^2 * h_s V = V_d + V_s = \frac{1}{6} \pi h (3r^2 + h^2) + l_s^2 \times h_s
\end{equation} \end{equation}
Numerically, considering a dome diameter of 28m, a dome height of 5.55m and a stem Numerically, considering a dome diameter of 28m, a dome height of 5.55m and a stem
@ -199,7 +199,7 @@ Exchange Rate, presented in Section~\ref{sec:Air_Exchange_Rate}.
The main function of the \pdome\ building is serving as a classroom for around The main function of the \pdome\ building is serving as a classroom for around
one hundred students. It has wood furniture consisting of chairs and tables, as one hundred students. It has wood furniture consisting of chairs and tables, as
well as a wooden stage in the center of the building, meant to be used for well as a wooden stage in the center of the building, meant to be used for
presentations. The building also contains a smaller room, housing the all the presentations. The building also contains a smaller room, housing all the
necessary technical equipment (cf. Figure~\ref{fig:Google_Maps_Streetview}). necessary technical equipment (cf. Figure~\ref{fig:Google_Maps_Streetview}).
The most accurate way of including information on the furniture in the CARNOT The most accurate way of including information on the furniture in the CARNOT
@ -233,7 +233,15 @@ has a surface area of:
\begin{equation} \begin{equation}
S_f = \frac{1}{4} \cdot 1.8 \left[\frac{\text{m}^2}{\text{m}^2}\right] S_f = \frac{1}{4} \cdot 1.8 \left[\frac{\text{m}^2}{\text{m}^2}\right]
\cdot 625\ \left[\text{m}^2\right] = 140\ \left[\text{m}^2\right] \cdot 625\ \left[\text{m}^2\right] = 281.25 \left[\text{m}^2\right]
\end{equation}
Since a wall has two large faces, the total surface gets divided between them
for a surface of each face of:
\begin{equation}
S_{f, \text{face}} =
\frac{1}{2} \times S_f \approx 140 \left[\text{m}^2\right]
\end{equation} \end{equation}
\subsubsection*{Mass} \subsubsection*{Mass}
@ -270,8 +278,9 @@ The last parameter of the furniture equivalent wall is computed by dividing its
volume by the surface: volume by the surface:
\begin{equation} \begin{equation}
h_f = \frac{V_f}{S_f} = \frac{10.41}{140} \left[\frac{m^3}{m^2}\right] = h_f = \frac{V_f}{S_{f, \text{face}}}
0.075\ \left[\text{m}\right] = 7.5\ \left[\text{cm}\right] = \frac{10.41}{140} \left[\frac{m^3}{m^2}\right]
= 0.075\ \left[\text{m}\right] = 7.5\ \left[\text{cm}\right]
\end{equation} \end{equation}
@ -284,7 +293,7 @@ parameters for each of the CARNOT nodes' properties.
\subsubsection{Windows} \subsubsection{Windows}
The windows are supposed to be made of two glass panes of thickness 4mm each. The windows are made of two glass panes of thickness 4mm each.
The values of the heat transfer coefficient (U-factor) can vary greatly for The values of the heat transfer coefficient (U-factor) can vary greatly for
different window technologies, but an educated guess can be made on the lower different window technologies, but an educated guess can be made on the lower
@ -345,8 +354,8 @@ Table~\ref{tab:material_properties}:
\centering \centering
\begin{tabular}{||c c c c||} \begin{tabular}{||c c c c||}
\hline \hline
Material & Thermal Conductivity $[k]$ $[\frac{W}{mK}]$ & Specific Heat Material & Thermal Conductivity $k$ $[\frac{W}{mK}]$ & Specific Heat
Capacity $[c]$ $[\frac{J}{kgK}]$ & Density $[\rho]$ $[\frac{kg}{m^3}]$ Capacity $c$ $[\frac{J}{kgK}]$ & Density $\rho$ $[\frac{kg}{m^3}]$
\\ \\
\hline \hline \hline \hline
Plywood & 0.12 & 1210 & 540 \\ Plywood & 0.12 & 1210 & 540 \\
@ -375,7 +384,7 @@ the cases where the full cooling capacity is not required.
\subsubsection*{Ventilation} \subsubsection*{Ventilation}
According to the manufacturer manual \cite{aermecRoofTopManuelSelection}, the According to the manufacturer manual \cite{aermecRoofTopManuelSelection}, the
\acrshort{hvac} unit's external fan has an air debit ranging between 4900 \acrshort{hvac} unit's external fan has an air flow ranging between 4900
$\text{m}^3/\text{h}$ and 7000 $\text{m}^3/\text{h}$. $\text{m}^3/\text{h}$ and 7000 $\text{m}^3/\text{h}$.
\subsubsection*{Air Exchange Rate}\label{sec:Air_Exchange_Rate} \subsubsection*{Air Exchange Rate}\label{sec:Air_Exchange_Rate}
@ -467,12 +476,13 @@ compressors. The instruction manual of the
\acrshort{hvac}~\cite{aermecRoofTopManuelSelection} notes that in summer only \acrshort{hvac}~\cite{aermecRoofTopManuelSelection} notes that in summer only
one of the compressors is running. This allows for a larger \acrshort{eer} value one of the compressors is running. This allows for a larger \acrshort{eer} value
and thus better performance. We can see that this is the case for most of the and thus better performance. We can see that this is the case for most of the
experiment, where the power consumption caps at around 6~kW. There are, however, experiment, where the power consumption caps at around 6~kW, which is the
moments during the first part of the experiment where the power momentarily maximum power consumption of one compressor. There are, however, moments during
peaks over the 6~kW limit, and goes as high as around 9~kW. This most probably the first part of the experiment where the power momentarily peaks over the 6~kW
happens when the \acrshort{hvac} decides that the difference between the set limit, and goes as high as around 9~kW. This most probably happens when the
point temperature and the actual measured values is too large to compensate with \acrshort{hvac} decides that the difference between the set point temperature
a single compressor. and the actual measured values is too large to compensate with a single
compressor.
Figure~\ref{fig:Polydome_exp7_settemp} presents the values of the set point Figure~\ref{fig:Polydome_exp7_settemp} presents the values of the set point
temperature and the measured internal temperature. temperature and the measured internal temperature.
@ -506,7 +516,7 @@ pressure, wind speed and direction, etc. A detailed overview of each
measurement necessary for a simulation is given in the CARNOT user measurement necessary for a simulation is given in the CARNOT user
manual~\cite{CARNOTManual}. manual~\cite{CARNOTManual}.
In order to compare the CARNOT model's performance to that of the real \pdome\, In order to compare the CARNOT model's performance to that of the real \pdome,
it is necessary to simulate the CARNOT model under the same set of conditions as it is necessary to simulate the CARNOT model under the same set of conditions as
the ones present during the experimental data collection. In order to do this, the ones present during the experimental data collection. In order to do this,
all the missing values that are required by the simulation have to be filled. In all the missing values that are required by the simulation have to be filled. In
@ -540,12 +550,12 @@ All the other parameters related to solar irradiance, such as the in-plane
irradiance components, in-plane diffuse irradiance from the sky and the ground irradiance components, in-plane diffuse irradiance from the sky and the ground
are computed using the Python pvlib. are computed using the Python pvlib.
The values that cannot be either calculated or approximated from the available The values that can be neither calculated nor approximated from the available
data, such as the precipitation, wind direction, incidence angles in place of data, such as the precipitation, wind direction, incidence angles in place of
vertical and main/secondary surface axis, have been replaced with the default vertical and main/secondary surface axis, have been replaced with the default
CARNOT placeholder value of -9999. The relative humidity, cloud index, pressure CARNOT placeholder value of -9999. The relative humidity, cloud index, pressure
and wind speed have been fixed to 50\%, 0.5, 96300 Pa, 0 $\text{m}/\text{s}$ and wind speed have been fixed to the default CARNOT values of 50\%, 0.5, 96300
respectively. Pa, 0 $\text{m}/\text{s}$ respectively.
\subsubsection{\pdome\ and CARNOT model comparison}\label{sec:CARNOT_comparison} \subsubsection{\pdome\ and CARNOT model comparison}\label{sec:CARNOT_comparison}

59
50_Choice_of_Hyperparameters.tex
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@ -1,4 +1,4 @@
\section{Choice of Hyperparameters} \section{Choice of Hyperparameters}~\label{sec:hyperparameters}
This section will discuss and try to validate the choice of all the This section will discuss and try to validate the choice of all the
hyperparameters necessary for the training of a \acrshort{gp} model to capture hyperparameters necessary for the training of a \acrshort{gp} model to capture
@ -9,16 +9,14 @@ behaviour directly from data. This comes in contrast to white-box and grey-box
modelling techniques, which require much more physical insight into the plant's modelling techniques, which require much more physical insight into the plant's
behaviour. behaviour.
The advantage of black-box models lies in the lack of physical parameters to be On the flip side, this versatility comes at the cost of having to properly
fitted. On the flip side, this versatility of being able to fit much more define the model hyperparameters: the number of regressors, the number of
complex models purely on data comes at the cost of having to properly define the autoregressive lags for each class of inputs, the shape of the covariance
model hyperparameters: the number of regressors, the number of autoregressive function have to be taken into account when designing a \acrshort{gp} model.
lags for each class of inputs, the shape of the covariance function have to be These choices have direct influence on the resulting model behaviour and where
taken into account when designing a \acrshort{gp} model. These choices have it can be generalized, as well as indirect influence in the form of more time
direct influence on the resulting model behaviour and where it can be consuming computations in the case of larger number of regressors and more
generalized, as well as indirect influence in the form of more time consuming complex kernel functions.
computations in the case of larger number of regressors and more complex kernel
functions.
As described in Section~\ref{sec:gp_dynamical_system}, for the purpose of this As described in Section~\ref{sec:gp_dynamical_system}, for the purpose of this
project, the \acrlong{gp} model will be trained using the \acrshort{narx} project, the \acrlong{gp} model will be trained using the \acrshort{narx}
@ -64,7 +62,12 @@ always benefit the \acrshort{gp} model at least not comparably to the
additional computational complexity. additional computational complexity.
For the exogenous inputs the choice has therefore been made to take the For the exogenous inputs the choice has therefore been made to take the
\textit{Global Solar Irradiance} and \textit{Outside Temperature Measurement}. \textit{Global Solar Irradiance}\footnotemark and \textit{Outside Temperature
Measurement}.
\footnotetext{Using the \acrshort{ghi} makes sense in the case of the \pdome\
building which has windows facing all directions, as opposed to a room which
would have windows on only one side.}
\subsection{The Kernel} \subsection{The Kernel}
@ -104,7 +107,7 @@ three lengthscales apart.
\begin{tabular}{||c c ||} \begin{tabular}{||c c ||}
\hline \hline
$\norm{\mathbf{x} - \mathbf{x}'}$ & $\norm{\mathbf{x} - \mathbf{x}'}$ &
$\exp{(-\frac{1}{2}*\frac{\norm{\mathbf{x} - \mathbf{x}'}^2}{l^2})}$ \\ $\exp{(-\frac{1}{2}\times\frac{\norm{\mathbf{x} - \mathbf{x}'}^2}{l^2})}$ \\
\hline \hline \hline \hline
$1l$ & 0.606 \\ $1l$ & 0.606 \\
$2l$ & 0.135 \\ $2l$ & 0.135 \\
@ -133,6 +136,8 @@ and the variance for different combinations of the exogenous input lags ($l_w$),
the controlled input lags ($l_u$) and the output lags ($l_y$) for a classical the controlled input lags ($l_u$) and the output lags ($l_y$) for a classical
\acrshort{gp} model. \acrshort{gp} model.
% TODO: [Lengthscales] Explain lags and w1,1, w2, etc.
\begin{table}[ht] \begin{table}[ht]
%\vspace{-8pt} %\vspace{-8pt}
\centering \centering
@ -165,15 +170,15 @@ the controlled input lags ($l_u$) and the output lags ($l_y$) for a classical
\label{tab:GP_hyperparameters} \label{tab:GP_hyperparameters}
\end{table} \end{table}
In general, the results of Table~\ref{tab:GP_hyperparameters} show that the In general, the results of Table~\ref{tab:GP_hyperparameters} show that the past
past outputs are important when predicting future values. Of importance is also outputs are important when predicting future values. Of importance is also the
the past inputs, with the exception of the models with very high variance, where past inputs, with the exception of the models with very high variance, where the
the relative importances stay almost constant across all the inputs. For the relative importances stay almost constant across all the inputs. For example, it
exogenous inputs, the outside temperature ($w2$) is generally more important can be seen that for the exogenous inputs, the outside temperature ($w2$) is
than the solar irradiation ($w1$). In the case of more autoregressive lags for generally more important than the solar irradiation ($w1$). In the case of more
the exogenous inputs the more recent information is usually more important in autoregressive lags for the exogenous inputs the more recent information is
the case of the solar irradiation, while the second-to-last measurement is usually more important in the case of the solar irradiation, while the
preffered for the outside temperature. second-to-last measurement is preffered for the outside temperature.
For the classical \acrshort{gp} model the appropriate choice of lags would be For the classical \acrshort{gp} model the appropriate choice of lags would be
$l_u = 1$ and $l_y = 3$ with $l_w$ taking the values of either 1, 2 or 3, $l_u = 1$ and $l_y = 3$ with $l_w$ taking the values of either 1, 2 or 3,
@ -348,8 +353,6 @@ metrics, as well as being cheaper from a computational perspective. In order to
make a more informed choice for the best hyperparameters, the simulation make a more informed choice for the best hyperparameters, the simulation
performance of all three combinations has been analysed. performance of all three combinations has been analysed.
\clearpage
\begin{table}[ht] \begin{table}[ht]
%\vspace{-8pt} %\vspace{-8pt}
\centering \centering
@ -381,7 +384,6 @@ other combinations scoring much worse on the \acrshort{msll} and \acrshort{lpd}
loss functions. This has, therefore, been chosen as the model for the full year loss functions. This has, therefore, been chosen as the model for the full year
simulations. simulations.
\subsection{Validation of hyperparameters}\label{sec:validation_hyperparameters} \subsection{Validation of hyperparameters}\label{sec:validation_hyperparameters}
The validation step has the purpose of testing the viability of the trained The validation step has the purpose of testing the viability of the trained
@ -449,10 +451,13 @@ this proves to be the best simulation model.
Lastly, \model{3}{1}{3} has a much worse simulation performance than the other Lastly, \model{3}{1}{3} has a much worse simulation performance than the other
two models. This could hint at an over fitting of the model on the training data. two models. This could hint at an over fitting of the model on the training data.
\clearpage
This is consistent with the results found in Table~\ref{tab:GP_loss_functions} This is consistent with the results found in Table~\ref{tab:GP_loss_functions}
for the \acrshort{rmse} and \acrshort{smse}, as well as can be seen in for the \acrshort{rmse} and \acrshort{smse}, as well as can be seen in
Appendix~\ref{apx:hyperparams_gp}, Figure~\ref{fig:GP_313_test_validation}, Appendix~\ref{apx:hyperparams_gp}, Figure~\ref{fig:GP_313_test_validation},
where the model has much worse performance on the testing dataset predictions where \model{3}{1}{3} has much worse performance on the testing dataset predictions
than the other two models. than the other two models.
The performance of the three models in simulation mode is consistent with the The performance of the three models in simulation mode is consistent with the
@ -471,8 +476,6 @@ while still having good performance on all loss functions. In implementation,
however, this model turned out to be very unstable, and the more conservative however, this model turned out to be very unstable, and the more conservative
\model{1}{1}{3} model was used instead. \model{1}{1}{3} model was used instead.
\clearpage
\subsubsection{Sparse and Variational Gaussian Process} \subsubsection{Sparse and Variational Gaussian Process}
For the \acrshort{svgp} models, only the performance of \model{1}{2}{3} was For the \acrshort{svgp} models, only the performance of \model{1}{2}{3} was

5
60_The_MPC_Problem.tex
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@ -1,6 +1,6 @@
\section{The MPC Problem}\label{sec:mpc_problem} \section{The MPC Problem}\label{sec:mpc_problem}
The \acrlong{ocp} to be solved was chosen in such a way as to make The \acrlong{ocp} to be solved was chosen in such a way as to make the
analysis of the models' performances more straightforward. The objective is analysis of the models' performances more straightforward. The objective is
tracking a defined reference temperature as close as possible, while ensuring tracking a defined reference temperature as close as possible, while ensuring
the heat input stays within the HVAC capacity. The \textit{zero-variance} method the heat input stays within the HVAC capacity. The \textit{zero-variance} method
@ -42,7 +42,8 @@ buildings based on the rolling 48h average outside temperature.
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
\includegraphics[width = \textwidth]{Images/sia_180_2014.png} \includegraphics[width = \textwidth]{Images/sia_180_2014.png}
\caption{The SIA 180:2014 norm for residential building temperatures} \caption{The SIA 180:2014 norm for residential building
temperatures~\cite{sia180:2014ProtectionThermiqueProtection2014}}
\label{fig:sia_temperature_norm} \label{fig:sia_temperature_norm}
\end{figure} \end{figure}

18
70_Implementation.tex
View file

@ -21,13 +21,13 @@ in the \acrshort{wdb} object is given in Section~\ref{sec:CARNOT_WDB}.
\subsection{Simulink Model} \subsection{Simulink Model}
The secondary functions of the Simulink model is the weather prediction, as well The secondary functions of the Simulink model is the weather prediction, as well
as communication with the Python controller. A complete schema of the Simulink as communication with the Python controller. A complete schematic of the
setup is presented in Figure~\ref{fig:Simulink_complete}. Simulink setup is presented in Figure~\ref{fig:Simulink_complete}.
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
\includegraphics[width = 0.75\textwidth]{Images/polydome_python.pdf} \includegraphics[width = 0.75\textwidth]{Images/polydome_python.pdf}
\caption{Simulink Schema of the Complete Simulation} \caption{Simulink diagram of the Complete Simulation}
\label{fig:Simulink_complete} \label{fig:Simulink_complete}
\end{figure} \end{figure}
@ -158,7 +158,7 @@ Let $w_l$, $u_l$, and $y_l$ be the lengths of the state vector components
$\mathbf{w}$, $\mathbf{u}$, $\mathbf{y}$ (cf. Equation~\ref{eq:components}). $\mathbf{w}$, $\mathbf{u}$, $\mathbf{y}$ (cf. Equation~\ref{eq:components}).
Also, let X be the matrix of all the system states over the optimization horizon Also, let X be the matrix of all the system states over the optimization horizon
and W be the matrix of the predicted disturbances for all the future steps. The and W be the matrix of the predicted disturbances for all the future steps. The
original \acrlong{ocp} can be rewritten as: original \acrlong{ocp} can be rewritten using index notation as:
\begin{subequations}\label{eq:sparse_optimal_control_problem} \begin{subequations}\label{eq:sparse_optimal_control_problem}
\begin{align} \begin{align}
@ -174,7 +174,7 @@ original \acrlong{ocp} can be rewritten as:
\end{align} \end{align}
\end{subequations} \end{subequations}
\subsection{Python server} \subsection{Model Identification and Update using a Python server}
The Python server is responsible for the control part of the simulation. It The Python server is responsible for the control part of the simulation. It
delegates which controller is active, is responsible for training and updating delegates which controller is active, is responsible for training and updating
@ -182,16 +182,16 @@ the \acrshort{gp} and \acrshort{svgp} models, as well as keeping track of all
the intermediate results for analysis. the intermediate results for analysis.
In the beginning of the experiment there is no information available on the In the beginning of the experiment there is no information available on the
building's thermal behaviour. For this part of the simulation, the controller building's thermal behaviour. For this part of the simulation, the server
switches to a \acrshort{pi} controller with random disturbances until it gathers switches to a \acrshort{pi} controller with random disturbances until it gathers
enough data to train a \acrshort{gp} model. This ensured that the building is enough data to train a \acrshort{gp} model. This ensures that the building is
sufficiently excited to capture its dynamics, while maintaining the temperature sufficiently excited to capture its dynamics, while maintaining the temperature
within an acceptable range (~15 --- 25 $\degree$C). within an acceptable range (~15 --- 25 $\degree$C).
Once enough data has been captured all the features are first scaled to the Once enough data has been captured all the features are first scaled to the
range [-1, 1] in order to reduce the possibility of numerical instabilities. The range [-1, 1] in order to reduce the possibility of numerical instabilities. The
Python controller then trains the \acrshort{gp} model and switches to tracking Python server then trains the \acrshort{gp} model and switches to tracking the
the appropriate SIA 180:2014 reference temperature (cf. appropriate SIA 180:2014 reference temperature (cf.
Section~\ref{sec:reference_temperature}). Section~\ref{sec:reference_temperature}).
For the case of the \acrshort{svgp}, a new model is trained once enough data is For the case of the \acrshort{svgp}, a new model is trained once enough data is

42
80_Results.tex
View file

@ -1,17 +1,17 @@
\section{Results}\label{sec:results} \section{Full-year Simulation Results}\label{sec:results}
This section focuses on the presentation and interpretation of the year-long This section focuses on the presentation and interpretation of the year-long
simulation of the control schemes present previously. All the control schemes simulation of the control schemes presented previously. All the control schemes
analysed in this Section have been done with a sampling time of 15 minutes and a analysed in this Section have used a sampling time of 15 minutes and a control
control horizon of 8 steps. horizon of 8 steps.
Section~\ref{sec:GP_results} analyses the results of a conventional Section~\ref{sec:GP_results} analyses the results of a conventional
\acrlong{gp} Model trained on the first five days of gathered data. The models \acrlong{gp} Model trained on the first five days of gathered data. The model
is then used for the rest of the year, with the goal of tracking the defined is then used for the rest of the year, with the goal of tracking the defined
reference temperature. reference temperature.
Section~\ref{sec:SVGP_results} goes into details on the analysis of the Learning Section~\ref{sec:SVGP_results} goes into details on the analysis of the learning
scheme using a \acrshort{svgp} Model. In this scenario, the model is first scheme using a \acrshort{svgp} Model. In this scenario, the model is first
trained on the first five days of data, and updates every day at midnight with trained on the first five days of data, and updates every day at midnight with
the new information gathered from closed-loop operation. the new information gathered from closed-loop operation.
@ -19,7 +19,7 @@ the new information gathered from closed-loop operation.
\subsection{Conventional Gaussian Processes}\label{sec:GP_results} \subsection{Conventional Gaussian Processes}\label{sec:GP_results}
The first simulation, to be used as a baseline comparison with the The first simulation, to be used as a baseline comparison with the
\acrshort{svgp} Models developed further consists of using a `static' \acrshort{svgp} Models developed further, consists of using a `static'
\acrshort{gp} model trained on five days worth of experimental data. This model \acrshort{gp} model trained on five days worth of experimental data. This model
is then employed for the rest of the year. is then employed for the rest of the year.
@ -94,7 +94,7 @@ $\degree$C respectively.
\end{figure} \end{figure}
This large difference of performance could be explained by the change in outside This large difference of performance could be explained by the change in outside
weather (Solar Irradiance and Outside Temperature --- the exogenous inputs) from weather (solar irradiance and outside temperature --- the exogenous inputs) from
the one present during the training phase. It can be seen in the one present during the training phase. It can be seen in
Figure~\ref{fig:Dataset_outside_temperature} that already at 500 points in the Figure~\ref{fig:Dataset_outside_temperature} that already at 500 points in the
simulation both the GHI and the Outside Temperature are outside of the training simulation both the GHI and the Outside Temperature are outside of the training
@ -143,11 +143,11 @@ at midnight using the newly accumulated data from closed-loop operation.
The results of this setup are presented in The results of this setup are presented in
Figure~\ref{fig:SVGP_fullyear_simulation}. It can already be seen that this Figure~\ref{fig:SVGP_fullyear_simulation}. It can already be seen that this
setup performs much better than the initial one. The only large deviations from setup performs much better than the initial one. The only large deviations from
the reference temperature are due to cold --- when the \acrshort{hvac}'s limited the reference temperature are due to cold weather, when the \acrshort{hvac}'s
heat capacity is unable to maintain the proper temperature. Additionnaly, the limited heat capacity is unable to maintain the proper temperature.
\acrshort{svgp} controller takes around 250-300ms of computation time for each Additionnaly, the \acrshort{svgp} controller takes around 250 - 300ms of
simulation time, decreasing the computational cost of the original \acrshort{gp} computation time for each simulation time, decreasing the computational cost of
by a factor of six. the original \acrshort{gp} by a factor of six.
@ -161,7 +161,7 @@ by a factor of six.
\clearpage \clearpage
Comparing the Absolute Error of the Measured vs Reference temperature for the Comparing the absolute error of the measured vs reference temperature for the
duration of the experiment (cf. Figure~\ref{fig:SVGP_fullyear_abserr}) with the duration of the experiment (cf. Figure~\ref{fig:SVGP_fullyear_abserr}) with the
one of the original experiment, the average absolute error is reduced from 1.33 one of the original experiment, the average absolute error is reduced from 1.33
$\degree$C to only 0.05 $\degree$C, with the majority of the values being lower $\degree$C to only 0.05 $\degree$C, with the majority of the values being lower
@ -210,7 +210,7 @@ behaviour of the plant over all the experimental steps in the first two cases.
It still has a noticeable error when predicting the behaviour of the plant on It still has a noticeable error when predicting the behaviour of the plant on
new data (i.e. simulations starting at steps 10750 and 11000), but it is much new data (i.e. simulations starting at steps 10750 and 11000), but it is much
less than before. This gives a hint at the fact that the \acrshort{svgp} model's less than before. This gives a hint at the fact that the \acrshort{svgp} model's
performance ameliorates throughout the year, but it does require much more data performance improves throughout the year, but it does require much more data
than the classical \acrshort{gp} model to capture the building dynamics. than the classical \acrshort{gp} model to capture the building dynamics.
\begin{figure}[ht] \begin{figure}[ht]
@ -361,10 +361,10 @@ Analyzing the results of a simulation done on only one day's worth of initial
simulation data (cf. Figures~\ref{fig:SVGP_96pts_fullyear_simulation} simulation data (cf. Figures~\ref{fig:SVGP_96pts_fullyear_simulation}
and~\ref{fig:SVGP_96pts_abserr}) it is very notable that the model performs and~\ref{fig:SVGP_96pts_abserr}) it is very notable that the model performs
almost identically to the one identified in the previous sections. This almost identically to the one identified in the previous sections. This
nightlights one of the practical benefits of the \acrshort{svgp} implementations highlights one of the practical benefits of the \acrshort{svgp} implementations
compared to the classical \acrlong{gp} -- it is possible to start with a more compared to the classical \acrlong{gp} -- it is possible to start with a rougher
rough controller trained on less data and refine it over time, reducing the need controller trained on less data and refine it over time, reducing the need for
for cumbersome and potentially costly initial experiments for gathering data. cumbersome and potentially costly initial experiments for gathering data.
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
@ -400,7 +400,7 @@ based on closed-loop operation.
\end{figure} \end{figure}
As it can be seen in Figure~\ref{fig:SVGP_480window_fullyear_simulation}, this As it can be seen in Figure~\ref{fig:SVGP_480window_fullyear_simulation}, this
model is unable to properly track the reference temperature. In fact, five days model is unable to exhaustively track the reference temperature. In fact, five days
after the identification, the model forgets all the initial data and becomes after the identification, the model forgets all the initial data and becomes
unstable. This instability then generates enough excitation of the plant for the unstable. This instability then generates enough excitation of the plant for the
model, to again learn its behaviour. This cycle repeats every five days, when the model, to again learn its behaviour. This cycle repeats every five days, when the
@ -470,7 +470,7 @@ models can be deployed with less explicit identification data, but they will
continue to improve over the course of the year, as the building passes through continue to improve over the course of the year, as the building passes through
different regions of the state space and more data is collected. different regions of the state space and more data is collected.
These results do not, however, discredit the use of \acrlong{gp} for employment However, these results do not discredit the use of \acrlong{gp} for employment
in a multi-seasonal situation. As shown before, given the same amount of data in a multi-seasonal situation. As shown before, given the same amount of data
and ignoring the computational cost, they perform better than the alternative and ignoring the computational cost, they perform better than the alternative
\acrshort{svgp} models. The bad initial performance could be mitigated by \acrshort{svgp} models. The bad initial performance could be mitigated by

85
90_Conclusion.tex Normal file
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@ -0,0 +1,85 @@
\section{Conclusion}~\label{sec:conclusion}
The aim of this project was to analyse the performance of \acrshort{gp} based
controllers for use in longer lasting implementations, where differences in
building behaviour become important compared to the initially available data.
{\color{red}
First, the performance of a classical \acrshort{gp} model trained on 5 days
worth of experimental data was analysed. This model turned out to be unable to
correctly extrapolate building behaviour as the weather changed throughout the
year.
}
First, the performance of a classical \acrshort{gp} model trained on five days
worth of experimental data was analysed. Initially, this model performed very
well both in one step ahead prediction and multi-step ahead simulation over new,
unseen, data. With the change in weather, however, the model shifted
Several \acrshort{svgp} implementations were then analysed. They turned out to
provide important benefits over the classical models, such as the ability to
easily scale when new data is being added and the much reduced computational
effort required. They do however present some downsides, namely increasing the
number of hyperparameters by having to choose the number of inducing locations,
as well as performing worse than then classical \acrshort{gp} implementation
given the same amount of data.
%Finally, the possible improvements to the current implementations have been
%addressed, noting that classical \acrshort{gp} implementations could also be
%adapted to the \textit{learning control} paradigm, even if their implementation
%could turn out to be much more involved and more computationally expensive than
%the \acrshort{svgp} alternative.
\subsection{Further Research}~\label{sec:further_research}
Section~\ref{sec:results} has presented and compared the results of a full-year
simulation for a classical \acrshort{gp} model, as well as a few incarnations of
\acrshort{svgp} models. The results show that the \acrshort{svgp} have much
better performance, mainly due to the possibility of updating the model
throughout the year. The \acrshort{svgp} models also present a computational
cost advantage both in training and in evaluation, due to several approximations
shown in Section~\ref{sec:gaussian_processes}.
Focusing on the \acrlong{gp} models, there could be several ways of improving
its performance, as noted previously: a more varied identification dataset and
smart update of a fixed-size data dictionary according to information gain,
could mitigate the present problems.
Using a Sparse \acrshort{gp} without replacing the maximum log likelihood
with the \acrshort{elbo} could improve performance of the \acrshort{gp} model at
the expense of training time.
An additional change that could be made is inclusion of the most amount of prior
information possible through setting a more refined kernel, as well as adding
prior information on all the model hyperparameters when available. This approach
however goes against the `spirit' of black-box approaches, since significant
insight into the physics of the plant is required in order to properly model and
implement this information.
On the \acrshort{svgp} side, several changes could also be proposed, which were
not properly addressed in this work. First, the size of the inducing dataset was
chosen experimentally until it was found to accurately reproduce the manually
collected experimental data. In order to better use the available computational
resources, this value could be found programmatically in a way to minimize
evaluation time, while still providing good performance. Another possibility is
the periodic re-evaluation of this value when new data comes in, since as more
and more data is collected the model becomes more complex, and in general more
inducing locations could be necessary to properly reproduce the training data.
Finally, none of the presented controllers take into account occupancy rates or
adapt to possible changes in the real building, such as adding or removing
furniture, deteriorating insulation and so on. The presented update methods only
deals with adding information on behaviour in different state space regions, i.e
\textit{learning}. Additionally, their ability to \textit{adapt} to changes in
the actual plant's behaviour should be further addressed.
\section*{Acknowledgements}
I would like to thank Manuel Koch for the great help provided during the course
of the project starting from the basics on CARNOT modelling, to helping me
better compare the performance of different controllers, as well as Prof.\ Colin
Jones, whose insights were always very guiding, while still allowing me to
discover everything on my own.
\clearpage

4
99A_GP_hyperparameters_validation.tex
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@ -43,14 +43,14 @@
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
\includegraphics[width = \textwidth]{Plots/GP_313_training_performance.pdf} \includegraphics[width = \textwidth]{Plots/GP_313_training_performance.pdf}
\caption{Prediction performance of \model{1}{1}{3} on the training dataset} \caption{Prediction performance of \model{3}{1}{3} on the training dataset}
\label{fig:GP_313_train_validation} \label{fig:GP_313_train_validation}
\end{figure} \end{figure}
\begin{figure}[ht] \begin{figure}[ht]
\centering \centering
\includegraphics[width = \textwidth]{Plots/GP_313_test_performance.pdf} \includegraphics[width = \textwidth]{Plots/GP_313_test_performance.pdf}
\caption{Prediction performance of \model{1}{1}{3} on the test dataset} \caption{Prediction performance of \model{3}{1}{3} on the test dataset}
\label{fig:GP_313_test_validation} \label{fig:GP_313_test_validation}
\end{figure} \end{figure}

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7
main.tex
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@ -98,6 +98,7 @@ temperature control}
\usepackage{fancyhdr} \usepackage{fancyhdr}
\pagestyle{fancy} \pagestyle{fancy}
\setlength\headheight{35pt} \setlength\headheight{35pt}
\setlength\footskip{13.6pt}
\fancyhf{Multi-season GP performance for buildings} \fancyhf{Multi-season GP performance for buildings}
\rhead{\includegraphics[width=2cm]{Logo-EPFL.png}} \rhead{\includegraphics[width=2cm]{Logo-EPFL.png}}
\lhead{} \lhead{}
@ -127,16 +128,14 @@ temperature control}
\clearpage \clearpage
\input{10_Introduction.tex} \input{10_Introduction.tex}
\input{20_Previous_Research.tex}
\input{30_Gaussian_Processes_Background.tex} \input{30_Gaussian_Processes_Background.tex}
\input{40_CARNOT_model.tex} \input{40_CARNOT_model.tex}
\input{50_Choice_of_Hyperparameters.tex} \input{50_Choice_of_Hyperparameters.tex}
\input{60_The_MPC_Problem.tex} \input{60_The_MPC_Problem.tex}
\input{70_Implementation.tex} \input{70_Implementation.tex}
\input{80_Results.tex} \input{80_Results.tex}
\input{90_Further_Research.tex} \input{90_Conclusion.tex}
\input{95_Conclusion.tex} \printbibliography[heading=bibintoc]
\printbibliography
\appendix \appendix
\input{99A_GP_hyperparameters_validation.tex} \input{99A_GP_hyperparameters_validation.tex}
\input{99C_hyperparameters_results.tex} \input{99C_hyperparameters_results.tex}