**Stochastic Distributed Model Predictive Control for Energy Efficient Building: A Second Order Cone Programming Approach**

Advisors : Asso. Prof. Hoai-Nam Nguyen, Prof. Nel Samama – Electronics and Physics Department, Telecom-SudParis, Institut Polytechnique de Paris (hoai-nam.nguyen@telecom-sudparis.eu, nel.samama@telecom-sudparis.eu )

**1. ****Context**

Buildings nowadays contribute to roughly 40% of the global energy use, of which a large portion is used for heating, ventilation, and air-conditioning (HVAC) [1]. Thus, energy savings become a priority in the design and operation of modern HVAC systems. Several studies shown that advanced HVAC control can notably mitigate greenhouse gas emissions and reduce energy use with average energy savings of 13% to 28% [2].

The majority of buildings today still adopt simple rule-based control techniques with only limited energy saving capabilities [2]. The digital age comes with decreasing costs in sensing and computation, which is paving the way for the adoption of advanced control strategies, such as model predictive control (MPC). In the last decade, MPC has become a dominant control strategy in research on energy efficient building. The strength of MPC lies in the use of a mathematical model of the building to predict its future behavior. Using these predictions, MPC can optimally select the control actions based on a given objective while taking into account the comfort and technological constraints, and weather forecasts in a systematic and flexible way [3, 4].

Despite the abundance of research papers, MPC is still in its early stages. There are two main challenges in using MPC for building control.

1) MPC requires a relatively good mathematical building model. It is well known that this is a difficult and time consuming problem.

2) Stochastic MPC is generally considered in building control due to the uncertainties in the weather forecasts, and in the environment such as occupancy presence. However at the moment, the computational complexity of stochastic MPC is very important especially for medium/large size buildings. This has limited the application of MPC only to small size buildings.

**2. ****Objectives**

The main objective of this PhD thesis is to design a real-time optimal control law for energy efficient building. The aim is to maximize the mean and to minimize the covariance matrix of the performance index. For this purpose, stochastic distributed MPC framework will be adopted. We use stochastic MPC to cope with the uncertainties in the building systems, as well as with the time-varying electricity price. We use distributed MPC to deal with the fact that building is a large scale multi-input and multi-output system.

It is well known that the stochastic MPC optimization problem can be formulated as a second order cone program (SOCP). The traditional way to solve a SOCP problem is to employ a primal-dual interior point method (IPM), which has attractive polynomial time complexity. However, the IPM requires sophisticated and heavy computational tasks to be performed at each iteration, for example, solving Newton’s type systems. For real-time embedded SOCP problems, a single iteration of such a polynomial time algorithm is often too expensive to be of practical use. Hence, the classical way to tackle the stochastic MPC problem is to combine the scenario-based approach with the quadratic programming (QP). In the result, a very large scale QP problem that needs to solve online is obtained.

In this PhD, we will follow the SOCP approach. To avoid the high online computational complexity of the primal-dual interior point method, we will use the first-order splitting approaches, similar to the one that was recently proposed in [5] for the quadratically constrained quadratic program. The main idea is to exploit the inner structures of the optimization problem.

**3. ****Methods**

** **The PhD consists of 3 phases.

In Phase 1, we aim to design a stochastic centralized MPC algorithm. The main purpose here is to analyze the tradeoff between the performance and the computational complexity of the stochastic MPC. In this phase, we will also propose a SOCP solver for the MPC optimization problem.

In Phase 2, we aim to design the stochastic distributed MPC algorithm. We will also consider the problem of implementing a SOCP solver in embedded systems. This is generally a challenging due to the low computational and memory resources of the embedded systems. The results in Phase 1 will be used to take advantages of inner structures of the optimization algorithms.

In Phase 3, we validate the proposed methods on a real system. This phase is structured in two steps.

+) Step 1 : validate the algorithms on a simulated systems and the external hardware (hardware in the loop). In this case, we can have a complete control environment.

+) Step 2 : validate the algorithms on a real system.

**4. ****Expected Results**

+) A SOCP Matlab/Simulink Toolbox.

+) A demontration of a real time implementation of the stochastic distributed MPC algorithms in a low-cost microcontroller/microprocessor.

+) A comparison between the new methods and currently used methods.

**5. **** References**

[1] Manfren, M., Nastasi, B., Tronchin, L., Groppi, D., & Garcia, D. A. (2021). « *Techno-economic analysis and energy modelling as a key enablers for smart energy services and technologies in buildings »*. Renewable and Sustainable Energy Reviews, *150*, 111490.

[2] Drgoňa, J., Arroyo, J., Figueroa, I. C., Blum, D., Arendt, K., Kim, D., & Helsen, L. (2020). « *All you need to know about model predictive control for buildings »*. Annual Reviews in Control, 50, 190-232.

[3] Hou, J., Li, H., Nord, N., & Huang, G. (2022). « *Model predictive control under weather forecast uncertainty for HVAC systems in university buildings »*. Energy and Buildings, 257, 111793.

[4] Zhan, S., Lei, Y., Jin, Y., Yan, D., & Chong, A. (2022). « *Impact of occupant related data on** identification and model predictive control for buildings »*. Applied Energy, *323*, 119580.

[5] Hoai-Nam Nguyen, *« **Improved Prediction** Dynamics for** Robust** MPC** », *IEEE Transaction on Automatic Control, 2022.