# Robust model predictive control for an uncertain smart thermal grid

TL;DR: The focus of this paper is on modeling and control of Smart Thermal Grids in which the uncertainties in the demand and/or supply are included and it is shown that the robust MPC approach successfully keeps the supply-demand balance in the STG while satisfying the constraints of the production units in the presence of uncertainty in the heat demand.

Abstract: The focus of this paper is on modeling and control of Smart Thermal Grids (STGs) in which the uncertainties in the demand and/or supply are included. We solve the corresponding robust model predictive control (MPC) optimization problem using mixed-integer-linear programming techniques to provide a day-ahead prediction for the heat production in the grid. In an example, we compare the robust MPC approach with the robust optimal control approach, in which the day-ahead production plan is obtained by optimizing the objective function for entire day at once. There, we show that the robust MPC approach successfully keeps the supply-demand balance in the STG while satisfying the constraints of the production units in the presence of uncertainties in the heat demand. Moreover, we see that despite the longer computation time, the performance of the robust MPC controller is considerably better than the one of the robust optimal controller.

## Summary (2 min read)

### Introduction

- The authors solve the corre- sponding robust model predictive control (MPC) optimization problem using mixed-integer-linear programming techniques to provide a day-ahead prediction for the heat production in the grid.
- Moreover, the authors see that despite the longer computation time, the performance of the robust MPC controller is considerably better than the one of the robust optimal controller.
- As about half of a neighborhood’s electricity consumption is typically used for thermal purposes [1], introducing STG neighborhoods could have substantial benefits, such as: 1) less transport of energy, less energy loss, and lower transportation costs, and 2) using the produced heat at the neighborhood level as an energy source to avoid wasting heat.
- The authors consider robust MPC for STGs in the presence of uncertainties in the grid to provide a day-ahead heat production plan for the thermal grid.
- The uncertainties in the network can be due to the uncertainty in the demand and/or in the production because of using different resources such as solar energy or biogas.

### II. SMART THERMAL GRIDS

- Each of these greenhouses is considered as an agent and the full information of each agent, such as the production resources, the demand request for the next day, etc., is assumed to be available to the whole network.
- In addition to the local heat generation, there are one or more external parties that can provide heat to the network.
- The authors consider all the external parties as one single agent.
- To model the physical system, the authors discretize the system with sampling time of one hour.
- Let Hexchi j denotes the exchanged heat between two adjacent greenhouses i and j.

### III. MODEL PREDICTIVE CONTROL FOR STGS

- The authors aim is to reduce the overall production costs of the network while providing the network’s required heat under different operational constraints such as the limits for the generators and the buffers.
- The control objective will be focused on demand response [9], [17], which is the ability of domestic net-consumption of heat to respond to real-time1 electricity prices.
- 1The real-time electricity price is the one that varies almost every 15 minutes in the electricity market on the exact day of the electricity production.
- The cost of importing heat by greenhouse j at time step k is Cimp(HimpEx j(k)) = HimpEx j(k) ·HbuyingEx(k), (13) where HbuyingEx(k) is the price that greenhouse j pays for buying heat from external parties at time step k.
- Accordingly, the authors can rewrite (15) as a linear equation by introducing new binary and continuous 2As mentioned by experts at Eneco, a Dutch utility company and their project sponsor, boilers do not require a time-on/off constraints.

### IV. SOLVING THE WORST-CASE MPC

- At the beginning of each time step k, the controller measures the system state of the previous step.
- Then, using the information regarding the demand and the energy price, the controller determines the decision variables PGCHP j,HGBoil j,HimpEx j,µ stop u j , and µstartu j .
- To this end, the authors solve the inner optimization problem first.
- Note that the available mp-MILP algorithms are not very efficient when the size of the vector of parameters and the prediction horizon Np increases.

### V. EXAMPLE

- The authors solve robust MPC optimization problem to obtain a day-ahead prediction for the heat production plan for a small network of greenhouses and they compare the results with the ones obtained using robust optimal control approach.
- To solve the optimization problem (28)-(29), the authors chose M = 500 different uncertainty vectors e to obtain a 0.95% confidence level with accuracy error of 1% and they use the MILP solver from IBM CPLEX.
- The first plot of Figure 3 shows the heat demand of each greenhouse for one day.
- Here also, the CHPs are mainly used during the hours that the electricity price is quite high and they can also sell the extra electricity in the market .
- Moreover, robust MPC is a better control choice than robust optimal control although it requires more computation time.

### VI. CONCLUSIONS

- The authors have considered control of a typical smart thermal grid, namely a network of greenhouses, under uncertainties in demand and/or response.
- The authors assumed the uncertainty to be bounded and hence, a worst-case MPC optimization problem was solved.
- Since both the cost function and the constraints are linear, the optimization problem was formulated as a mixed-integer linear programming (MILP) problem; the authors have discussed three approaches to solve the obtained optimization problem.
- In a case study, the authors compared the MPC approach with the optimal control approach to obtain a day-ahead production plan for a sample network of greenhouses.
- The efficient control approach in this case is a distributed model predictive control approach in which the agents can only have partial information about the network.

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### Cites methods from "Robust model predictive control for..."

...STGs with uncertain thermal energy demands have been considered in [27], where a MPC strategy...

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### Cites background or methods from "Robust model predictive control for..."

...A deterministic demand profile without uncertainty is modelled, meaning that the optimization method also does not have to take into account any uncertain demand predictions during optimization, as is done in [23], [24] and [25]....

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...In [19] and [24], the profit from selling electricity generated by the CHP is also included....

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...In [23] and [24], a MAS MPC control framework is presented that additionally focuses on dealing with the uncertainty in heat energy demand [25], whereas in this thesis a straightforward deterministic heat demand is assumed [26]....

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...Also, the mixed-integer constraints found in [23], [24], [29] for the production capacity, ramping and minimum required on/off times are slightly adjusted with the help of hybrid modelling theory found in [30]....

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...For CHP specifically, this can include the coupling between heat and electricity production [24]....

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##### References

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### "Robust model predictive control for..." refers methods in this paper

...To this end, model predictive control (MPC) [21], [25] is a control method that has been proved to be a useful tool in both simulations and real-life applications [22], [23]....

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### "Robust model predictive control for..." refers background in this paper

...Because of this capability, STG implementation could contribute to a further decrease in carbon emissions, improved energy efficiency, and renewable energy implementation [7], [18]....

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##### Frequently Asked Questions (2)

###### Q2. What are the future works mentioned in the paper "Robust model predictive control for an uncertain smart thermal grid" ?

In a case study, the authors compared the MPC approach with the optimal control approach to obtain a day-ahead production plan for a sample network of greenhouses. Moreover, in future work, the authors will also consider the constraints of the physical network ’ s model to be able to take the dependencies and the possible delays into account. An alternative scenario to the centralized control architecture is that each greenhouse tries to maximize its own benefit and hence, they will sell heat to the other greenhouses in the network.