# Energy-Aware Model Predictive Control of Assembly Lines

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## Abstract

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## 1. Introduction

#### 1.1. Review of ALBPs and Solution Methods

- SALBPs of type 1 (SALBP-1) deal with minimization of the number of workstations required to execute a given set of tasks.
- SALBPs of type 2 (SALBP-2) aim at minimizing the cycle time, i.e., at maximizing the production rate.
- SALBPs of type E (SALBP-E), where E stands for efficiency, focus on the minimization of the product between the cycle time and the number of workstations, i.e., the objective is to minimize the total idle time.
- SALBPs of type F (SALBP-F), where E stands for feasibility, are aimed at understanding if, for a given number of workstations and a desired cycle time c, it is possible to execute all the tasks.

#### 1.2. Energy-Aware Control Frameworks for Industrial Assembly Lines

#### 1.3. Paper Contributions

#### 1.4. Paper Structure

## 2. Problem Description

## 3. Problem Formulation

- ${e}_{t,w,i}$, which is a Boolean control variable equal to one if and only if the algorithm commands to execute tasks t on workstation w at time i;
- ${a}_{r,t,i}$, a Boolean control variable equal to one if and only if the algorithm commands to assign resource r to task t at time i;
- ${d}_{t,i}$, a real variable that represents the expected duration of tasks t at time i (that is, the time left to complete tasks t at time i). After the task is completed, it is clearly ${d}_{t,i}=0$. We define ${d}_{t}$ as the total time to complete task t (it is obviously ${d}_{t,i}\le {d}_{t}$);
- ${o}_{w,i}$, a real variable that captures the occupancy level of workstation w at time i. For instance, the occupancy level could be referred to the maximum number of tasks that can simultaneously run on the workstation (as in this paper), or to other metrics as well, depending on the specific case;

- ${P}_{i}^{grid}$, a real variable that represents the power flowing at the point of connection of the industry with the grid, at time i. It is by convention positive, when the industry absorbs power, and negative instead when the industry injects power into the grid. Thresholds are given for both the maximum allowed power withdrawal (${P}^{grid,max}>0$) and the maximum power injection (${P}^{grid,min}\le 0$):$${P}^{grid,min}\le {P}_{i}^{grid}\le {P}^{grid,max}$$
- To allow proper computation of the energy bill, we introduce a non-negative real variable, ${P}^{grid,in}\ge 0$, to capture the power that flows from the grid to the industry, and a second non-negative real variable, ${P}^{grid,out}\ge 0$, to capture the power that flows in the opposite direction, from the industry to the grid. Given these definitions, ${P}^{grid}$ can then be defined as:$${P}_{i}^{grid}={P}_{i}^{grid,in}-{P}_{i}^{grid,out}.$$Obviously, at any time, only one of the two components of ${P}_{i}^{grid}$ can be different from zero. To enforce this, we need to introduce two auxiliary Boolean variables, ${\delta}_{i}^{in}$, which should be equal to one if ${P}_{i}^{grid,in}>0$ (i.e., when the industry is taking power from the grid), and ${\delta}_{i}^{out}$, which should be equal to one if ${P}_{i}^{grid,out}>0$ (i.e., when the industry is injecting power into the grid). This behavior for the auxiliary variables can be enforced by adding the following constraints:$${P}_{i}^{grid,in}\le {\delta}_{i}^{in}{P}^{grid,max},$$$${P}_{i}^{grid,out}\le -{\delta}_{i}^{out}{P}^{grid,min}.$$Then, the following constraint ensures that at any time, only one between ${P}_{i}^{grid,in}$ and ${P}_{i}^{grid,out}$ can be different (i.e., greater) from zero.$${\delta}_{i}^{in}+{\delta}_{i}^{out}\le 1.$$
- Next, the real variable ${P}_{i}^{ess}$ is introduced, which represents the charging/discharging power (kW) of the battery at time i, which is limited between a maximum possible charging level and a maximum possible discharging level:$${P}^{ess,min}\le {P}_{i}^{ess}\le {P}^{ess,max}.$$
- The real variable ${x}_{i}^{ess}$ represents the energy level (kWh) of energy stored in the battery at time i. At any time, it must be:$${x}^{ess,min}\le {x}_{i}^{ess}\le {x}^{ess,max},$$$${x}_{i+1}^{ess}={x}_{i}^{ess}+T{P}_{i}^{ess},$$
- ${P}_{i}^{T}$, the aggregated power consumption of the tasks running at time i. It is defined as the sum of the power consumed by all the tasks currently executing (i.e., for which ${e}_{t,w,k}=1$):$${P}_{k}^{T}=\sum _{t}{e}_{t,w,k}{P}_{t,{d}_{t}-{d}_{t,k}+1},$$
- Finally, ${P}_{i}^{PV}\ge 0$ is the forecast of the power generated by the renewable plant at time i.

#### 3.1. Objective Function

- (1)
- The term ${V}_{1,k}$ is related to the tasks’ control and pushes the minimization of the time left to complete the tasks:$${V}_{1,k}=\sum _{i\in {\mathcal{H}}_{k}}\sum _{t\in {\mathcal{T}}_{k}}{d}_{t,i}.$$
- (2)
- The second term is to the energy cost. It is added in order to minimize, at each instant of time, the cost related to the energy consumption required by the tasks and to maximize the profit when the power is injected into the grid. We consider a scenario with a time-varying time-of-use tariff, where ${c}_{i}$ is the cost (EUR/kWh) of energy consumption at time i and ${p}_{i}$ is the remuneration (EUR/kWh) of the energy injected into the grid at time i. The term is:$${V}_{2,k}=\sum _{i}T({c}_{i}{P}_{i}^{grid,in}-{p}_{i}{P}_{i}^{grid,out}),$$
- (3)
- The third term is also energy related. It pushes the minimization of the peaks in the power exchanged between the industry and the grid. To avoid nonlinear formulations, which make the computation time of the algorithms higher, we minimized the H-infinity norm of the injected and absorbed power vectors, i.e., ${P}^{grid,in}$ and ${P}^{grid,out}$ (we recall that the H-infinity norm of a vector is defined as the largest component of the vector, so that we seek in practice to minimize the greater absorption and injection power peak). To capture the H-infinity norm of ${P}^{grid,in}$ and ${P}^{grid,out}$, we introduced two auxiliary variables, ${h}^{in}$ and ${h}^{out}$. By definition, the H-infinity norm is greater than or equal to any component of the vector, i.e.:$${P}_{i}^{grid,in}\le {h}^{in},$$$${P}_{i}^{grid,out}\le {h}^{out}.$$Finally, we minimized ${h}^{in}$ and ${h}^{out}$ in the objective function (so that, at the optimum, ${h}^{in}$ and ${h}^{out}$ are actually the H-infinity norms of the vectors ${P}^{grid,in}$ and ${P}^{grid,out}$).$${V}_{3,k}={h}^{in}+{h}^{out}.$$

#### 3.2. Remarks on Practical Implementation and Possible Disadvantages of the Solution

## 4. Simulation Results

#### 4.1. Simulation Scenario

- Minimization of cycle time. In this first simulation, we only sought to optimize the cycle time, while leaving out of the optimization all the energy-related considerations (i.e., we set to ${\alpha}_{2}$ and ${\alpha}_{3}$ zero in the objective function). This simulation serves as a baseline for the next one;
- Energy-aware task control. In this scenario, the proposed algorithm is tested, with all the terms, including also the energy-related ones. The goal is to show that the energy-related performance can be improved (i.e., energy bill savings and reduction of power peaks).

#### 4.2. Minimization of Cycle Time

#### 4.3. Energy-Aware Task Scheduling Optimization

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

ALBP | Assembly Line Balancing Problem |

ESS | Energy Storage System |

MPC | Model Predictive Control |

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Task ID | ${\mathit{d}}_{\mathit{t}}$ (h) | ${\mathit{S}}_{\mathit{t}}$ (Day) | ${\mathit{F}}_{\mathit{t}}$ (Day) | w | Task Precedence Relations | Input Resources |
---|---|---|---|---|---|---|

1 | 6 | 1 | 4 | 2 | 3; 4; 11 | |

2 | 3 | 1 | 4 | 2 | 3; 12 | |

3 | 4 | 1 | 4 | 2 | 1; 2 | 4; 5 |

4 | 8 | 1 | 4 | 1 | 2; 6 | |

5 | 0.25 | 1 | 4 | 2 | 1; 2; 3; 4 | 2 |

6 | 5 | 1 | 4 | 2 | 5 | 6; 11; 1 |

7 | 5 | 1 | 4 | 2 | 5 | 12 |

8 | 10 | 1 | 4 | 2 | 5; 6 | 14; 1; 4 |

9 | 24 | 3 | 8 | 2 | 6; 7 | 15 |

10 | 6 | 3 | 8 | 2 | 6; 7 | 15; 4 |

11 | 8 | 3 | 8 | 2 | 6; 7; 8; 9; 10 | 3; 14 |

12 | 6 | 7 | 12 | 3 | 6; 7; 8 | 1; 11 |

13 | 6 | 7 | 12 | 3 | 6; 7; 8 | 1; 12 |

14 | 12 | 7 | 12 | 3 | 6; 7; 8 | 3; 13 |

15 | 5 | 7 | 12 | 1 | 4 | 2; 4; 12 |

16 | 2 | 7 | 12 | 3 | 12; 13; 14; 15 | 1; 15 |

17 | 6 | 7 | 12 | 3 | 16 | 6; 5; 7; 12 |

18 | 6 | 7 | 13 | 3 | 16 | 11 |

19 | 12 | 7 | 13 | 3 | 16; 17; 18 | 14 |

20 | 3 | 10 | 14 | 1 | 15 | 1; 12 |

21 | 5 | 10 | 14 | 4 | 17; 18; 19; 20 | 3; 7; 11 |

22 | 10 | 10 | 14 | 4 | 17; 18; 19; 20 | 14 |

23 | 3 | 10 | 14 | 4 | 17; 18; 19; 20; 21; 22 | 15 |

24 | 6 | 12 | 17 | 1 | 20 | 2; 12 |

25 | 14 | 12 | 17 | 4 | 24 | 1; 11 |

26 | 1 | 12 | 17 | 4 | 21; 22; 23; 25 | 1; 13 |

27 | 3 | 12 | 17 | 4 | 21; 22; 23; 25 | 14 |

28 | 8 | 12 | 17 | 4 | 21; 22; 23; 25; 26; 27 | 15; 5; 7 |

29 | 5 | 15 | 20 | 5 | 28 | 3; 13 |

30 | 6 | 15 | 20 | 5 | 28; 29 | 12; 8; 6 |

31 | 6 | 15 | 20 | 5 | 28; 29 | 13; 4; 5 |

32 | 12 | 15 | 20 | 5 | 28; 29; 30; 31 | 11 |

33 | 3 | 15 | 20 | 1 | 24 | 2; 12 |

34 | 3 | 15 | 20 | 4 | 33 | 1; 11 |

35 | 1 | 15 | 20 | 4 | 34 | 1; 12 |

36 | 6 | 15 | 20 | 4 | 34 | 4; 14 |

37 | 8 | 15 | 20 | 4 | 35; 36 | 12; 4 |

38 | 5 | 15 | 20 | 5 | 37 | 3; 13 |

39 | 15 | 15 | 20 | 5 | 37 | 15; 7 |

40 | 10 | 15 | 20 | 5 | 37; 38; 39 | 14; 4 |

41 | 15 | 15 | 20 | 5 | 40 | 5; 4; 11 |

42 | 5 | 15 | 20 | 5 | 41 | 15 |

43 | 10 | 15 | 20 | 5 | 42 | 14 |

44 | 0.25 | 15 | 20 | 5 | 43 | 11 |

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**MDPI and ACS Style**

Liberati, F.; Cirino, C.M.F.; Tortorelli, A.
Energy-Aware Model Predictive Control of Assembly Lines. *Actuators* **2022**, *11*, 172.
https://doi.org/10.3390/act11060172

**AMA Style**

Liberati F, Cirino CMF, Tortorelli A.
Energy-Aware Model Predictive Control of Assembly Lines. *Actuators*. 2022; 11(6):172.
https://doi.org/10.3390/act11060172

**Chicago/Turabian Style**

Liberati, Francesco, Chiara Maria Francesca Cirino, and Andrea Tortorelli.
2022. "Energy-Aware Model Predictive Control of Assembly Lines" *Actuators* 11, no. 6: 172.
https://doi.org/10.3390/act11060172