# A Coalitional Model Predictive Control for the Energy Efficiency of Next-Generation Cellular Networks

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## Abstract

**:**

## 1. Introduction

## 2. Problem Description

#### 2.1. Base Station Dynamics

#### 2.2. Coalition Dynamics

#### 2.3. Energy Consumption Model

#### 2.4. Data Traffic Model

#### 2.5. On-Grid Energy Consumption Optimization Problem

## 3. Control Methods

#### 3.1. Coalitional MPC

Algorithm 1: Coalitional MPC. |

Initial data:$N,\phantom{\rule{3.33333pt}{0ex}}{T}_{\mathrm{up}},\phantom{\rule{3.33333pt}{0ex}}Q,\phantom{\rule{3.33333pt}{0ex}}{F}_{i,j},\phantom{\rule{3.33333pt}{0ex}}{z}_{i}^{\mathrm{max}},\phantom{\rule{3.33333pt}{0ex}}{b}_{i}^{\mathrm{max}},\phantom{\rule{3.33333pt}{0ex}}{c}_{\mathrm{link}},\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{\mathcal{C}}$.Starting point:$k=0,\phantom{\rule{3.33333pt}{0ex}}\mathrm{\Lambda}={\mathrm{\Lambda}}_{\mathrm{cen}},\phantom{\rule{3.33333pt}{0ex}}{x}_{i}={x}_{i}\left(0\right)$.Main program:for$k\le N-1$ doif$\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.277778em}{0ex}}(k,\phantom{\rule{3.33333pt}{0ex}}{T}_{\mathrm{up}})=0$ then1: The upper control layer measures states ${x}_{\mathcal{C}}\left(k\right)$ and disturbances ${w}_{\mathcal{C}}\left(k\right)$, $\forall \mathcal{C}\in \mathcal{B}/\mathrm{\Lambda}$.2: The upper layer computes the set ${\mathcal{T}}_{\mathrm{new}}$ (Equation (10)) and solves Equation (11) subject to Equation (2) for each ${\mathcal{T}}_{\mathrm{new}}\subseteq \mathcal{T}$. Afterwards, the topology $\mathrm{\Lambda}\in {\mathcal{T}}_{\mathrm{new}}$ with the lowest cost is selected. 3: Send the new topology $\mathrm{\Lambda}$ to the lower control layer. 4: Apply the first element of ${U}_{\mathcal{C}}^{*}\left(k\right)$ to coalition system (Equation (2)), for all $\mathcal{C}\in \mathcal{B}/\mathrm{\Lambda}$.else5: In the lower layer, each coalition $\mathcal{C}$ solves Equation (11) subject to its dynamics and constraints. 6: Apply the first element of ${U}_{\mathcal{C}}^{*}\left(k\right)$ to coalition system (Equation (2)), and obtain ${x}_{\mathcal{C}}(k+1)$, $\forall \mathcal{C}\in \mathcal{B}/\mathrm{\Lambda}$. end if7: Set $k\leftarrow k+1$.end for |

#### 3.2. Centralized and Decentralized MPC

#### 3.3. Energy-Aware Heuristic

#### 3.4. Best-Signal-Level Policy

## 4. Illustrative Examples

#### 4.1. Case Studies

- An academic scenario with a set of $\mathcal{B}=\{1,\phantom{\rule{3.33333pt}{0ex}}\dots ,\phantom{\rule{3.33333pt}{0ex}}9\}$ base stations (one MBS and eight SCBSs), which is sophisticated enough to show the potential of the proposed method (see Figure 2a). The complexity of the association process is caused by the number of BSs and active users.
- A large scenario with a set of $\mathcal{B}=\{1,\phantom{\rule{3.33333pt}{0ex}}\dots ,\phantom{\rule{3.33333pt}{0ex}}37\}$ base stations (one MBS and 36 SCBSs), which represents the complexity of real problems (see Figure 2b). In this case, it is only possible to compare the proposed method with a heuristic approach.

#### 4.2. Simulation Parameters

#### 4.3. Disturbances

#### 4.4. Key Performance Indicators

- Grid consumption (kWatts-hour).
- Reduction of grid consumption (%) in comparison with the best-signal level mechanism.
- Average stored energy (Watts-hour).
- Users served (%).
- Average transmission rate (%) in comparison with the best-signal level mechanism.
- Performance cost, which is defined as the sum of the accumulated stage cost over the simulation:$$\begin{array}{c}{J}_{\mathrm{perf}}=\sum _{k=1}^{N}\sum _{\mathcal{C}\in \mathcal{B}/\mathrm{\Lambda}}{l}_{\mathcal{C}}\left(\right)open="("\; close=")">{x}_{1}^{z}\left(k\right).\end{array}$$
- Cooperation cost, which is defined as the sum of the cooperation cost over the simulation:$$\begin{array}{c}{J}_{\mathrm{coop}}=\sum _{k=1}^{N}\sum _{\mathcal{C}\in \mathcal{B}/\mathrm{\Lambda}}{g}_{\mathcal{C}}\left(\right)open="("\; close=")">{L}_{\mathrm{\Lambda}}\left(k\right),\end{array}$$

#### 4.5. Results and Discussion

#### 4.5.1. Academic Scenario

#### 4.5.2. Large Scenario

## 5. Conclusions

## 6. Materials and Methods

^{®}R2017a on Windows 10 with a PC Intel

^{®}Core™ i7-8700 CPU at $3.20$ GHz and 16 GB RAM.

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

BS | Base Station |

SCBS | Small-cell Base Station |

MBS | Macro-cell Base Station |

LTE | Long-Term Evolution |

MPC | Model Predictive Control |

MILP | Mixed-Integer Linear Problem |

NGCN | Next-Generation Cellular Network |

HetNet | Heterogeneous Cellular Network |

NHPP | Non-Homogeneous Poisson Process |

HPP | Homogeneous Poisson Process |

SINR | Signal-to-Interference-plus-Noise Ratio |

RAN | Radio Access Network |

ITU | International Telecommunication Union |

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**Figure 2.**Simulated scenarios: (

**a**) 9-BS network, and (

**b**) 37-BS network. Blue lines refer fixed communication links between the SCBSs and the MBS; and green lines refer to dynamic cooperation links between the SCBSs.

**Figure 5.**Average users served for the MPC-based methods in the 9-BS network: (

**a**) SCBSs, and (

**b**) MBS.

**Figure 6.**Comparison of four SCBSs from the 9-BS network. In each subfigure, the first row represents the user served by the SCBS; the second one is the user flow expected; the third row depicts the renewable energy arrival; and the fourth one shows the energy stored in the battery.

**Figure 7.**Average users served for the MPC-based methods in the 37-BS network: (

**a**) SCBSs, and (

**b**) MBS.

**Figure 8.**Comparison of several SCBSs from the 37-BS network. In each subfigure, the first row represents the user served by the SCBS; the second one is the user flow expected; the third row depicts the renewable energy arrival; and the fourth one show the energy stored in the battery.

Description | Value | Units | |
---|---|---|---|

$\mathcal{A}$ | Coverage area | $3.5$ | ${\mathrm{km}}^{2}$ |

$BW$ | Bandwidth of the LTE system | 20 | MHz |

${d}_{i,j}$ | Inter-site distance | 500 | m |

${T}_{1}$ | Transmission power of the MBS | 43 | dBm |

${T}_{i}$ | Transmission power of the SCBSs | 22 | dBm |

${E}_{1}^{\mathrm{S}}$ | Static power consumption of the MBS | 130 | W.h |

${E}_{i}^{\mathrm{S}}$ | Static power consumption of the SCBSs | $6.8$ | W.h |

${\Delta}_{1}$ | Consumption slope of the MBS | $4.7$ | – |

${\Delta}_{i}$ | Consumption slope of the SCBSs | $4.0$ | – |

${F}_{i,j}$ | Max. users flow between the BSs | 100 | – |

${z}_{i}^{\mathrm{max}}$ | Max. users served by the SCBSs | 200 | – |

${b}_{i}^{\mathrm{max}}$ | Max. battery capacity of the SCBSs | 200 | W.h |

${N}_{\mathrm{p}}$ | Prediction horizon | 5 | – |

${T}_{\mathrm{up}}$ | Upper layer execution | 10 | – |

Q | Weighting parameter | 100 | – |

${\alpha}_{i}$ | Weighting factor | ${10}^{7}$ | – |

${c}_{\mathrm{link}}$ | Cost per cooperation link | ${10}^{4}$ | – |

- | Path loss between the MBS and z | Cost 231 model | – |

- | Mobility model | Random walk | – |

- | Mobility speed | 4 | km/h |

Energy Management | User Experience | MPC Costs | ||||||
---|---|---|---|---|---|---|---|---|

Control Scheme | Grid Consumption (kW.h) | Grid Consumption Improvement (%) | Average Stored Energy (W.h) | Users Served (%) | Transmission Rate (Mbps) | ${\mathit{J}}_{\mathbf{perf}}$ | ${\mathit{J}}_{\mathbf{coop}}$ | ${\mathit{J}}_{\mathbf{total}}={\mathit{J}}_{\mathrm{perf}}+{\mathit{J}}_{\mathrm{coop}}$ |

Best-signal-level | $938.1$ | − | $65.5$ | $95.2$ | $4.2$ | − | − | − |

Energy-aware heuristic | $830.3$ | $11.5$ | $61.2$ | $94.7$ | $3.9$ | − | − | − |

Centralized MPC | 130 | $86.1$ | $56.6$ | $96.1$ | $4.0$ | $4.44\xb7{10}^{4}$ | $1.25\xb7{10}^{7}$ | $1.25\xb7{10}^{7}$ |

Decentralized MPC | $550.7$ | $41.3$ | $58.5$ | $96.5$ | $3.8$ | $5.34\xb7{10}^{6}$ | $4.00\xb7{10}^{6}$ | $9.34\xb7{10}^{6}$ |

Coalitional MPC | $272.8$ | $70.9$ | $57.9$ | $96.3$ | $4.0$ | $1.64\xb7{10}^{5}$ | $1.14\xb7{10}^{7}$ | $1.16\xb7{10}^{7}$ |

Energy Management | User Experience | MPC Costs | ||||||
---|---|---|---|---|---|---|---|---|

Control Scheme | Grid Consumption (kW.h) | Grid Consumption Improvement (%) | Average Stored Energy (W.h) | Users Served (%) | Transmission Rate (Mbps) | ${\mathit{J}}_{\mathbf{perf}}$ | ${\mathit{J}}_{\mathbf{coop}}$ | ${\mathit{J}}_{\mathbf{total}}={\mathit{J}}_{\mathrm{perf}}+{\mathit{J}}_{\mathrm{coop}}$ |

Best-signal-level | $252.8$ | − | $63.5$ | $95.3$ | $3.3$ | − | − | − |

Energy-aware heuristic | $227.5$ | $10.1$ | $60.2$ | $94.3$ | $2.8$ | − | − | − |

Centralized MPC | $130.8$ | $48.3$ | $71.1$ | $95.1$ | $3.1$ | $4.5\times {10}^{4}$ | $1.37\times {10}^{8}$ | $1.3705\times {10}^{8}$ |

Decentralized MPC | $4732.8$ | $-1772$ | $70.8$ | $95.5$ | $2.9$ | $3.24\times {10}^{8}$ | $1.8\times {10}^{7}$ | $3.42\times {10}^{8}$ |

Coalitional MPC | $182.2$ | $27.9$ | $70.5$ | $95.1$ | $3.1$ | $5.02\times {10}^{5}$ | $1.36\times {10}^{8}$ | $1.365\times {10}^{8}$ |

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**MDPI and ACS Style**

Masero, E.; Fletscher, L.A.; Maestre, J.M.
A Coalitional Model Predictive Control for the Energy Efficiency of Next-Generation Cellular Networks. *Energies* **2020**, *13*, 6546.
https://doi.org/10.3390/en13246546

**AMA Style**

Masero E, Fletscher LA, Maestre JM.
A Coalitional Model Predictive Control for the Energy Efficiency of Next-Generation Cellular Networks. *Energies*. 2020; 13(24):6546.
https://doi.org/10.3390/en13246546

**Chicago/Turabian Style**

Masero, Eva, Luis A. Fletscher, and José M. Maestre.
2020. "A Coalitional Model Predictive Control for the Energy Efficiency of Next-Generation Cellular Networks" *Energies* 13, no. 24: 6546.
https://doi.org/10.3390/en13246546