# A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty

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## Abstract

**:**

## 1. Introduction

## 2. Theorical Background

#### 2.1. Affine Arithmetic

#### 2.2. Affinely Adjustable Robust Optimization (AARO)

## 3. Deterministic Model for Security Constrained Unit Commitment

## 4. AARO Model for Security Constrained Unit Commitment (SCUC)

#### 4.1. AARO Formulation to Solve the SCUC Problem

Algorithm 1: Method for Adding N − 1 Security Constraints as user cuts |

#### 4.2. Uncertainty Sets

## 5. Tests and Results

#### 5.1. 6-Bus System

#### 5.2. IEEE RTS 24-Bus Power System

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AA | Affine arithmetic |

AARO | Affinely adjustable robust optimization |

AAROUC | Affinely adjustable robust optimization unit commitment |

AGC | Automatic generation control |

BD | Benders decomposition |

DRO | Distributionally robust optimization |

LODF | Line outage distribution factor |

LR | Lagrangian relaxation |

LSF | Linear sensitivity factors |

MCS | Monte Carlo Simulations |

MILP | Mixed-integer linear programming |

OF | Objective function |

OTDF | Outage transfer distribution factors |

PTDF | Power transfer distribution factor |

SCUC | Security constrained unit commitment |

ROUC | Robust optimization of the UC |

SOUC | Stochastic optimization for the unit commitment |

SSCUC | Stochastic security constraint unit commitment |

UC | Unit commitment |

## Appendix A. Nomenclature

#### Appendix A.1. Indexes and Notations

i | Index of conventional generators, 1 to I |

w | Index of Wind generators, 1 to W |

p | Index of PV generators, 1 to P |

l, k | Index of lines and contingencies, respectively, 1 to L |

s, ss | Index of buses, 1 to S |

t, tt, $\tau $ | Index of time periods, 1 to T |

#### Appendix A.2. Parameters

${A}_{i}^{s}$ | Generation map for conventional generators |

${A}_{w}^{s}$ | Generation map for wind generators |

${A}_{p}^{s}$ | Generation map for PV generators |

$F{C}_{i}$ | Fixed production cost of thermal generator [USD] |

${C}_{i}$ | Operating variable cost of generator [USD] |

$Cs{u}_{i}$ | Startup cost of conventional generator [USD] |

$Cs{d}_{i}$ | Shut down cost of thermal generator [USD] |

${d}_{s,t}$ | Demand in bus s at time t [MW] |

${g}_{i}^{down}$ | Minimum down time of thermal generator i [h] |

${g}_{i}^{up}$ | Minimum up time of thermal generator i [h] |

${g}_{i}^{down,init}$ | Time that thermal generator i has been down before $t=0$ [h] |

${g}_{i}^{up,init}$ | Time that thermal generator i has been up before $t=0$ [h] |

${g}_{i}^{max}$ | Rated capacity of thermal generator i [MW] |

${g}_{i}^{min}$ | Minimum output of thermal generator i [MW] |

${g}_{i}^{on-off}$ | On-Off status of generator i at $t=0$ (equal to 1 if ${g}_{i}^{up,init}>0$ |

and 0 otherwise) | |

${\widehat{g}}_{w,t}$ | Power Forecast of wind generator w, at time t |

${\widehat{g}}_{p,t}$ | Power Forecast of PV generator p, at time t |

${\widehat{d}}_{s,t}$ | Load Forecast of demand at bus s, at time t |

${F}_{l}^{max}$ | Maximum Capacity of the line l [MW] |

$TCF$ | Transmission capacity factor of the line l |

${L}_{i}^{down,min}$ | Length of time the thermal generator i has to be off at the start |

time of the planning horizon [h] | |

${L}_{i}^{up,min}$ | Length of time the thermal generator i has to be on at the start |

time of the planning horizon [h] | |

${ramp}_{i}^{down}$ | Ramp-down limit of thermal generator i [MW/h] |

${ramp}_{i}^{up}$ | Ramp-up limit of thermal generator i [MW/h] |

$PTD{F}_{l,s}$ | Matrix of Power transfer distribution factors |

$LOD{F}_{l,k}$ | Matrix of Line Outage distribution factors |

$OTD{F}_{l,k}$ | Matrix of Outage transfer distribution factors |

${\widehat{e}}_{w,t}$ | Materialized forecast error of the wind generator w, at time t [MW] |

${\widehat{e}}_{p,t}$ | Materialized forecast error of the PV generator p, at time t [MW] |

${\widehat{e}}_{s,t}$ | Materialized forecast error of the the demand located at bus s, |

at time t [MW] | |

${\widehat{e}}_{w,t}^{max}$ | Maximum forecast error of the wind generator w, at time t [MW] |

${\widehat{e}}_{p,t}^{max}$ | Maximum forecast error of the PV generator p, at time t [MW] |

${\widehat{e}}_{s,t}^{max}$ | Maximum forecast error of the demand located at bus s, at time |

t [MW] |

#### Appendix A.3. Variables

${g}_{i,t}$ | Conventional generator power output, of the generator i at time t [MW] |

${x}_{i,t}$ | Binary variable equal to 1 if the thermal generator i is producing at time |

t, and 0 otherwise | |

${y}_{i,t}$ | Binary variable equal to 1 if the thermal generator i is started at the |

beginning of time t and 0 otherwise | |

${z}_{i,t}$ | Binary variable equal to 1 if the thermal generator i is shutdown at the |

beginning of time t and 0 otherwise | |

$Pne{t}_{s,t}$ | Net power injection in bus s, at time t [MW] |

${f}_{l,t}$ | Power flow of the line l, at time t, under normal operation [MW] |

${f}_{l,k,t}$ | Power flow of line l under the contingency k, at time t [MW] |

${g}_{i,t}^{0}$ | Central value of power output, of the generator i at time t [MW] |

$Pne{t}_{s,t}^{0}$ | Central value for net power injection in bus s, at time t [MW] |

$\lambda $ | Upper bound of the highest cost for the dispatch problem [USD] |

${\gamma}_{i,t,w,\tau}$ | Adjustment of the generator i at time t given by the deviation of forecast |

error of the wind generator w in the past $\tau $ periods [p.u.] | |

${\gamma}_{i,t,p,\tau}$ | Adjustment of the generator i at time t given by the deviation of forecast |

error of the PV generator p in the past $\tau $ periods [p.u.] | |

${\gamma}_{i,t,\tau}$ | Adjustment of the generator i at time t given by the deviation |

of forecast error of total demand in the past $\tau $ periods [p.u.] | |

${\eta}_{w,t}^{1}$ | Adjust variable for the objective function given by the deviation of |

forecast error of the wind generator w at time t [USD/MW] | |

${\eta}_{p,t}^{1}$ | Adjust variable for the objective function given by |

the deviation of forecast error of the PV generator p at time t | |

$[USD$ | Adjust variable for the objective function given by |

$/MW]{\eta}_{t}^{1}$ | the deviation of forecast error of the total demand at time t [USD/MW]. |

${\eta}_{i,t,w,\tau}^{2}$ | Adjust variable for power limits of the generator i at time t given by |

the deviation of forecast error of the wind generator w in the past | |

$\tau $ periods [p.u.] | |

${\eta}_{i,t,p,\tau}^{2}$ | Adjust variable for power limits of the generator i at time t given by |

the deviation of forecast error of the PV generator p in the past $\tau $ | |

periods [p.u.] | |

${\eta}_{i,t,\tau}^{2}$ | Adjust variable for power limits of the generator i at time t given by |

the deviation of forecast error of total demand in the past $\tau $ periods [p.u.] | |

${\eta}_{i,t,w,\tau}^{3}$ | Adjust variable for ramping limits of the generator i at time t given by |

the deviation of forecast error of the wind generator w in the past $\tau $ | |

periods [p.u.] | |

${\eta}_{i,t,p,\tau}^{3}$ | Adjust variable for ramping limits of the generator i at time t given by |

the deviation of forecast error of PV generator p in the past $\tau $ | |

periods [p.u.] | |

${\eta}_{i,t,\tau}^{3}$ | Adjust variable for ramping limits of the generator i at time t given by |

the deviation of forecast error of the total demand in the past $\tau $ | |

periods [p.u.] | |

${\eta}_{l,t,w,\tau}^{4}$ | Adjust variable for power flow limits of the line l at time t given by the |

deviation of forecast error of the wind generator w in the past $\tau $ | |

periods [p.u.] | |

${\eta}_{l,t,p,\tau}^{4}$ | Adjust variable for power flow limits of the line l at time t given by the |

deviation of forecast error of the PV generator p in the past $\tau $ | |

periods [p.u.] |

${\eta}_{l,t,s,\tau}^{4}$ | Adjust variable for power flow limits of the line l at time t given by |

deviation of forecast the demand at bus s in the past $\tau $ periods [p.u.] | |

${\eta}_{l,k,t,w,\tau}^{5}$ | Adjust variable for power flow limits of the line l under contingency of |

the line k at time t given by the deviation of forecast error of the wind | |

generator w in the past $\tau $ periods [p.u.] | |

${\eta}_{l,k,t,p,\tau}^{5}$ | Adjust variable for power flow limits of the line l under contingency of |

line k at time t given by the deviation of forecast error of the PV generator | |

p in the past $\tau $ periods [p.u.] | |

${\eta}_{l,k,t,s,\tau}^{5}$ | Adjust variable for power flow limits of the line l under contingency of |

line k at time t given by the deviation of forecast error of the demand | |

at bus s in the past $\tau $ periods [p.u.] |

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**Figure 5.**Zoom of Figure 4.

**Figure 8.**Zoom of Figure 7.

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## Share and Cite

**MDPI and ACS Style**

Sierra-Aguilar, J.E.; Marín-Cano, C.C.; López-Lezama, J.M.; Jaramillo-Duque, Á.; Villegas, J.G.
A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty. *Appl. Sci.* **2021**, *11*, 3987.
https://doi.org/10.3390/app11093987

**AMA Style**

Sierra-Aguilar JE, Marín-Cano CC, López-Lezama JM, Jaramillo-Duque Á, Villegas JG.
A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty. *Applied Sciences*. 2021; 11(9):3987.
https://doi.org/10.3390/app11093987

**Chicago/Turabian Style**

Sierra-Aguilar, Juan Esteban, Cristian Camilo Marín-Cano, Jesús M. López-Lezama, Álvaro Jaramillo-Duque, and Juan G. Villegas.
2021. "A New Affinely Adjustable Robust Model for Security Constrained Unit Commitment under Uncertainty" *Applied Sciences* 11, no. 9: 3987.
https://doi.org/10.3390/app11093987