# A Novel Strategy to Reduce Computational Burden of the Stochastic Security Constrained Unit Commitment Problem

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## Abstract

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## 1. Introduction

## 2. Stochastic SCUC Formulation

#### 2.1. Background

#### 2.2. Mathematical Model of the SSCUC Problem

- Binary variable logic:$${y}_{i}\left(t\right)-{z}_{i}\left(t\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{x}_{i}\left(t\right)-{x}_{i}(t-1)\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I$$$${y}_{i}\left(t\right)+{z}_{i}\left(t\right)\le 1\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I$$
- Operating costs of thermal plants:$${C}_{i}(t,\xi )\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{a}_{i}{x}_{i}\left(t\right)+\sum _{b=1}^{B}{k}_{i,b}{g}_{i,b}(t,\xi )+{SUC}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I,\xi \le \Xi $$
- Total power output of the thermal generator i, expressed as the total sum of the generation level in each segment b of the cost curve:$${g}_{i}(t,\xi )\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{b=1}^{B}{g}_{i,b}(t,\xi )\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I,\xi \le \Xi $$
- Minimum generator output constraint:$${g}_{i}(t,\xi )\ge {g}_{i}^{min}{x}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I,\xi \le \Xi $$
- Maximum generator output constraint:$${g}_{i,b}(t,\xi )\le {g}_{i}^{max}{x}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I,b\le B,\xi \le \Xi $$
- Initial on–off status of generator i at $t=0$ (15), minimum up (16) and down time constraints (17):$${x}_{i}\left(t\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{g}_{i}^{on--off}\phantom{\rule{1.em}{0ex}}\forall i\le I,t\le {L}_{i}^{up,min}+{L}_{i}^{down,min}$$$$\sum _{tt=t-{g}_{i}^{up}+1}^{t}{y}_{i}\left(tt\right)\le {x}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall i\le I,\phantom{\rule{1.em}{0ex}}\forall t\ge {L}_{i}^{up,min}$$$$\sum _{tt=t-{g}_{i}^{down}+1}^{t}{z}_{i}\left(tt\right)\le 1-{x}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall i\le I,\phantom{\rule{1.em}{0ex}}\forall t\ge {L}_{i}^{down,min}$$$${L}_{i}^{up,min}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}min[T,({g}_{i}^{up}-{g}_{i}^{up,init}){g}_{i}^{on--off}]$$$${L}_{i}^{down,min}\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}min[T,({g}_{i}^{down}-{g}_{i}^{down,init})(1-{g}_{i}^{on--off})]$$
- Ramping constraints:$$-{ramp}_{i}^{down}\le {g}_{i}(t,\xi )-{g}_{i}(t-1,\xi )\phantom{\rule{1.em}{0ex}}\forall i\le I,\phantom{\rule{1.em}{0ex}}2\le t\le T,\xi \le \Xi $$$${ramp}_{i}^{up}\ge {g}_{i}(t,\xi )-{g}_{i}(t-1,\xi )\phantom{\rule{1.em}{0ex}}\forall i\le I,\phantom{\rule{1.em}{0ex}}2\le t\le T,\xi \le \Xi $$$$-{ramp}_{i}^{down}\le {g}_{i}(t=1,\xi )-{g}_{i}^{0}\phantom{\rule{1.em}{0ex}}\forall i\le I,\xi \le \Xi $$$${ramp}_{i}^{up}\ge {g}_{i}(t=1,\xi )-{g}_{i}^{0}\phantom{\rule{1.em}{0ex}}\forall i\le I,\xi \le \Xi $$
- Generator off counter set-up constraints. In this case, symbols | and $\wedge $ indicate the logical conditions IF and AND, respectively:$$\begin{array}{cc}\hfill {w}_{i,j}\left(t\right)\le & \sum _{tt=SU{C}_{i,j}^{lim}}^{min\{t-1,SU{C}_{i,j+1}^{lim}-1\}}{z}_{i}(t-j)\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +1\left|\right\{j=J-1\wedge SU{C}_{i,j}^{lim}\le {g}_{i}^{down,init}+t-1<SU{C}_{i,j+1}^{lim}\}\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& +1\left|\right\{j=J\wedge SU{C}_{i,j}^{lim}\le {g}_{i}^{down,init}+t-1\}\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I,j\le J\hfill \end{array}$$$$\sum _{j=1}^{J}{w}_{i,j}\left(t\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}{y}_{i}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I$$$${SUC}_{i}\left(t\right)\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}\sum _{j=1}^{J}{w}_{i,j}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall t\le T,i\le I$$
- Net power balance constraints:$${P}_{s}^{Net}(t,\xi )=\sum _{i}{A}_{i}^{s}\xb7{g}_{i}(t,\xi )-{D}_{s}\left(t\right)+{L}_{s}^{sh}(t,\xi )+\sum _{w}{A}_{w}^{s}\xb7[{\widehat{g}}_{w}(t,\xi )-{Q}_{w}(t,\xi )]\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{2.em}{0ex}}\phantom{\rule{1.em}{0ex}}$$$$\forall ,s\le S,t\le T,\xi \le \Xi $$$$\sum _{s}{P}_{s}^{Net}(t,\xi )\phantom{\rule{3.33333pt}{0ex}}=\phantom{\rule{3.33333pt}{0ex}}0\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}t\le T,\xi \le \Xi $$
- Power flow constraints under normal (27) and post-contingency (28) operation conditions. In this case, ${\overline{PTDF}}_{l,s}$ is the power transfer distribution factor of line l for a power injection at bus s, and ${\overline{LODF}}_{l,k}$ is the line outage distribution factor for line l when line k is out of service.$$-{F}_{l}^{\mathrm{max}}\xb7TCF\le \sum _{s}{\overline{PTDF}}_{l,s}\xb7{P}_{s}^{Net}(t,\xi )\le {F}_{l}^{\mathrm{max}}\xb7TCF\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}l\le L,t\le T,\xi \le \Xi $$$$-{F}_{l}^{max}\xb7TCF\le {F}_{l}(t,\xi )+{\overline{LODF}}_{l,k}\xb7{F}_{k}(t,\xi )\le {F}_{l}^{max}\xb7TCF\phantom{\rule{1.em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}l\le L,k\le K,t\le T,\xi \le \Xi $$
- Constraints (29) and (30), represent the limit of involuntary load shedding ${L}_{s}^{sh}(t,\xi )$ and wind power curtailment ${Q}_{w}(t,\xi )$, respectively:$$0\le {L}_{s}^{sh}(t,\xi )\le {D}_{s}\left(t\right)\phantom{\rule{1.em}{0ex}}\forall s\le S,t\le T,\xi \le \Xi $$$$0\le {Q}_{w}(t,\xi )\le {\widehat{g}}_{w}(t,\xi )\phantom{\rule{1.em}{0ex}}\forall w\le W,t\le T,\xi \le \Xi $$

## 3. Solution Strategy

#### 3.1. Method for Adding Binding N − 1 Security Constraints

Algorithm 1: Method for Adding N − 1 Security Constraints. |

1. Set: $SC{R}_{l,k,t}\left(\xi \right)=0,\forall l\le L,k\le K,t\le T,\xi \le \Xi $ |

#### 3.2. Progressive Hedging Algorithm

Algorithm 2: Progressive Hedging Algorithm. |

1 initialize: $\nu =0,{w}^{\nu}\left(\xi \right)=0,\forall \xi \in \Xi $ |

11 if$x\left(\xi \right)$ is the same for all scenarios $\xi \in \Xi $then stop and report ${\overline{x}}^{\nu}$;$\phantom{(}$ |

12 else go to 5;$\phantom{(}$ |

#### 3.2.1. Adjustment of $\rho $ Parameter and Convergence Improvement

#### 3.3. Integrating N − 1 Security Constraints Within the PHA

Algorithm 3: Method for Adding $N-1$ Constraints Embedded Within the PHA. |

1 Set: $SC{R}_{l,k,t}\left(\xi \right)=0$, $\forall l\le L,k\le K,t\le T,\xi \le \Xi $ |

2 Initialize:$\nu =0,{w}^{\nu}\left(\xi \right)=0,{\rho}^{0}=0,\overline{x}=0,\phantom{\rule{1.em}{0ex}}\forall \xi \in \Xi $ |

3 Update Iteration:$\nu =\nu +1$ |

19 Compute:${\overline{x}}^{\nu}$ = ${\sum}_{\xi \in \Xi}{p}_{\xi}\left({x}^{\nu}\right)$ |

20 Update:${\rho}^{\nu}$ according to the strategies presented in Section 3.2.1 |

21 Update:${w}^{\nu}\left(\xi \right)={w}^{\nu}\left(\xi \right)+\rho ({x}^{\nu}\left(\xi \right)-{\overline{x}}^{\nu})$, $\forall \xi \in \Xi $ |

22 if$x\left(\xi \right)$ is equal in every scenario $\xi $then stop and report ${\overline{x}}^{\nu}$;$\phantom{(}$ |

22 else go to 3;$\phantom{(}$ |

## 4. Results

#### 4.1. Parameters Setting for the PHA

#### 4.2. Formulations

**Formulation A:**

**Formulation B:**

**Formulation C:**

**Formulation A**). This gives some key metrics to compare the performance of the other formulations: the optimality gap and the objective function of the optimal solution of the problem (i.e., the best relaxed solution). As stopping criterion, the minimum relative optimality gap defined in GAMS was set to 1% for all tested formulations. The optimality gap was calculated as the difference between the solution of a formulation and the best relaxed solution, divided by the best relaxed solution. Table 1 presents the values of the objective function, the computing time, and the optimality gap of

**Formulation A**, obtained with the server.

#### 4.3. Analysis and Comparison of Results

## 5. Expansion Transmission Indices

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

BD | Benders decomposition |

COO | Constrained ordinal optimization |

EF | Extensive formulation |

EFS | Extensive Form of the Scenario |

LODF | Line outage distribution factor |

LR | Lagrangian relaxation |

LSF | Linear sensitivity factors |

MIP | Mixed integer program |

PHA | Progressive hedging algorithm |

PSCOPF | Preventive security-constrained optimal power flow |

PTDF | Power transfer distribution factor |

SCUC | Security constrained unit commitment |

SSCUC | Stochastic security constraint unit commitment |

UC | Unit commitment |

## Appendix A. Nomenclature

#### Appendix A.1. Indices

b | Index of generating unit cost curve segments, 1 to B |

i | Index of thermal generators, 1 to I |

j | Index of thermal generator star-up costs, 1 to K |

w | Index of Wind generators, 1 to W |

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

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

t, tt | Index of hours, 1 to T |

$\xi $ | Index of scenarios, 1 to $\Xi $ |

#### Appendix A.2. Parameters

${A}_{i}^{s}$ | Generation map for thermal generator i located at bus s |

${A}_{w}^{s}$ | Generation map for wind generator w located at bus s |

${a}_{i}$ | Fixed production cost of thermal generator ($) |

${B}_{sm}$ | Admittance of line l connecting nodes s-m (S) |

${D}_{s}\left(t\right)$ | Demand at bus s (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) |

${g}_{i}^{0}$ | Output if thermal generator i at $t=0$ (MW) |

${g}_{i}^{max}$ | Rated capacity of thermal generator i (MW) |

${g}_{i}^{min}$ | Minimum output of thermal generator i (MW) |

${g}_{i,b}^{max}$ | Capacity of segment b of the cost curve of 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) |

${\widehat{g}}_{w}(t,\xi )$ | forecasted output power of wind generator w, at time t and scenario $\xi $ |

${k}_{i,b}$ | Slope of the segment b of the cost curve of thermal generator i ($/MW) |

${c}^{sh}$ | Cost of non-attended demand ($/MW) |

${c}_{w}$ | Cost of wind power curtailment ($/MW) |

${F}_{l}^{max}$ | 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) |

${SUC}_{i,j}^{cost}$ | Cost steps in start-up cost curve of thermal generator i ($) |

${SUC}_{i,j}^{lim}$ | Time steps in start-up cost curve of thermal generator i (h) |

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

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

$SC{R}_{l,k}\left(t\right)$ | Security Constraint Recorder |

${L}_{l}^{V}$ | Vector of vulnerable lines |

${L}_{k}^{S}$ | Vector of critical contingencies |

${p}_{\xi}$ | Probability of each scenario $\xi $ |

#### Appendix A.3. Variables

${C}_{i}(t,\xi )$ | Operating cost of generator i, at time t and scenario $\xi $ ($) |

${count}_{i}^{down}$ | Thermal generatori down time period counter |

${g}_{i}(t,\xi )$ | Thermal generator i output, at time t and scenario $\xi $ (MW) |

${g}_{i,b}(t,\xi )$ | Output of thermal generator i of segment b, at time t and scenario $\xi $ (MW) |

${L}_{s}^{sh}(t,\xi )$ | Unserved load at bus s, at time t and scenario $\xi $ (MW) |

${SUC}_{i}\left(t\right)$ | Start-up cost of generator i at time t ($) |

${w}_{i,j}\left(t\right)$ | Binary variable equal to 1 if generator i is started at time t after being off for j hours, and 0 otherwise |

${x}_{i}\left(t\right)$ | Binary variable equal to 1 if the thermal generator i is producing at time t, and 0 otherwise |

${y}_{i}\left(t\right)$ | Binary variable equal to 1 if the thermal generator i is started at the beginning of time t, and 0 otherwise |

${z}_{i}\left(t\right)$ | Binary variable equal to 1 if the thermal generator i is shutdown at the beginning of time t, and 0 otherwise |

${P}_{s}^{Net}(t,\xi )$ | Net power injection in bus s, at time t and scenario $\xi $ (MW) |

${Q}_{w}(t,\xi )$ | Wind power curtailment of the wind generator w, at time t and scenario $\xi $ (MW) |

${f}_{l}(t,\xi )$ | Power flow of the line l, at time t and scenario $\xi $, under normal operation (MW) |

${f}_{k}(t,\xi )$ | Power flow of the contingency k, at time t and scenario $\xi $, under normal operation (MW) |

## Appendix B. Detailed Results

Server | Desktop | |||
---|---|---|---|---|

Formulation | Gap(%) | Running Time (s) | Gap(%) | Running Time(s) |

A | $1.10$ | 61222 | - | - |

B | $0.95$ | 7245 | - | - |

${C}_{{\rho}_{1}}$ | $1.62$ | 1222 | $2.12$ | 1564 |

${C}_{{\rho}_{2}}$ | $1.85$ | 1194 | $2.28$ | 1552 |

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**Figure 5.**${L}_{l}^{V}$: The average number of times that line l is overloaded, given a k single contingency.

Objective Function [MUS$] | Running Time [s] | Optimality Gap [%] |
---|---|---|

$1.1110$ | 61222 | $1.10$ |

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**MDPI and ACS Style**

Marín-Cano, C.C.; Sierra-Aguilar, J.E.; López-Lezama, J.M.; Jaramillo-Duque, Á.; Villegas, J.G. A Novel Strategy to Reduce Computational Burden of the Stochastic Security Constrained Unit Commitment Problem. *Energies* **2020**, *13*, 3777.
https://doi.org/10.3390/en13153777

**AMA Style**

Marín-Cano CC, Sierra-Aguilar JE, López-Lezama JM, Jaramillo-Duque Á, Villegas JG. A Novel Strategy to Reduce Computational Burden of the Stochastic Security Constrained Unit Commitment Problem. *Energies*. 2020; 13(15):3777.
https://doi.org/10.3390/en13153777

**Chicago/Turabian Style**

Marín-Cano, Cristian Camilo, Juan Esteban Sierra-Aguilar, Jesús M. López-Lezama, Álvaro Jaramillo-Duque, and Juan G. Villegas. 2020. "A Novel Strategy to Reduce Computational Burden of the Stochastic Security Constrained Unit Commitment Problem" *Energies* 13, no. 15: 3777.
https://doi.org/10.3390/en13153777