# Designing a Profit-Maximizing Critical Peak Pricing Scheme Considering the Payback Phenomenon

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## Abstract

**:**

## 1. Introduction

## 2. Backgrounds

#### 2.1. Price Responsiveness Model

#### 2.2. Designing a Critical Peak Pricing (CPP) Scheme without Payback

## 3. Characterization of the Payback Phenomenon

**Figure 1.**Exponentially decreasing payback functions for several payback duration values of 1, 2, 3, 5, 7, and 9.

## 4. Payback Effects on CPP Design

#### 4.1. Payback Effects on the Event Scheduling Problem

**Figure 2.**(

**a**) Demand and real-time market clearing prices (RTMCPs); (

**b**) A simple example to show how payback affects the optimal schedule of critical events.

**Table 1.**Profits of the load serving entity (LSE) with and without payback for two event schedules in the example.

Critical Event Period | Without Payback ($) | With Payback ($) | |
---|---|---|---|

$k=2$ | revenue | 84 | 88 |

cost | 54 | 61 | |

profit | 30 | 27 | |

$k=3$ | revenue | 84 | 88 |

cost | 55 | 60 | |

profit | 29 | 28 |

#### 4.2. Payback Effects on the Optimal Peak Rate

## 5. Simulations and Verification

#### 5.1. Simulation Methods

#### 5.2. Results: Payback Effects on the Optimal Event Schedule

${f}_{PB}\text{(}n\text{)}=\left\{{f}_{PB}\text{(}1\text{)},{f}_{PB}\text{(}2\text{)},{f}_{PB}\text{(}3\text{)}\right\}$ | Optimal Schedule of Critical Events | Profit (Million Dollars) |
---|---|---|

Without payback | {154, 561, 668} | 46.476 |

Exponentially decreasing payback {0.54, 0.30, 0.16} | {157, 561, 644} | 40.265 |

Uniformly distributed payback {1/3, 1/3, 1/3} | {157, 561, 704} | 40.843 |

#### 5.3. Result: Payback Effects on the Optimal Peak Rate

**Figure 4.**Simulation results of the optimal peak rate with respect to the payback duration for (

**a**) Exponentially decreasing payback (EDP); (

**b**) Uniformly distributed payback (UDP).

**Figure 5.**Simulation results of the optimal peak rate with respect to the payback ratio for selected values of ${D}_{PB}$.

**Figure 6.**Profit of the load serving entities (LSE) with respect to the peak rate for the case without payback and several cases with payback.

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

${\text{\rho}}_{BASE}$ | off-peak rate of a critical peak pricing scheme |

${\text{\rho}}_{PEAK}$ | peak rate of a critical peak pricing scheme |

${\text{\rho}}_{RTMCP,k}$ | real-time market clearing price in period $k$ |

${\text{\rho}}_{N,PEAK}^{*}$ | optimal peak rate for a normal situation without payback |

${\text{\rho}}_{PB,PEAK}^{*}$ | optimal peak rate considering payback effects |

$\Delta {\text{\rho}}_{PB,PEAK}$ | difference of ${\text{\rho}}_{PB,PEAK}^{*}$ from ${\text{\rho}}_{N,PEAK}^{*}$ |

${q}_{0,k}$ | nominal consumption of customers in period $k$ |

${q}_{CR,k}$ | consumption of customers in period $k$ when a critical event is triggered |

${q}_{PB,k}$ | recovered demand due to payback in period $k$ |

${Q}_{0,k}$ | cumulative consumption during the critical event periods starting from the period $k$ |

${Q}_{CUR,k}$ | cumulative curtailed demand for a critical event starting in period $k$ |

${Q}_{PB,k}$ | paid-back demand for the critical event in period $k$ |

${R}_{k}$ | revenue of a load serving entity in period $k$ |

${C}_{k}$ | cost of a load serving entity in period $k$ |

$P{I}_{k}$ | profit index in period $k$ |

$P{I}_{N,k}$ | profit index in a normal situation without payback in period $k$ |

$P{I}_{PB,k}$ | profit index considering payback effects in period $k$ |

${u}_{k}$ | binary event decision variable in period $k$ |

${N}_{CPP}$ | maximum number of critical events |

${D}_{CPP}$ | duration of the critical event |

${D}_{PB}$ | payback duration |

${f}_{PB}(n)$ | payback function |

${f}_{PB}^{U}(n)$ | normalized payback function |

${\text{\alpha}}_{PB}$ | payback ratio |

$\text{\beta}$ | price elasticity of customers |

$H$ | scheduling time horizon of the event scheduling problem |

$\Delta k$ | minimum interval between successive events |

$\text{\lambda}$ | constant for the exponentially decreasing payback function |

$c$ | constant for the uniformly distributed payback function |

${\rm K}|{\text{\rho}}_{PEAK}$ | solution of the events scheduling problem for a given peak rate ${\text{\rho}}_{PEAK}$ |

${{\rm K}}^{*}$ | solution of the events scheduling problem for the optimal peak rate |

## Abbreviations

ARMA | autoregressive moving average |

CPP | critical peak pricing |

DR | demand response |

EDP | exponentially decreasing payback |

LSE | load serving entity |

RTMCP | real-time market clearing price |

RTP | real-time pricing |

TOU | time-of-use |

UDP | uniformly distributed payback |

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**MDPI and ACS Style**

Park, S.C.; Jin, Y.G.; Yoon, Y.T. Designing a Profit-Maximizing Critical Peak Pricing Scheme Considering the Payback Phenomenon. *Energies* **2015**, *8*, 11363-11379.
https://doi.org/10.3390/en81011363

**AMA Style**

Park SC, Jin YG, Yoon YT. Designing a Profit-Maximizing Critical Peak Pricing Scheme Considering the Payback Phenomenon. *Energies*. 2015; 8(10):11363-11379.
https://doi.org/10.3390/en81011363

**Chicago/Turabian Style**

Park, Sung Chan, Young Gyu Jin, and Yong Tae Yoon. 2015. "Designing a Profit-Maximizing Critical Peak Pricing Scheme Considering the Payback Phenomenon" *Energies* 8, no. 10: 11363-11379.
https://doi.org/10.3390/en81011363