# Dynamic Pricing for Demand Response Considering Market Price Uncertainty

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Objectives

#### 1.3. Literature Review

#### 1.4. Contributions

- It considers the retail rates as variables for the short-term scheduling horizon. Dynamic sale prices are determined for different customer groups based on their short run price elasticities. The elastic behavior of end-users towards the blocks of real-time prices is taken into account in the model. Retail rates can be calculated for a lead time of one day.
- The model considers involvement in bilateral contracts and the pool market to determine the optimal electricity procurement policy, without knowing the precise values of price and demand.
- It enables the REPs to tune their level of conservatism through a flexible decision-making framework. The optimal solutions are immune against the forecast errors to some controlled extent.

#### 1.5. Paper Organization

## 2. Problem Formulation

- Customers are equipped with smart meters. Thus the dynamic prices offered by the REP can be employed at the end-users’ points.
- REPs have sufficient data about the DG units’ cost functions, bilateral contracts, and the price elasticity of demand for each time period.
- Customers behave elastically in the short term. They are assumed to be flexible. If the price increases, they shift demand to periods with lower prices.
- Each node refers to zones with similar market prices. Therefore, this model can be used by REPs that serve loads at several distribution networks.
- The expected payoff of the REP is calculated based on price and load demand forecasts rather than the actual values of these inputs.

#### 2.1. Forward Contracts

#### 2.2. Call Options

#### 2.3. DGs

## 3. Robust Optimization Model

## 4. Case Studies

- Case 1: call options, forward contracts, and DG units as risk hedging tools.
- Case 2: call options and forward contracts as risk hedging tools.
- Case 3: call options and DG units as risk hedging tools.
- Case 4: forward contracts and DG units as risk hedging tools.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Nomenclature

Indices | |

$t$ | Time periods |

$l$ | Loads |

$b$ | Blocks |

$n$ | Nodes |

$f$ | Bilateral agreements |

$j$ | Call options |

$i$ | DG units |

Parameters | |

$\tau $ | Duration of each time period (h) |

${\lambda}^{M}$ | Electricity price at the pool market (€/kWh) |

${R}_{l}^{P}$ | Average retail rate for customer group l at block b (€/kWh) |

${\lambda}_{f}$ | Energy price in bilateral agreement f (€/kWh) |

${R}_{l}^{Max/Min}$ | Maximum/minimum retail rates offered to load group l (€/kWh) |

${\lambda}_{j}$ | Price of call option j (€/kWh) |

${D}_{l}^{P}$ | Predicted base load for customer group l (kW) |

$Pe{r}_{j}$ | Premium of call option j (€/kWh) |

${a}_{l},{b}_{l}$ | Coefficients of hourly demand function of the customers (kW,(kW)^{2}h/€) |

${P}_{f}^{Max/Min}$ | Maximum/minimum power of bilateral agreement f (kW) |

${D}_{l}^{Max/Min}$ | Maximum/minimum demand of load group l (€/kWh) |

$M{R}_{i}^{Up/Dn}$ | Ramping up/down limit of DG unit i (kW/h) |

${P}_{j}^{b}$ | Power blocks in call option j (kW) |

${p}_{i,b}^{Max}$ | Generation limit in block b of the piecewise linear cost function (kW) |

${p}_{i}^{Min}$ | Minimum power output of DG unit i (kW) |

$M{R}_{i}^{Up/Dn}$ | Ramping up/down limit of DG unit i (kW/h) |

$F{C}_{i}^{DG}$ | Fixed cost of DG unit i (€/h) |

${C}_{i,b}^{DG}$ | Generation cost of DG i in block b of the piecewise linear cost function (kW) |

Variables | |

${r}_{l}$ | Retail rates for the load group l (€/kWh) |

${d}_{l}$ | Aggregated demand of customers in load group l (kW) |

${p}_{j}$ | Power of call option j (kW) |

${p}^{M}$ | Power purchase from the pool market (kW) |

${p}_{i,b}^{DG}$ | Power generation of DG i in block b of the piecewise linear cost function (kW) |

${p}_{i}^{DG}$ | Power generation of DG i (kW) |

${p}_{f}$ | Power procured through bilateral agreement f (kW) |

${x}_{f}$ | Binary decision variable which is 1 when the bilateral agreement f is selected |

${x}_{j}^{b}$ | Binary decision variable which is 1 if power block b of call option j is selected |

${x}_{l}^{b}$ | Binary decision variable which is 1 if block b of the retail price for customer l is selected |

Sets | |

$T$ | Time periods |

$N$ | Nodes that the REP is serving the loads. |

$L$ | Loads served by the REP |

$J$ | Available call options |

$F$ | Available bilateral contracts |

$I$ | DG units owned by the REP |

${I}_{n}$ | DG units located at node n |

${L}_{n}$ | Loads connected to node n |

${F}_{n}^{t}$ | Bilateral contracts available at node n at time period t |

${B}_{l}^{t}$ | Demand function blocks of load group l at time period t |

${B}_{j}$ | Blocks of call option j |

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Customer Groups | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Number of residential consumers | 1496 | 375 | 996 | 927 | 611 | 256 | 86 | 661 | 317 | 1103 | 587 | 1271 | 1115 | 878 |

Nodes | 1 | 5 | 3 | 4 | 1 | 2 | 6 | 4 | 3 | 6 | 2 | 4 | 1 | 5 |

Forward Contracts | Node | Available Time Periods | Minimum Power (kW) | Maximum Power (kW) | Price (€/kWh) |
---|---|---|---|---|---|

1 | 2 | 5, 6, 7, 8, 9, 10 | 20 | 350 | 0.045 |

2 | 3 | 13, 14, 15, 16, 17 | 30 | 430 | 0.05 |

3 | 2 | 1, 2, 3, 4, 5, 6, 7 | 40 | 360 | 0.035 |

4 | 4 | 1, 2, 19, 20, 21, 22, 23, 24 | 15 | 410 | 0.045 |

5 | 5 | 11, 12, 13, 14, 15, 16, 17 | 35 | 506 | 0.051 |

6 | 5 | 9, 10, 11, 12, 13, 14, 15 | 19 | 120 | 0.056 |

7 | 1 | 6, 7, 8, 9, 10, 11, 12, 13, 14 | 40 | 155 | 0.029 |

8 | 6 | 5, 6, 7, 8, 9, 10 | 45 | 419 | 0.038 |

Characteristics | Value | ||||||
---|---|---|---|---|---|---|---|

DG units | 1 | 2 | 3 | 4 | 5 | 6 | |

Nodes | 4 | 4 | 2 | 1 | 2 | 3 | |

Maximum output of DG units (kw) | Block 1 | 19 | 15 | 33 | 43 | 57 | 67 |

Block 2 | 52 | 62 | 44 | 97 | 68 | 89 | |

Block 3 | 72 | 89 | 73 | 118 | 100 | 110 | |

Block 4 | 93 | 127 | 88 | 139 | 118 | 124 | |

Block 5 | 120 | 143 | 118 | 173 | 129 | 141 | |

Generation cost (€/kWh) | Block 1 | 0.045 | 0.046 | 0.039 | 0.040 | 0.043 | 0.050 |

Block 2 | 0.049 | 0.048 | 0.042 | 0.046 | 0.049 | 0.053 | |

Block 3 | 0.052 | 0.053 | 0.044 | 0.055 | 0.052 | 0.058 | |

Block 4 | 0.055 | 0.056 | 0.050 | 0.060 | 0.055 | 0.062 | |

Block 5 | 0.065 | 0.071 | 0.055 | 0.062 | 0.059 | 0.069 | |

Ramping up limit (kW/h) | 27 | 38 | 40 | 28 | 43 | 29 | |

Ramping down limit (kW/h) | 29 | 40 | 43 | 31 | 47 | 31 |

Call Options | Nodes | Available Time Periods | Premium (€/kWh) | Price (€/kWh) | Blocks of Power (kW) | ||||
---|---|---|---|---|---|---|---|---|---|

Block 1 | Block 2 | Block 3 | Block 4 | Block 5 | |||||

1 | 2 | 3, 4, 5, 6, 7 | 0.014 | 0.023 | 0 | 117 | 234 | 351 | 468 |

2 | 3 | 12, 13, 14, 15 | 0.020 | 0.054 | 0 | 89 | 177 | 266 | 354 |

3 | 4 | 20, 21, 22, 23 | 0.009 | 0.018 | 0 | 68 | 135 | 203 | 271 |

4 | 6 | 1, 2, 3, 21, 22, 23, 24 | 0.016 | 0.027 | 0 | 39 | 78 | 117 | 156 |

Energy Procurement Sources | Deterministic Model | Robust Model | |||||||
---|---|---|---|---|---|---|---|---|---|

Case 1 | Case 2 | Case 3 | Case 4 | Case 1 | Case 2 | Case 3 | Case 4 | ||

Node 1 | Market | 97.83% | 99.15% | 98.68% | 97.83% | 97.50% | 99.15% | 98.35% | 97.50% |

DG units | 1.32% | 0.00% | 1.32% | 1.32% | 1.65% | 0.00% | 1.65% | 1.65% | |

Call options | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | |

Forward contracts | 0.85% | 0.85% | 0.00% | 0.85% | 0.85% | 0.85% | 0.00% | 0.85% | |

Node 2 | Market | 89.12% | 93.25% | 95.87% | 89.12% | 89.12% | 93.25% | 95.87% | 89.12% |

DG units | 4.13% | 0.00% | 4.13% | 4.13% | 4.13% | 0.00% | 4.13% | 4.13% | |

Call options | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | |

Forward contracts | 6.75% | 6.75% | 0.00% | 6.75% | 6.75% | 6.75% | 0.00% | 6.75% | |

Node 3 | Market | 91.92% | 94.94% | 96.98% | 91.92% | 91.92% | 94.94% | 96.98% | 91.92% |

DG units | 3.02% | 0.00% | 3.02% | 3.02% | 3.02% | 0.00% | 3.02% | 3.02% | |

Call options | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | |

Forward contracts | 5.06% | 5.06% | 0.00% | 5.06% | 5.06% | 5.06% | 0.00% | 5.06% | |

Node 4 | Market | 96.48% | 97.91% | 97.86% | 97.19% | 95.89% | 97.91% | 97.29% | 96.62% |

DG units | 1.43% | 0.00% | 1.43% | 1.43% | 2.02% | 0.00% | 2.00% | 2.00% | |

Call options | 0.71% | 0.71% | 0.71% | 0.00% | 0.71% | 0.71% | 0.71% | 0.00% | |

Forward contracts | 1.38% | 1.38% | 0.00% | 1.38% | 1.38% | 1.38% | 0.00% | 1.38% | |

Node 5 | Market | 92.54% | 92.54% | 100.00% | 92.54% | 92.54% | 92.54% | 100.00% | 92.54% |

DG units | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | |

Call options | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | |

Forward contracts | 7.46% | 7.46% | 0.00% | 7.46% | 7.46% | 7.46% | 0.00% | 7.46% | |

Node 6 | Market | 91.75% | 91.75% | 97.21% | 94.54% | 90.80% | 90.80% | 97.21% | 93.58% |

DG units | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | 0.00% | |

Call options | 2.79% | 2.79% | 2.79% | 0.00% | 2.79% | 2.79% | 2.79% | 0.00% | |

Forward contracts | 5.46% | 5.46% | 0.00% | 5.46% | 6.42% | 6.42% | 0.00% | 6.42% |

Case | Deterministic Model | Robust Model: Dynamic Rates | Robust Model: Fixed Rates |
---|---|---|---|

Case 1 | 37,363.54 | 35,957.07 | 33,729.00 |

Case 2 | 37,320.32 | 35,887.40 | 33,665.10 |

Case 3 | 37,301.88 | 35,886.23 | 33,661.75 |

Case 4 | 37,323.67 | 35,911.66 | 33,690.80 |

Case | Deterministic Approach | Robust Approach |
---|---|---|

Case 1 | 34,021.31 | 34,980.74 |

Case 2 | 33,979.05 | 34,882.30 |

Case 3 | 33,857.46 | 34,867.15 |

Case 4 | 34,006.43 | 34,908.16 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Fotouhi Ghazvini, M.A.; Soares, J.; Morais, H.; Castro, R.; Vale, Z.
Dynamic Pricing for Demand Response Considering Market Price Uncertainty. *Energies* **2017**, *10*, 1245.
https://doi.org/10.3390/en10091245

**AMA Style**

Fotouhi Ghazvini MA, Soares J, Morais H, Castro R, Vale Z.
Dynamic Pricing for Demand Response Considering Market Price Uncertainty. *Energies*. 2017; 10(9):1245.
https://doi.org/10.3390/en10091245

**Chicago/Turabian Style**

Fotouhi Ghazvini, Mohammad Ali, João Soares, Hugo Morais, Rui Castro, and Zita Vale.
2017. "Dynamic Pricing for Demand Response Considering Market Price Uncertainty" *Energies* 10, no. 9: 1245.
https://doi.org/10.3390/en10091245