# Equilibrium Pricing with Duality-Based Method: Approach for Market-Oriented Capacity Remuneration Mechanism

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Works

#### 1.2. Overview of Used Methods and Optimization Techniques

#### 1.3. Novelty and Originality

## 2. Formulation of Primal and Dual Problem as a Method for Pricing Firm Capacity

#### 2.1. Discussion

#### 2.1.1. Goal Function in Primal

#### 2.1.2. Equilibrium Pricing of Firm Capacity

#### 2.1.3. More Than One Node and Cross-Border Participation

#### 2.1.4. Cross-Border Participation

#### 2.1.5. Issues in Cross-Border Participation

#### 2.1.6. Practical Implementation

## 3. Case Study

- (1st) The first event of the possible future state consists of outcomes: prolongation of the current situation, which translated in the extension of a decision of decommissioning of thermal power plants and nuclear power plants Krško and Paks.
- (2nd) The second event of the possible future state consists of the outcomes: decommissioning of nuclear power plant Krško and decommissioning of nuclear power plant Paks.
- (3rd) The third event of the possible future state consists of outcomes: decommissioning of thermal power plants which will be at the end of life by the year 2025, and there is no commissioning of new power plants by the year 2025.
- (4th) The fourth event of the possible future state consists of outcomes: decommissioning of thermal power plants which will be at the end of life by the year 2025; no commissioning of new power plants by the year 2025; decommissioning of nuclear power plant Krško, and decommissioning of nuclear power plant Paks.
- (5th) The fifth event of the possible future state consists of outcomes: decommissioning of thermal power plants which will be at the end of life by the year 2025; no commissioning of new power plants by the year 2025; decommissioning of nuclear power plant Krško; decommissioning of nuclear power plant Paks; and dry year.

## 4. Results

## 5. Discussion

#### 5.1. Exponential Price Decrease

#### 5.2. Law of Diminishing Marginal Utility

#### 5.3. Capacity Factor

#### 5.4. Expected Price and Revenue

#### 5.5. Practical Application

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Nomenclature

Sets | |

$i\in {\mathcal{I}}_{k}$ | Set of generating units in node $k$, ${\mathcal{I}}_{k}\subset \mathbb{N}$ |

$k\in \mathcal{K}$ | Set of nodes, $\mathcal{K}\subset \mathbb{N}$ |

$m\in {\mathsf{\Omega}}_{k}$ | Set of incidence power lines with node $k$, ${\mathsf{\Omega}}_{k}\subset \mathbb{N}$ |

$t\in \left[0,T\right]$ | Planning interval $\left[0,T\right]\subset \mathbb{R}$ |

Parameters | |

$AT{C}_{mk}$ | Available transfer capacity between node $m$ and node $k$ (MW) |

${c}_{i}\left({y}_{i}\right)$ | Short-run total cost of electricity generation, ${y}_{i}$, of the generator $i$ (€) |

${d}_{k}\left(t\right)$ | Electricity demand in time $t$ in node $k$ (MWh) |

$DS{M}_{i}\left(t\right)$ | Capacity reserved in unit i and time t of demand side |

${f}_{mk}\left(t\right)$ | Power flow on the line $m$ to node $k$ in time $t$ (MW) |

$F{C}_{i}\left(t\right)$ | Firm capacity reserved in unit $i$ in time step $t$ (MW) |

${k}_{Tu}^{i}$ | Maximal power output of the generator $i$ (MW) |

$M{C}_{i}\left({y}_{i}\right)$ | Short-run marginal cost of electricity generation, ${y}_{i}$, of the generator $i$ (€/MWh) |

${n}_{Tu}^{i}$ | Minimal power output of the generator $i$ (MW) |

$NG{C}_{k}$ | Net generation capacity in the node $k$ (MW) |

$NT{C}_{mk}$ | Net transfer capacity between nodes $m$ and $k$ (MW) |

$NT{F}_{mk}$ | Notified transmission flow between nodes $m$ and $k$ (MW) |

$Nu{C}_{k}$ | Non-usable capacity in the node $k$ (MW) |

${R}_{k}$ | System service margin for regulation reserve in node $k$ (MW) |

$RA{C}_{k}\left(t\right)$ | Reliable available capacity in node $k$ and time $t$ (MW) |

$R{M}_{k}\left(t\right)$ | Remaining margin in node $k$ and time $t$ (MW) |

${S}_{k}$ | System service margin for spinning reserve in node $k$ (MW) |

$SS{R}_{k}\left(t\right)$ | Spinning reserve and regulation in time $t$ and node $k$ (MW) |

${\overline{SSR}}_{k}$ | System service margin for spinning reserve and regulation (MW) |

$S{C}_{k}\left(t\right)$ | Spare capacity in time $t$ and node $k$ (MW) |

$TR{M}_{mk}$ | Transmission reliability margin between nodes $m$ and $k$ (MW) |

$TT{C}_{mk}$ | Total transfer capacity between nodes $m$ and $k$ (MW) |

$U{C}_{k}\left(t\right)$ | Unavailable capacity in time $t$ and node $k$ (MW) |

Variables | |

${\kappa}_{i}^{Tu}\left(t\right)$ | Shadow price in time $t$ of unit $i$ maximum power output constraint (€/MW) |

${\lambda}_{mk}\left(t\right)$ | Value of the Lagrange multiplier or shadow price in (€/MW) associated with the $AT{C}_{mk}$ |

${\nu}_{i}^{Tu}\left(t\right)$ | Shadow price in time $t$ of unit $i$ minimum power output constraint (€/MW) |

${\pi}_{k}^{e}\left(t\right)$ | Shadow price in time $t$ of electricity generation in node $k$ (€/MW∙h) |

${\pi}_{k}^{r}\left(t\right)$ | Shadow price in time $t$ of regulation reserve in node $k$(€/MW) |

${\pi}_{k}^{s}\left(t\right)$ | Shadow price in time $t$ of spinning reserve in time $t$ in node $k$(MW) |

${\pi}_{k}^{fc}\left(t\right)$ | Shadow price in time $t$ of firm capacity in node $k$(MW) |

${r}_{i}\left(t\right)$ | Regulation reserve of generator $i$ in time $t$ (MW) |

${s}_{i}\left(t\right)$ | Spinning reserve of generator $i$ in time $t$ (MW) |

${y}_{i}\left(t\right)$ | Power generation of generator $i$ in time $t$ (MW) |

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**Figure 2.**Depiction of the regional block consisting of Austria (AT), Slovenia (SI), Hungary (HU), and Croatia (HR) and possible power flows between countries.

**Figure 3.**Flow diagram depicts possible power transactions and power flows between power systems and energy markets in (

**a**) reference regional model; (

**b**) regional model; and (

**c**) model of isolated power systems.

**Figure 4.**The firm capacity (FC) prices ${\pi}_{k}^{fc}$ in the Croatian power system assessed for the worst-case scenario.

**Figure 5.**The capacity factor (CF) of a newly installed FC in the Croatian power system for different $M{C}_{fc}$ assessed for model outcomes.

**Figure 6.**The probabilistic reliability indexes (

**a**) expected energy not served (EENS); (

**b**) loss of load expectation (LOLE); and (

**c**) loss of load probability (LOLP) for the Croatian power system assessed for the model of isolated power systems and worst-case scenario.

**Table 1.**Comparison of FC prices for different countries and capacity remuneration mechanisms (CRMs).

Country | FC Price €/MW/Year | VoLL €/MWh [64] | Comment |
---|---|---|---|

Croatia | (1000–7000) 3100–21,700 | (1000) 3100 | Calculated FC price in this paper is in range (1000–7000) €/MW/year, with the use of referent VoLL value of (1000) €/MWh Value Adjusted to VoLL as in [64] gives FC price in range of 3100–21,700 €/MW/year |

UK | 6000–24,000 [68] | 15,900 | T-4 auctions. High volatility of prices between auctions. |

France | 0–20,000 [37,68] | 6920 | 0 €/MW/year in 2019. undermined confidence in CRM mechanism |

Ireland | 46,000 [68] | 11,520 | Similar to UK. But new build contracts are limited to 10 years. Closing coal and old gas fired power plants. Isolated system. |

Poland | 45,000–60,000 [68] | 6260 | Similar to UK. Need for significant new build. |

Italy | 75,000 [68] | 11,340 | Coal power plants excluded from CRM scheme. Reached price cap |

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**MDPI and ACS Style**

Ilak, P.; Herenčić, L.; Rajšl, I.; Raos, S.; Tomšić, Ž.
Equilibrium Pricing with Duality-Based Method: Approach for Market-Oriented Capacity Remuneration Mechanism. *Energies* **2021**, *14*, 567.
https://doi.org/10.3390/en14030567

**AMA Style**

Ilak P, Herenčić L, Rajšl I, Raos S, Tomšić Ž.
Equilibrium Pricing with Duality-Based Method: Approach for Market-Oriented Capacity Remuneration Mechanism. *Energies*. 2021; 14(3):567.
https://doi.org/10.3390/en14030567

**Chicago/Turabian Style**

Ilak, Perica, Lin Herenčić, Ivan Rajšl, Sara Raos, and Željko Tomšić.
2021. "Equilibrium Pricing with Duality-Based Method: Approach for Market-Oriented Capacity Remuneration Mechanism" *Energies* 14, no. 3: 567.
https://doi.org/10.3390/en14030567