# Shadow Pricing of Electric Power Interruptions for Distribution System Operators in Finland

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

_{2}, SO

_{2}and NO

_{x}in the United States coal power industry [23]. On the other hand, ref. [24] adopts parametric distance function approach to calculate the value of power outages for French DSOs. The purpose of this paper is to use shadow pricing techniques to assess customer interruption costs from the DSO perspective. This paper aims to provide a reliable CIC estimation method for the DSOs, so that they can make careful arrangements in operational and capital costs. Another crucial contribution of this paper is to show the weaknesses of the customer compensation scheme in Finland. We propose that a fairer compensation scheme should be designed to reflect the true costs of the power interruptions incurred from the DSO point of view. We should note that VoLL or the worth of ENS are not in the scope of this paper. The following paper is organized as follows: Section 2 introduces the methodology of the directional distance function and shadow pricing of a production technology. Section 3 presents the empirical study and the results of the shadow pricing of power interruption analysis for 78 DSOs from Finland. Section 4 and Section 5 include our discussion remarks and conclusion.

## 2. Directional Distance Function and the Shadow Pricing of Electric Power Interruptions

_{o}[27]. Let g = (g

_{y}, g

_{b}) be a directional vector and β be the maximum expansion of good outputs in the direction of g

_{y}and the minimum contraction of the bad outputs in the direction of g

_{b}, then D

_{o}is defined as:

_{y}> 0 mean the expansion of desirable output, while g

_{b}> 0 mean the contraction of the undesirable output. The relationship between the directional distance function and the revenue function reveals the shadow price for the undesirable outputs [18]. Let p indicate the good output prices and q indicate the bad output prices. These are represented as:

_{o}as:

- l: the constant of the quadratic directional distance function,
- α
_{n}: the input coefficients, - β
_{m}: the desirable output coefficients, - γ
_{j}: the undesirable output coefficients, - α
_{mn′}: the quadratic of input coefficients, - β
_{mm′}: the quadratic of desirable output coefficients, - γ
_{jj′}: the quadratic of undesirable output coefficients, - δ
_{nm}: the product of the inputs and desirable outputs coefficients, - η
_{nj}: the product of the inputs and undesirable outputs coefficients, - μ
_{mj}: the coefficients of the product of the desirable and undesirable outputs.

_{n}, α

_{mn′}, β

_{m}, β

_{mm′}, γ

_{j}, γ

_{jj′}, δ

_{nm}, η

_{nj}, μ

_{mj}, are chosen to minimize the sum of the deviations of the directional distance function value from the frontier technology (in our case the electric power supply). The coefficients of Equation (16) are calculated via solving Equation (17) with Python, by adopting the directional vector as g = (1, 1). Equation (18) requires the output–input vector to be feasible. Equations (19) and (20) impose the monotonicity conditions of Equations (13) and (14). Equation (21) imposes positive monotonicity on the inputs for the mean level of input usage. That is, at the mean level of inputs, x¯, an increase in input usage holding good and bad outputs constant, causes the directional output distance function to increase, implying greater inefficiency. Equation (22) is due to the translation property of Equation (6).

_{n}, α

_{mn′}, β

_{m}, β

_{mm′}, γ

_{j}, γ

_{jj′}, δ

_{nm}, η

_{nj}, μ

_{mj}and are solved with Python.

## 3. Empirical Study and Results

#### 3.1. Empirical Study

#### 3.2. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

DSO | 2013 | 2014 | 2015 | DSO | 2013 | 2014 | 2015 |
---|---|---|---|---|---|---|---|

Äänekosken Energia Oy | 0.454 | 0.449 | 0.457 | Lehtimäen Sähkö Oy | 0.457 | 0.461 | 0.443 |

Alajärven Sähkö Oy | 0.447 | 0.466 | 0.278 | Leppäkosken Sähkö Oy | 0.457 | 0.470 | 0.468 |

Caruna Espoo Oy | 0.462 | 0.472 | 0.474 | LE-Sähköverkko Oy | 0.477 | 0.481 | 0.482 |

Caruna Oy | 0.348 | 0.447 | 0.439 | Mäntsälän Sähkö Oy | 0.458 | 0.474 | 0.467 |

Ekenäs Energi Ab | 0.477 | 0.470 | 0.471 | Muonion Sähköosuuskunta | 0.426 | 0.343 | 0.035 |

Elenia Oy | 0.295 | 0.445 | 0.209 | Naantalin Energia Oy | 0.479 | 0.479 | 0.481 |

Enontekiön Sähkö Oy | 0.000 | 0.357 | 0.337 | Nurmijärven Sähköverkko Oy | 0.463 | 0.474 | 0.469 |

ESE-Verkko Oy | 0.480 | 0.482 | 0.479 | Nykarleby Kraftverk Ab | 0.429 | 0.422 | 0.426 |

Esse Elektro-Kraft Ab | 0.416 | 0.452 | 0.286 | Oulun Energia Siirto ja Jakelu Oy | 0.468 | 0.470 | 0.472 |

Etelä-Suomen Energia Oy | 0.364 | 0.399 | 0.433 | Oulun Seudun S. Verkkopalvelut Oy | 0.452 | 0.467 | 0.443 |

Forssan Verkkopalvelut Oy | 0.481 | 0.482 | 0.482 | Outokummun Energia Oy | 0.457 | 0.432 | 0.462 |

Haminan Energia Oy | 0.478 | 0.480 | 0.476 | Paneliankosken Voima Oy | 0.414 | 0.472 | 0.466 |

Haukiputaan Sähköosuuskunta | 0.471 | 0.474 | 0.478 | Parikkalan Valo Oy | 0.179 | 0.444 | 0.349 |

Helen Sähköverkko Oy | 0.483 | 0.483 | 0.482 | Pellon Sähkö Oy | 0.465 | 0.449 | 0.440 |

Herrfors Nät-Verkko Oy Ab | 0.347 | 0.433 | 0.384 | PKS Sähkönsiirto Oy | 0.376 | 0.376 | 0.066 |

Iin Energia Oy | 0.477 | 0.478 | 0.459 | Pori Energia Sähköverkot Oy | 0.386 | 0.440 | 0.449 |

Imatran Seudun Sähkönsiirto Oy | 0.268 | 0.448 | 0.394 | Porvoon Sähköverkko Oy | 0.399 | 0.439 | 0.449 |

Järvi-Suomen Energia Oy | 0.242 | 0.438 | 0.319 | Raahen Energia Oy | 0.482 | 0.480 | 0.481 |

Jeppo Kraft Andelslag | 0.454 | 0.453 | 0.445 | Rantakairan Sähkö Oy | 0.465 | 0.466 | 0.456 |

JE-Siirto Oy | 0.481 | 0.480 | 0.482 | Rauman Energia Oy | 0.445 | 0.475 | 0.463 |

Jylhän Sähköosuuskunta | 0.457 | 0.474 | 0.435 | Rovakaira Oy | 0.413 | 0.452 | 0.431 |

Karhu Voima Oy | 0.481 | 0.477 | 0.464 | Rovaniemen Verkko Oy | 0.480 | 0.480 | 0.474 |

Kemin Energia Oy | 0.478 | 0.479 | 0.467 | Sallila Sähkönsiirto Oy | 0.434 | 0.466 | 0.465 |

Keminmaan Energia Oy | 0.446 | 0.478 | 0.371 | Savon Voima Verkko Oy | 0.175 | 0.287 | 0.146 |

KENET Oy | 0.469 | 0.473 | 0.476 | Seiverkot Oy | 0.470 | 0.475 | 0.476 |

Keravan Energia Oy | 0.472 | 0.468 | 0.479 | Tampereen Sähköverkko Oy | 0.449 | 0.478 | 0.450 |

Keuruun Sähkö Oy | 0.355 | 0.414 | 0.378 | Tenergia Oy | 0.405 | 0.427 | 0.391 |

Koillis-Lapin Sähkö Oy | 0.371 | 0.445 | 0.447 | Tornion Energia Oy | 0.469 | 0.473 | 0.447 |

Koillis-Satakunnan Sähkö Oy | 0.441 | 0.443 | 0.415 | Tornionlaakson Sähkö Oy | 0.418 | 0.463 | 0.436 |

Kokemäen Sähkö Oy | 0.425 | 0.466 | 0.468 | Tunturiverkko Oy | 0.443 | 0.451 | 0.412 |

Köyliön-Säkylän Sähkö Oy | 0.452 | 0.469 | 0.473 | Turku Energia Sähköverkot Oy | 0.469 | 0.481 | 0.481 |

Kronoby Elverk Ab | 0.440 | 0.410 | 0.459 | Vaasan Sähköverkko Oy | 0.417 | 0.430 | 0.434 |

KSS Verkko Oy | 0.454 | 0.455 | 0.470 | Vakka-Suomen Voima Oy | 0.352 | 0.458 | 0.430 |

Kuopion Sähköverkko Oy | 0.481 | 0.481 | 0.478 | Valkeakosken Energia Oy | 0.476 | 0.476 | 0.108 |

Kuoreveden Sähkö Oy | 0.445 | 0.468 | 0.418 | Vantaan Energia Sähköverkot Oy | 0.479 | 0.480 | 0.481 |

Kymenlaakson Sähköverkko Oy | 0.375 | 0.426 | 0.391 | Vatajankosken Sähkö Oy | 0.419 | 0.455 | 0.451 |

Lammaisten Energia Oy | 0.457 | 0.473 | 0.475 | Verkko Korpela Oy | 0.323 | 0.253 | 0.378 |

Lankosken Sähkö Oy | 0.274 | 0.410 | 0.384 | Vetelin Sähkölaitos Oy | 0.431 | 0.474 | 0.135 |

Lappeenrannan Energiaverkot Oy | 0.401 | 0.455 | 0.437 | Vimpelin Voima Oy | 0.359 | 0.446 | 0.360 |

DSO | 2013 | 2014 | 2015 | DSO | 2013 | 2014 | 2015 |
---|---|---|---|---|---|---|---|

Äänekosken Energia Oy | 1.87 | 2.20 | 1.68 | Lehtimäen Sähkö Oy | 1.66 | 1.44 | 2.57 |

Alajärven Sähkö Oy | 2.29 | 1.10 | 13.29 | Leppäkosken Sähkö Oy | 1.67 | 0.87 | 1.02 |

Caruna Espoo Oy | 1.37 | 0.74 | 0.60 | LE-Sähköverkko Oy | 0.42 | 0.17 | 0.11 |

Caruna Oy | 8.69 | 2.30 | 2.85 | Mäntsälän Sähkö Oy | 1.62 | 0.65 | 1.07 |

Ekenäs Energi Ab | 0.41 | 0.85 | 0.80 | Muonion Sähköosuuskunta | 3.35 | 9.00 | 30.21 |

Elenia Oy | 12.16 | 2.41 | 17.93 | Naantalin Energia Oy | 0.32 | 0.29 | 0.20 |

Enontekiön Sähkö Oy | 32.80 | 8.10 | 9.35 | Nurmijärven Sähköverkko Oy | 1.32 | 0.60 | 0.96 |

ESE-Verkko Oy | 0.22 | 0.09 | 0.31 | Nykarleby Kraftverk Ab | 3.46 | 3.91 | 3.65 |

Esse Elektro-Kraft Ab | 4.31 | 2.01 | 12.78 | Oulun Energia Siirto ja Jakelu Oy | 0.98 | 0.84 | 0.76 |

Etelä-Suomen Energia Oy | 7.62 | 5.39 | 3.22 | Oulun Seudun S. Verkkopalvelut Oy | 1.99 | 1.05 | 2.56 |

Forssan Verkkopalvelut Oy | 0.19 | 0.09 | 0.10 | Outokummun Energia Oy | 1.69 | 3.25 | 1.34 |

Haminan Energia Oy | 0.35 | 0.27 | 0.49 | Paneliankosken Voima Oy | 4.38 | 0.73 | 1.10 |

Haukiputaan Sähköosuuskunta | 0.82 | 0.59 | 0.38 | Parikkalan Valo Oy | 19.99 | 2.49 | 8.62 |

Helen Sähköverkko Oy | 0.08 | 0.06 | 0.11 | Pellon Sähkö Oy | 1.18 | 2.18 | 2.74 |

Herrfors Nät-Verkko Oy Ab | 8.71 | 3.22 | 6.31 | PKS Sähkönsiirto Oy | 6.86 | 6.85 | 27.96 |

Iin Energia Oy | 0.44 | 0.40 | 1.56 | Pori Energia Sähköverkot Oy | 6.22 | 2.76 | 2.20 |

Imatran Seudun Sähkönsiirto Oy | 13.98 | 2.26 | 5.67 | Porvoon Sähköverkko Oy | 5.38 | 2.80 | 2.21 |

Järvi-Suomen Energia Oy | 15.73 | 2.87 | 10.54 | Raahen Energia Oy | 0.12 | 0.27 | 0.18 |

Jeppo Kraft Andelslag | 1.90 | 1.95 | 2.41 | Rantakairan Sähkö Oy | 1.16 | 1.11 | 1.74 |

JE-Siirto Oy | 0.20 | 0.26 | 0.13 | Rauman Energia Oy | 2.44 | 0.55 | 1.30 |

Jylhän Sähköosuuskunta | 1.67 | 0.59 | 3.09 | Rovakaira Oy | 4.44 | 2.00 | 3.33 |

Karhu Voima Oy | 0.19 | 0.40 | 1.24 | Rovaniemen Verkko Oy | 0.22 | 0.24 | 0.63 |

Kemin Energia Oy | 0.39 | 0.33 | 1.03 | Sallila Sähkönsiirto Oy | 3.12 | 1.13 | 1.21 |

Keminmaan Energia Oy | 2.36 | 0.37 | 7.16 | Savon Voima Verkko Oy | 20.28 | 12.66 | 22.32 |

KENET Oy | 0.92 | 0.69 | 0.49 | Seiverkot Oy | 0.86 | 0.57 | 0.52 |

Keravan Energia Oy | 0.73 | 0.97 | 0.32 | Tampereen Sähköverkko Oy | 2.17 | 0.37 | 2.12 |

Keuruun Sähkö Oy | 8.20 | 4.38 | 6.71 | Tenergia Oy | 5.00 | 3.60 | 5.86 |

Koillis-Lapin Sähkö Oy | 7.20 | 2.42 | 2.32 | Tornion Energia Oy | 0.95 | 0.68 | 2.29 |

Koillis-Satakunnan Sähkö Oy | 2.71 | 2.59 | 4.37 | Tornionlaakson Sähkö Oy | 4.17 | 1.28 | 3.01 |

Kokemäen Sähkö Oy | 3.68 | 1.15 | 1.00 | Tunturiverkko Oy | 2.57 | 2.08 | 4.52 |

Köyliön-Säkylän Sähkö Oy | 1.99 | 0.91 | 0.69 | Turku Energia Sähköverkot Oy | 0.96 | 0.17 | 0.16 |

Kronoby Elverk Ab | 2.77 | 4.69 | 1.57 | Vaasan Sähköverkko Oy | 4.24 | 3.40 | 3.17 |

KSS Verkko Oy | 1.86 | 1.81 | 0.86 | Vakka-Suomen Voima Oy | 8.41 | 1.64 | 3.36 |

Kuopion Sähköverkko Oy | 0.19 | 0.19 | 0.35 | Valkeakosken Energia Oy | 0.48 | 0.47 | 25.00 |

Kuoreveden Sähkö Oy | 2.45 | 0.98 | 4.14 | Vantaan Energia Sähköverkot Oy | 0.31 | 0.27 | 0.17 |

Kymenlaakson Sähköverkko Oy | 6.88 | 3.63 | 5.87 | Vatajankosken Sähkö Oy | 4.08 | 1.78 | 2.09 |

Lammaisten Energia Oy | 1.67 | 0.67 | 0.53 | Verkko Korpela Oy | 10.33 | 14.93 | 6.73 |

Lankosken Sähkö Oy | 13.56 | 4.67 | 6.31 | Vetelin Sähkölaitos Oy | 3.33 | 0.65 | 23.08 |

Lappeenrannan Energiaverkot Oy | 5.27 | 1.78 | 2.96 | Vimpelin Voima Oy | 7.95 | 2.36 | 7.91 |

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Inputs | Desirable Output | Undesirable Output | ||
---|---|---|---|---|

SC (%) | OPEX (k €) | ES (GWh) | CML (k mins) | |

2013 | ||||

Mean | 47.27 | 3015.51 | 619.92 | 14,299.67 |

Stdev. | 25.60 | 5674.63 | 1200.07 | 45,908.75 |

Minimum | 3.04 | 35.35 | 16.67 | 0.81 |

Maximum | 100.00 | 32,156.33 | 7492.00 | 300,711.21 |

2014 | ||||

Mean | 48.65 | 2891.61 | 616.07 | 5367.14 |

Stdev. | 25.46 | 5021.57 | 1189.88 | 14,184.51 |

Minimum | 3.23 | 55.30 | 16.38 | 1.90 |

Maximum | 100.00 | 25,616.35 | 7425.00 | 85,712.50 |

2015 | ||||

Mean | 50.34 | 3134.97 | 613.64 | 14,575.97 |

Stdev. | 25.27 | 5857.64 | 1177.45 | 56,013.63 |

Minimum | 3.30 | 71.00 | 15.84 | 6.18 |

Maximum | 100.00 | 29,906.08 | 7283.00 | 448,823.76 |

Variables | 2013 | 2014 | 2015 |
---|---|---|---|

l | 16.3252 | 0.007764978 | 0.017882857 |

α_{1} | 0 | 0 | 0 |

α_{2} | −7 × 10^{−10} | 0.24284221 | 0.27902876 |

β_{1} | −1 | −0.99987881 | −1 |

γ_{1} | 0 | 0.000121188 | 0 |

α_{11} | 0 | 0 | 0 |

α_{22} | −1 × 10^{−10} | 0.050119522 | 0.020527845 |

β_{11} | 0 | −1.977 × 10^{−7} | 0 |

γ_{11} | 0 | −1.977 × 10^{−7} | 0 |

α_{12} | 0 | 0 | 0 |

δ_{11} | 0 | 0 | 0 |

δ_{21} | 0 | 0.000204027 | 0 |

η_{11} | 0 | 0 | 0 |

η_{21} | 0 | 0.000204027 | 0 |

μ_{11} | 0 | −1.977 × 10^{−7} | 0 |

Standard Customer Compensation | |
---|---|

Outage Duration (h) | Compensation (%) |

12–24 | 10 |

24–72 | 25 |

72–120 | 50 |

120–192 | 100 |

192–288 | 150 |

>288 | 200 |

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## Share and Cite

**MDPI and ACS Style**

Küfeoğlu, S.; Gündüz, N.; Chen, H.; Lehtonen, M. Shadow Pricing of Electric Power Interruptions for Distribution System Operators in Finland. *Energies* **2018**, *11*, 1831.
https://doi.org/10.3390/en11071831

**AMA Style**

Küfeoğlu S, Gündüz N, Chen H, Lehtonen M. Shadow Pricing of Electric Power Interruptions for Distribution System Operators in Finland. *Energies*. 2018; 11(7):1831.
https://doi.org/10.3390/en11071831

**Chicago/Turabian Style**

Küfeoğlu, Sinan, Niyazi Gündüz, Hao Chen, and Matti Lehtonen. 2018. "Shadow Pricing of Electric Power Interruptions for Distribution System Operators in Finland" *Energies* 11, no. 7: 1831.
https://doi.org/10.3390/en11071831