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Optimizing the Electricity Consumption with a High Degree of Flexibility Using a Dynamic Tariff and Stackelberg Game

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Abstract

Recent advancements in the sensor industry, smart metering systems and communication technology have led to interesting electricity consumption optimization opportunities that contribute to both peak reduction and bill savings and better integration of flexible appliances (including e-mobility). In combination with advanced tariffs, there has been a promising demand side management strategy devised from different perspectives: consumers interested in cost minimization, retailers and grid operators interested in peak minimization or hybrid solutions which reduce the costs to the extent of a certain peak level. In this paper, we propose an optimization algorithm that is significantly enhanced by a Stackelberg-type dynamic nonzero-sum game in which the consumers optimize and send their 24-h consumption schedules to the electricity retailer and receive the hourly tariff rates until their savings and the Flattening Index are maximized. Thus, it has been demonstrated that the one-iteration optimization is not as rewarding as the proposed game-optimization algorithm and that the results are heavily influenced by the degree of flexibility of the appliances. The algorithm is tested and validated using a large real input dataset, recorded at 15-min interval for a period of one year from a small residential community that consists of 11 modern houses with more than 300 appliances and high flexibility in terms of shifting, and the results highlight the consumers’ gain, FI and peak to average ratio indicators.

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Abbreviations

h :

Hour, \(h = \overline{1,24}\)

H :

Hour for shifting, \(H = \overline{1,24}\)

\(A^{h}\) :

Action or consumption of all consumers at hour h

\(A_{i}^{h}\) :

Hourly electricity consumption of a consumer i

\(A_{ij}^{h}\) :

Hourly electricity consumption of an appliance j that belongs to consumer i

\({\text{PA}}_{i}^{h}\) :

Hourly consumption of all programmable appliances that belong to consumer i

\({\text{PASW}}_{i}^{h}\) :

Hourly consumption of all shiftable without interruption appliances that belong to consumer i

\({\text{PASI}}_{i}^{h}\) :

Hourly consumption of all shiftable interruptible appliances that belong to consumer i

\({\text{MA}}_{i}\) :

Maximum hourly consumption imposed as a shifting condition to avoid new peaks

\(C_{i}\) :

Electricity cost for consumer i

\(C^{0}\) :

Initial electricity cost for community (unoptimized)

\(C_{i}^{0}\) :

Initial electricity cost for consumer i (unoptimized)

\({\text{CI}}_{ij}\) :

Daily cost of programmable interruptible appliance \(j\)

\({\text{CW}}_{ij}\) :

Daily cost of programmable non-interruptible appliance \(j\)

\({\text{CW}}_{ijH}\) :

Daily cost for a possible shift of a non-interruptible appliance j

\({\text{DA}}_{i}\) :

Daily actions or hourly consumption schedule array of consumer i

\({\text{TDA}}\) :

Vector of total hourly consumption of all consumers, \({\text{TDA}} = \left\{ {A^{h} } \right\}, h = \overline{1,24}\)

\(R_{i}\) :

Consumer’s i utility function

\(R_{{{\text{ret}}}}\) :

Retailer’s utility function

\(G\) :

Community gain

\(G_{i}\) :

Consumer’s gain

\(t^{{h_{{{\text{offpeak\_rate}}}} }}\) :

Tariff rate for off-peak hours

\(t^{{h_{{{\text{peak\_rate}}}} }}\) :

Tariff rate for peak hours

\(t^{{h_{{{\text{shoulder\_rate}}}} }}\) :

Tariff rate for day hours

\(t^{{h_{{{\text{rate}}}} }}\) :

Tariff rate for a specific hour of consumption, \(t^{{h_{{{\text{rate}}}} }} \in \{\) \(t^{{h_{{{\text{peak\_rate}}}} }}\), \(t^{{h_{{{\text{offpeak\_rate}}}} }}\), \(t^{{h_{{{\text{shoulder\_rate}}}} }} \}\)

\(t^{h}\) :

Tariff rate for a specific hour of consumption that is a function of \(A^{h}\)

\(a,b,c\) :

Coefficients of the tariff rate

DSM:

Demand side management

FI:

Flattening index

\(k_{j} \) :

Vector of the operating hours of appliance j

\(\alpha_{j}\) :

Vector of the hourly consumption of appliance j

\(Lk_{j}\) :

Length of vector \(k_{j}\) representing the total operating hours of appliance j

PAR:

Peak to average ratio

\(k_{{{\text{SI}}}}^{h}\) :

Vector of hours from which consumption of an interruptible appliance will be shifted

\({\text{NPA}}\) :

Non-programmable appliance

\({\text{NPA}}_{i}^{h}\) :

Hourly total consumption of all NPA for consumer i

RES:

Renewable energy sources

\(N_{i}^{h}\) :

Hourly total consumption of NPA for consumer i

\(sN_{i}\) :

Sorted \(N_{i}\)

\({\text{PA}}\) :

Programmable appliance that can be an interruptible or non-interruptible appliance

\({\text{PACM}}_{i}\) :

Constraints matrix \(\left( {24xm} \right)\) for PA of consumer i

\({\text{PACM}}_{ij}^{h}\) :

Element of matrix PACMi

\({\text{SI}}\) :

Shiftable interruptible appliance

\({\text{SI}}_{i}\) :

Matrix of SI appliances of consumer i

\({\text{SI}}_{ij}^{h}\) :

Element of matrix \({\text{SI}}_{i}\)

\(t{\text{SI}}_{i}\) :

Temporary matrix of SI appliances of consumer i

\({\text{SW}}\) :

Shiftable without interruption or non-interruptible appliance

\({\text{SW}}_{ij}^{h}\) :

Hourly consumption of a SW appliance

\({\text{SWA}}_{ij}\) :

Matrix with all possible shifts of an appliance j

\({\text{SWA}}_{ijH}^{h}\) :

Element of matrix \({\text{SWA}}_{ij}\)

\({\text{ToU}}\) :

Time-of-use tariff

\({\text{th}}\) :

Threshold for shifting

\(j\) :

A programmable appliance

\(j_{{{\text{mCW}}}}\) :

SW Appliance with the lowest cost selected for shifting

\(j_{{{\text{mCI}}}}\) :

SI appliance with the lowest cost selected for shifting

\(m\) :

Number of PA

\(m_{{{\text{SW}}}}\) :

Number of SW appliances

\(m_{{{\text{SI}}}}\) :

Number of SI appliances

\({\text{mCW}}\) :

Minimum cost of SW

\({\text{mCI}}\) :

Minimum cost of a SI appliance

\(n\) :

Number of consumers

\({\text{niter}}\) :

Number of iterations, \({\text{iter}} = \overline{{1,{\text{niter}}}}\)

DLC:

Direct load control

MILP:

Mixed integer linear programming

IoT:

Internet of Things

DER:

Distributed energy resources

EV:

Electric vehicle

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Acknowledgements

This paper presents the scientific results of the project “Intelligent system for trading on wholesale electricity market” (SMARTRADE), co-financed by the European Regional Development Fund (ERDF), through the Competitiveness Operational Programme (COP) 2014-2020, priority axis 1—Research, technological development and innovation (RD&I) to support economic competitiveness and business development, Action 1.1.4-Attracting high-level personnel from abroad in order to enhance the RD capacity, contract ID P_37_418, no. 62/05.09.2016, beneficiary: The Bucharest University of Economic Studies. This work was supported by a grant from the Romanian Ministry of Research and Innovation, CCCDI-UEFISCDI, project number 462PED/28.10.2020, project code PN-III-P2-2.1-PED-2019-1198, within PNCDI III. We thank Professor Virginia Mărăcine for the helpful and insightful discussions during revision.

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Authors and Affiliations

  1. Department of Economic Informatics and Cybernetics, The Bucharest University of Economic Studies, no. 6 Piaţa Romană, Bucharest, Romania

    Simona-Vasilica Oprea & Adela Bâra

  2. Department of Automatic Control and Electrical Engineering, Dunărea de Jos University, no. 47 Domnească, Galați, Romania

    George Adrian Ifrim

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  1. Simona-Vasilica Oprea
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Correspondence to Simona-Vasilica Oprea.

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Oprea, SV., Bâra, A. & Ifrim, G.A. Optimizing the Electricity Consumption with a High Degree of Flexibility Using a Dynamic Tariff and Stackelberg Game. J Optim Theory Appl 190, 151–182 (2021). https://doi.org/10.1007/s10957-021-01876-1

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