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Real-Time Pricing of Smart Grid Based on Piece-Wise Linear Functions

  • Zhihong Xu , Liangyu Guo , Yan Gao EMAIL logo , Muhammad Hussain and Panhong Cheng
Published/Copyright: September 18, 2019
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Journal of Systems Science and Information
From the journal Journal of Systems Science and Information Volume 7 Issue 4

Abstract

In a power grid system, utility is a measure of the satisfaction of users’ electricity consumption; cost is a monetary value of electricity generated by the supplier. The utility and cost functions represent the satisfaction of different users and the supplier. Quadratic utility, logarithmic utility, and quadratic cost functions are widely used in social welfare maximization models of real-time pricing. These functions are not universal; they have to be discussed in detail for individual models. To overcome this problem, a piece-wise linear utility function and a piece-wise linear cost function with general properties are proposed in this paper. By smoothing the piece-wise linear utility and cost functions, a social welfare maximization model can be transformed into a differentiable convex optimization problem. A dual optimization method is used to solve the smoothed model. Through mathematical deduction and numerical simulations, the rationality of the model and the validity of the algorithm are verified as long as the elastic and cost coefficients take appropriate values. Thus, different user types and the supplier can be determined by selecting different elastic and cost coefficients.

Keywords: smart grid; real-time pricing; piece-wise linear utility and cost functions; dual optimization method

Supported by the Natural Science Foundation of China (11171221)

Acknowledgements

The authors gratefully acknowledge the editor and two anonymous referees for their insightful comments and helpful suggestions that led to a marked improvement of the article.

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Appendix 1

The error analysis between the smooth function and the piece-wise linear utility function is as follows:

  1. When 0 < x ≤ a1,

    |U^ε1(x)−U(x)|=|U^ε1(x)−U1(x)|=(ω1+ω2)x+(ω1−ω2)a12+(x−a1)2(ω2−ω1)2(x−a1)2+ε12−ω1x=(x−a1)2+ε12+(x−a1)2(x−a1)2+ε12(x−a1)(ω2−ω1)=12(x−a1)(ω2−ω1)ε(x−a1)2+ε12(x−a1)2+ε12+a1−x≤12(x−a1)(ω2−ω1)ε(x−a1)2+ε12(x−a1)2+ε12=12(x−a1)(ω2−ω1)ε(x−a1)2+ε≤12(x−a1)(ω2−ω1)ε(x−a1)2=12(ω2−ω1)εx−a1.

  2. When a1 < x ≤ a2,

    |U^ε2(x)−U(x)|=|U^ε2(x)−U2(x)|=(ω2+ω3)x+2(ω1−ω2)a1+(ω2−ω3)a22+(x−a2)2(ω3−ω2)2(x−a2)2+ε12−ω2x−(ω1−ω2)a1=(x−a2)2+ε12+(x−a2)2(x−a2)2+ε12(x−a2)(ω3−ω2)=12(x−a2)(ω3−ω2)ε(x−a2)2+ε12(x−a2)2+ε12+a2−x≤12(x−a2)(ω3−ω2)ε(x−a2)2+ε12(x−a2)2+ε12=12(x−a2)(ω3−ω2)ε(x−a2)2+ε≤12(x−a2)(ω3−ω2)ε(x−a2)2=12(ω3−ω2)εx−a2.

  3. When a2 < x ≤ a3,

    |U^ε3(x)−U(x)|=|U^ε3(x)−U3(x)|=|(ω3+ω4)x+2(ω1−ω2)a1+2(ω2−ω3)a2+(ω3−ω4)a32+(x−a3)2(ω4−ω3)2(x−a3)2+ε12−ω3x−(ω1−ω2)a1−(ω2−ω3)a2|=(x−a3)2+ε12+(x−a3)2(x−a3)2+ε12(x−a3)(ω4−ω3)=12(x−a3)(ω4−ω3)ε(x−a3)2+ε12(x−a3)2+ε12+a3−x≤12(x−a3)(ω4−ω3)ε(x−a3)2+ε12(x−a3)2+ε12=12(x−a3)(ω4−ω3)ε(x−a3)2+ε≤12(x−a3)(ω4−ω3)ε(x−a3)2=12(ω4−ω3)εx−a3.

  4. When x > a3,

    |U^ε3(x)−U(x)|=|U^ε3(x)−U4(x)|=|(ω3+ω4)x+2(ω1−ω2)a1+2(ω2−ω3)a2+(ω3−ω4)a32+(x−a3)2(ω4−ω3)2(x−a3)2+ε12−ω4x−(ω1−ω2)a1−(ω2−ω3)a2−(ω3−ω4)a3|=−(x−a3)2+ε12(x−a3)+(x−a3)22(x−a3)2+ε12(ω4−ω3)=(x−a3)2+ε12(x−a3)−(x−a3)22(x−a3)2+ε12(ω4−ω3)=(x−a3)2+ε12−(x−a3)2(x−a3)2+ε12(x−a3)(ω4−ω3)=12(x−a3)(ω4−ω3)ε(x−a3)2+ε12(x−a3)2+ε12+x−a3≤12(x−a3)(ω4−ω3)ε(x−a3)2+ε12(x−a3)2+ε12=12(x−a3)(ω4−ω3)ε(x−a3)2+ε≤12(x−a3)(ω4−ω3)ε(x−a3)2=12(ω4−ω3)εx−a3.

Appendix 2

The error analysis between the smooth function and the piece-wise linear cost function is as follows:

  1. When 0 < L ≤ L1,

    |C^ε1(L)−C(L)|=|C^ε1(L)−C1(L)|=C1(L)+C2(L)2+(L−L1)C2(L)−C1(L)2(L−L1)2+ε12−C1(L)=(L−L1)C2(L)−C1(L)2(L−L1)2+ε12+C2(L)−C1(L)2=(C2(L)−C1(L))(L−L1)2+ε12+L−L12(L−L1)2+ε12=(C2(L)−C1(L))ε2(L−L1)2+ε12(L−L1)2+ε12+L1−L≤(C2(L)−C1(L))ε2(L−L1)2+ε12(L−L1)2+ε12=(C2(L)−C1(L))ε2(L−L1)2+ε≤(C2(L)−C1(L))ε2(L−L1)2=(L−L1)(L1C2−C1L2)ε2(L−L1)2(L2−L1)L1=L1C2−C1L22(L−L1)(L2−L1)L1ε

  2. When L1 < L ≤ L2,

    |C^ε2(L)−C(L)|=|C^ε2(L)−C2(L)|=C2(L)+C3(L)2+(L−L2)C3(L)−C2(L)2(L−L2)2+ε12−C2(L)=(L−L2)C3(L)−C2(L)2(L−L2)2+ε12+C3(L)−C2(L)2=(C3(L)−C2(L))(L−L2)2+ε12+L−L22(L−L2)2+ε12=(C3(L)−C2(L))ε2(L−L2)2+ε12(L−L2)2+ε12+L2−L≤(C3(L)−C2(L))ε2(L−L2)2+ε12(L−L2)2+ε12=(C3(L)−C2(L))ε2(L−L2)2+ε≤(C3(L)−C2(L))ε2(L−L2)2=(L−L2)(C3−C2)(L2−L1)−(C2−C1)(L3−L2)ε(L3−L2)(L2−L1)2(L−L2)2=(C3−C2)(L2−L1)−(C2−C1)(L3−L2)2(L−L2)(L3−L2)(L2−L1)ε.

  3. When L2 < L ≤ L3,

    |C^ε3(L)−C(L)|=|C^ε3(L)−C3(L)|=C3(L)+C4(L)2+(L−L3)C4(L)−C3(L)2(L−L3)2+ε12−C3(L)=(L−L3)C4(L)−C3(L)2(L−L3)2+ε12+C4(L)−C3(L)2=(C4(L)−C3(L))(L−L3)2+ε12+L−L32(L−L3)2+ε12=(C4(L)−C3(L))ε2(L−L3)2+ε12(L−L3)2+ε12+L3−L≤(C4(L)−C3(L))ε2(L−L3)2+ε12(L−L3)2+ε12=(C4(L)−C3(L))ε2(L−L3)2+ε≤(C4(L)−C3(L))ε2(L−L3)2=(L−L3)(C4−C3)(L3−L2)−(C3−C2)(L4−L3)ε(L4−L3)(L3−L2)2(L−L3)2=(C4−C3)(L3−L2)−(C3−C2)(L4−L3)2(L−L3)(L4−L3)(L3−L2)ε.

  4. When L > L3,

    |C^ε3(L)−C(L)|=|C^ε3(L)−C4(L)|=C3(L)+C4(L)2+(L−L3)C4(L)−C3(L)2(L−L3)2+ε12−C4(L)=(L−L3)C4(L)−C3(L)2(L−L3)2+ε12−C4(L)−C3(L)2=(C4(L)−C3(L))(L−L3)2+ε12+L3−L2(L−L3)2+ε12=(C4(L)−C3(L))ε2(L−L3)2+ε12(L−L3)2+ε12+L−L3≤(C4(L)−C3(L))ε2(L−L3)2+ε12(L−L3)2+ε12=(C4(L)−C3(L))ε2(L−L3)2+ε≤(C4(L)−C3(L))ε2(L−L3)2=(L−L3)(C4−C3)(L3−L2)−(C3−C2)(L4−L3)ε(L4−L3)(L3−L2)2(L−L3)2=(C4−C3)(L3−L2)−(C3−C2)(L4−L3)2(L−L3)(L4−L3)(L3−L2)ε.

Received: 2019-03-04
Accepted: 2019-05-16
Published Online: 2019-09-18
Published in Print: 2019-09-25

© 2019 Walter De Gruyter GmbH, Berlin/Boston

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https://doi.org/10.21078/JSSI-2019-295-22
Keywords for this article
smart grid; real-time pricing; piece-wise linear utility and cost functions; dual optimization method

Articles in the same Issue

  1. Real-Time Pricing of Smart Grid Based on Piece-Wise Linear Functions
  2. How the Investor’s Risk Preferences Influence the Optimal Allocation in a Credibilistic Portfolio Problem
  3. Bootstrap LM Tests for Spatial Dependence in Panel Data Models with Fixed Effects
  4. Precautionary Effort Investment under Cross Risk Aversion
  5. Analysis on Chinese Airline Network Invulnerability
  6. Research on Rumor Spreading Model with Time Delay and Control Effect
  7. Teaching Systems Theory/Thinking/Behavior: Systemic Behavior Instead of One-Sidedness: Making Bridges Among Specialists
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