# Dynamic Consensus-Based ADMM Strategy for Economic Dispatch with Demand Response in Power Grids

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## Abstract

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## 1. Introduction

- Design of distributed economic dispatch algorithm incorporating demand response in time-variant demands. The generation limit and ramp rate limit of the generating unit and power consumption limit of the responsive demand units are considered;
- Finding optimal generation, optimal demand, and real-time pricing of electrical energy and maximizing social welfare;
- Analyzing the effect of renewable sources in real-time pricing and social welfare maximization.
- The proposed algorithm prioritizes privacy by exclusively sharing the energy mismatch with neighboring agents, safeguarding the privacy of each individual. Furthermore, it ensures robustness and scalability, making it a reliable and adaptable solution.

## 2. Preliminaries

#### 2.1. Graph Theory

#### 2.2. Dynamic Consensus Algorithm

#### 2.3. KKT Conditions

- (i)
- Stationary: The gradient of the Lagrange function with respect to x must be zero:$$\nabla f\left({x}^{*}\right)+\sum _{i\in \u03f5}{\lambda}_{i}^{*}\nabla {g}_{i}\left({x}^{*}\right)+\sum _{i\in I}{\mu}_{i}^{*}\nabla {h}_{i}\left({x}^{*}\right)=0.$$
- (ii)
- Primal Feasibility: The solution ${x}^{*}$ must satisfy the constraints:$${g}_{i}\left({x}^{*}\right)=0,\phantom{\rule{1.em}{0ex}}\forall i\in \u03f5,;{h}_{i}\left({x}^{*}\right)\le 0,\phantom{\rule{1.em}{0ex}}\forall i\in I.$$
- (iii)
- Dual Feasibility: The Lagrange multipliers for inequality constraints must be non-negative:$${\mu}_{i}^{*}\ge 0,\phantom{\rule{1.em}{0ex}}\forall i\in I.$$
- (iv)
- Complementary Slackness: For each inequality constraint, either the constraint is active or the corresponding Lagrange multiplier is zero:$${\mu}_{i}^{*}{h}_{i}\left({x}^{*}\right)=0.$$

#### 2.4. ADMM

## 3. Problem Formulation

#### 3.1. Centralized Economic Dispatch with Demand Response

- (i)
- The marginal utility should be positive, indicating that as consumption increases, the utility also rises.
- (ii)
- As power consumption increases, the level of satisfaction should eventually reach a point of saturation.
- (iii)
- If there is no consumption, the utility for the consumer should be zero.

#### 3.2. Decentralized Economic Dispatch with Demand Response

**Case I:**${U}_{i}({P)}_{di}={\beta}_{i}{P}_{di}-{\alpha}_{i}{{P}_{di}}^{2}$- In this case, the optimal power consumption for the $i\mathrm{th}$ demand unit is given by$$\begin{array}{c}\hfill {P}_{gi}^{k+1}=\left[\frac{-\rho ({P}_{gi}^{k}-N{\overline{P}}_{gdi.mean}^{k}+{u}_{i}^{k})-{\beta}_{i}}{2{\alpha}_{i}+\rho}\right]\end{array}$$
**Case II:**${U}_{i}({P)}_{di}=\frac{{{\beta}_{i}}^{2}}{4{\alpha}_{i}}\mathrm{for}{P}_{di}\ge \frac{{\beta}_{i}}{2{\alpha}_{i}}$- For this case, the optimal power consumption for the $i\mathrm{th}$ demand unit is given by$$\begin{array}{c}\hfill {P}_{gi}^{k+1}=({P}_{gi}^{k}-N{\overline{P}}_{gdi.mean}^{k}+{u}_{i}^{k})\end{array}$$

Algorithm 1 Simplified Economic Dispatch with Demand Response |

- →
**Input:**Cost function variables (a, b, c), penalty parameter $\rho $ - →
**Input:**Utility function variables ($\alpha $, $\beta $) for each demand unit - →
**Define:**Generating range, ramp rate - →
**Define:**Minimum and maximum demand capacity - →
**Initialize:**Scaled dual updater u for all agents - →
**Initialize:**Generating capacity ${P}_{gi}^{0}$ for each generating unit - → Calculate initial power mismatch ${P}_{\overline{gd}i.\mathrm{mean}}^{0}$ for each agent
for
$k=0$
to
∞
doExecute for all agents:• Update ${P}_{gi}^{(k+1)}$ using Equation (31) • Update ${P}_{di}^{(k+1)}$ using Equations (33) or (34) • Update ${P}_{\overline{gd}i.\mathrm{mean}}^{(k+1)}$ using Equation (27) • Update scaled dual variable ${u}_{i}^{(k+1)}$ using Equation (28) • Update price updater ${\lambda}_{i}^{(k+1)}$ using Equation (29) end for |

## 4. Results

- →
- In Section 4.2, the case where the proposed algorithm does not incorporate demand response is explained.
- →
- In Section 4.3, the case where the proposed algorithm is utilized to its full capability, considering demand response is presented.
- →
- Lastly, in Section 4.4, the effect of renewable energy in the proposed algorithm is examined.

#### 4.1. Simulation Setup for the Algorithm Test

#### 4.2. Result and Discussion of the Proposed Algorithm with No Demand Response

#### 4.3. Result and Discussion of Proposed Algorithm with Demand Response

#### 4.4. Result and Discussion on Effect of Renewable Generation in Proposed Algorithm with Demand Response

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

**Table A1.**Considered Hourly power distribution factor from the indicated power of IEEE standard bus system.

Time (Hr) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|

Factor | 0.50 | 0.52 | 0.55 | 0.58 | 0.62 | 0.66 | 0.71 | 0.78 | 0.85 | 0.92 | 0.85 | 0.78 |

Time (Hr) | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 |

Factor | 0.72 | 0.68 | 0.73 | 0.77 | 0.84 | 0.90 | 0.98 | 1.08 | 1.17 | 1.24 | 1.00 | 0.70 |

## References

- Kabindra, P.; Li, W.; Sonee, S.; Zhang, Y.; Zhao, H.; Umar, S. Autonomous transient power management strategy based on improved droop control for DC microgrid. Electr. Eng.
**2022**, 104, 4321–4334. [Google Scholar] - Wood, A.J.; Wollenberg, B.F.; ShebleB, G.B. Power Generation, Operation, and Control, 3rd ed.; John Wiley and Sons: Hoboken, NJ, USA, 2013. [Google Scholar]
- Arnold, G.; Batten, D.; Bose, A.; Hauser, C.; Whitehead, D.; Zweigle, G.; Gomez-Exposito, A.; Abur, A.; delaVillajaen, A.; Gomez-Quiles, C. Smart grid: The electric energy system of the future. Proc. IEEE
**2011**, 99, 1–11. [Google Scholar] - Shen, J.; Jiang, C.; Li, B. Controllable Load Management Approaches in Smart Grids. Energies
**2015**, 8, 11187–11202. [Google Scholar] [CrossRef] - Xu, Y.; Zhang, W.; Liu, W.; Wang, X. Distributed Subgradient-Based Coordination of Multiple Renewable Generators in a Microgrid. IEEE Trans. Power Syst.
**2014**, 29, 23–33. [Google Scholar] [CrossRef] - Zhang, Z.; Ying, X.; Chow, M.Y. Decentralizing the economic dispatch problem using a two-level incremental cost consensus algorithm in a smart grid environment. In Proceedings of the IEEE North American Power Symposium (NAPS), Boston, MA, USA, 4–6 August 2011; pp. 1–7. [Google Scholar]
- Wang, T.; O’Neill, D.; Kamath, H. Dynamic Control and Optimization of Distributed Energy Resources in a Micro-grid. IEEE Trans. Smart Grid
**2015**, 29, 2884–2894. [Google Scholar] [CrossRef] - Guo, F.; Li, G.; Wen, C.; Wang, L. An Accelerated Distributed Gradient-Based Algorithm for Constrained Optimization With Application to Economic Dispatch in a Large-Scale Power System. IEEE Trans. Smart Grid
**2021**, 51, 2041–2053. [Google Scholar] [CrossRef] - Wang, R.; Li, Q.; Zhang, B.; Wang, L. Distributed Consensus Based Algorithm for Economic Dispatch in a Microgrid. IEEE Trans. Smart Grid
**2019**, 10, 3630–3640. [Google Scholar] [CrossRef] - Zhang, R.; Xu, Y.; Liu, W.; Zang, C. Optimal Demand Response Based on Utility Maximization in Power Networks. IEEE Trans. Ind. Inform.
**2015**, 11, 717–727. [Google Scholar] [CrossRef] - Li, N.; Chen, L.; Liu, W.; Zang, C. Distributed Online Optimal Energy Management for Smart Grids; Engineering & Applied Science Division, California Institute of Technology: Pasadena, CA, USA, 2011. [Google Scholar]
- Rahbari-Asr, N.; Ojha, U.; Zhang, Z.; Chow, M.Y. Incremental Welfare Consensus Algorithm for Cooperative Distributed Generation/Demand Response in Smart Grid. IEEE Trans. Smart Grid
**2014**, 5, 2836–2845. [Google Scholar] [CrossRef] - Samadi, P.; Mohsenian-Rad, H.; Schober, R.; Vincent, W.S. Optimal Real-Time Pricing Algorithm Based on Utility Maximization for Smart Grid. In Proceedings of the First IEEE International Conference on Smart Grid Communications, Gaithersburg, MD, USA, 4–6 October 2010; pp. 415–420. [Google Scholar]
- Samadi, P.; Mohsenian-rad, H.; Schober, R.; Vincent, W.S. Advanced Demand Side Management for the Future Smart Grid Using Mechanism Design. IEEE Trans. Smart Grid
**2012**, 3, 1170–1180. [Google Scholar] [CrossRef] - Hug, G.; Kar, S.; Schober, R. Consensus + Innovations Approach for Distributed Multiagent Coordination in a Microgrid. IEEE Trans. Smart Grid
**2015**, 5, 1893–1903. [Google Scholar] [CrossRef] - Demetri, P.S.; Reza, O.; Richard, M.M. Dynamic consensus for mobile network. In Proceedings of the 16th IFAC World Congress, Prague, Czech Republic, 3–8 July 2005. [Google Scholar]
- Hug, G.; Kar, S.; Schober, R. Fast linear iterations for distributed averaging. Syst. Control Lett.
**2004**, 53, 65–78. [Google Scholar] - Boyd, S.; Vandenberghe, L. Convex Optimization, 1st ed.; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Nocedal, J.; Wright, S.J. Wright. Numerical Optimization, 2nd ed.; Springer: New York, NY, USA, 2006. [Google Scholar]
- Nguyan, D.H.; Narikiyo, T.; Kawanishi, M. Optimal Demand Response and Real-time Pricing by a Sequential Distributed Consensus-Based ADMM Approach. IEEE Trans. Smart Grid
**2018**, 9, 4964–4974. [Google Scholar] [CrossRef] - Stephen, B.; Neal, P.; Eric, C.; Borja, P.; Jonathan, E. Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers. Found. Trends Mach. Learn.
**2010**, 3, 1–122. [Google Scholar] - Ray, D.Z.; Carlos, E.M.; Robert, J.T. MATPOWER: Steady-State Operations, Planning and Analysis Tools for Power Systems Research and Education. IEEE Trans. Power Syst.
**2010**, 26, 12–19. [Google Scholar] - Dhamala, B.; Karki, N.R.; Mishra, A. Economic Dispatch in Electric Grid Considering Demand Response using Dynamic Consensus-Based ADMM approach. In Proceedings of the 11th IOE Graduate Conference, Pokhara, Nepal, 10–11 March 2022; pp. 36–43. [Google Scholar]

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**MDPI and ACS Style**

Dhamala, B.; Pokharel, K.; Karki, N.R.
Dynamic Consensus-Based ADMM Strategy for Economic Dispatch with Demand Response in Power Grids. *Electricity* **2024**, *5*, 449-470.
https://doi.org/10.3390/electricity5030023

**AMA Style**

Dhamala B, Pokharel K, Karki NR.
Dynamic Consensus-Based ADMM Strategy for Economic Dispatch with Demand Response in Power Grids. *Electricity*. 2024; 5(3):449-470.
https://doi.org/10.3390/electricity5030023

**Chicago/Turabian Style**

Dhamala, Bhuban, Kabindra Pokharel, and Nava Raj Karki.
2024. "Dynamic Consensus-Based ADMM Strategy for Economic Dispatch with Demand Response in Power Grids" *Electricity* 5, no. 3: 449-470.
https://doi.org/10.3390/electricity5030023