Key words: Demand response / Microgrid energy management / Circle search algorithm / Load curtailment / Distributed energy resources
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons /licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of this study is formulated is as follows:
Pi is the power of the ith unit. While xi, yi, and zi are the ith generator’s cost coefficients, mi and ni are the valve point effect coefficients.
The uncertainty of wind production is characterized by , while the divergence of wind output is represented by dWi, n2 represents the standard distribution function. represents the projected wind energy production at time t.
Fluctuations in the price of electrical power will elicit one of the following responses in demand. Certain loads, like lighting loads, are not adjustable between periods and may only be activated or deactivated. Consequently, these loads exhibit sensitivity only at a certain moment, known as “self-elasticity”, which is negative consistently. Consumption may be moved from high-demand to low-demand times, such as process loads. Multi-period sensitivity refers to this phenomenon and is quantified by “cross elasticity”, which is consistently positive [47].
x2, mΔ term categories clients by means of Δ.
As Δ grows, slightly the cost drops. The client with the utmost willingness to pay (Δ = 1) has the lowest increment of cost and hence the highest marginal benefit, while the clients with the lowest willingness to pay (Δ = 0) have the highest increments in cost and therefore the lowest marginal benefit.
.
Non-negative/positive change in cost.
The marginal cost is inverse to cost function.
Avoiding power waste: eliminating unnecessary energy expenses should cost (c(Δ, 0) = 0).
UB represents the utility’s total budget and CMj is the daily limit of interruptible electricity for customer j. Constraint (18) guarantees that the total daily reward a client receives is more than or equal to their day-to-day cost of disruption. Customer power is restricted and ensured by constraint (19), and greater customer benefits are realised. The incentive offered by the utility is not supposed to be more than the budget of the utility and is ensured by constraint (20). The overall daily power reduction of each customer is not supposed to fall below their daily interruptible power limit and that ensured by constraint (21).
The CSA aims to find the best solution by investigating random circles to broaden the search area. The main motivations for selecting CSA as the optimization method for this study are [48]:
Recently developed, swift and popular.
Only one governing equation; No complex stages and phases within the algorithm making it easy to code and execute.
Using the centre of the circle as a reference point, the angle formed between the tangent line at its contact point and the circumference of the circle decreases progressively as it nears the center. The angle at which the tangent line meets the point can vary unpredictably since the circle may have been positioned within the local solution. The CSA search agent is considered to be at the contact point Xt, while the algorithm’s optimal position is anticipated to be at the centre point Xc. Below, we detail the key steps of the CSA optimizer.
This phase in the CSA is vital, as it guarantees that every dimension of the search agent is assigned randomly. Many existing codes randomise dimensions unevenly, leading to algorithms sometimes achieving optimal solutions unexpectedly. Equation (22) states that the search agents are first set up inside the search space’s upper limit values (ULV) and lower limit values (LLV) as:(22)
Fig. 1
Load demand and electricity market price.
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