Economic energy scheduling through chaotic gorilla troops optimizer

Abstract

This research proposes a novel solution to the power generation scheduling problem based on the chaotic gorilla troop's optimizer algorithm (CGTO). Power system operational planning is a large-scale, highly constrained combinatorial optimization problem known as the Energy Generation Scheduling Problem. The Gorilla Troops optimizer is a bio-inspired heuristic optimizer that uses gorilla hierarchy and hunting notions to resolve challenging scheduling issues. The gorilla update method is initially obtain binary string of generators in order to determine the global best solution (s), which is followed by a chaotic operation. Chaotic search avoids local minima while GTO seeks out global optima, resulting in a better balance of exploitation and exploration. A simple and effective strategy for wind power integration placement and sizing is presented in this research. The messy behavior of the wind is predicted to follow the Weibull PDF. Examining the viability and effectiveness of units between 10 and 100, the results are compared to those attained using various techniques mentioned in the literature. The results unequivocally demonstrate that the recommended technique provides superior solutions than competing alternatives. The solution was improved further in the wind power sharing scenario. Convergence curve amply demonstrates the optimizer's robustness.

Introduction

A big electric power system today typically comprises of thousands of transmission lines and hundreds of generators. Such a large-scale system's functioning should take into consideration the ongoing fluctuation in customer needs. Power system scheduling is based on two crucial decision-making processes in order to guarantee there is liability and the efficient running of this type of systems. First, each time interval of the scheduling time horizon should be used to establish the commitment states (on or off) of the systems generating units. The "Unit Commitment" problem is the name for this issue. The power demand will be distributed among the committed units according to another choice, known as the "Economic Dispatch." Such a distribution is chosen based on the traits of the component units. For a least-cost solution, the results of these two choices must be taken into account concurrently. The scheduled power generation should be able to meet the anticipated demand in the scheduling problem, where the time horizon typically runs from one day to one week. However, there is always a chance that some of the committed units would fail owing to the unique outages or unexpected challenges in the operation of a power system; therefore, a hot reserve which is not in operation must be considered. In this case, the extra capacity (spinning reserve) might be able to supply the mismatch power and deliver electricity to the portion of the load that was originally attributed to the failing units. Another system need that must be met in the scheduling problem is the supply of the spinning reserve. A deterministic spinning reserve constraint is typically used to simulate such dependability requirements, and this constraint necessitates a specific amount of spinning reserve capacity. This, however, is an incorrect formulation of system limitations [I] since it does not accurately reflect the risk that the system actually faces. The ideal timetable for a power system's operation should also take into account the system's component producing units' characteristics. According to how they use energy resources, generating units in a power system can also be categorized. The term "thermal units" refers to the generators that convert heat energy into electricity and have an endless supply of fuel. The majority of the units in the current power systems belong to this type of generating units. A different kind of generating units consists of fuel-constrained units that operate similarly to thermal units. However, there is a maximum amount of gasoline that can be made accessible to any fuel-constrained unit at any given time. A unit is subject to this constraint for a number of reasons, including the scarcity of natural resources, utility and fuel supplier contracts, and transportation issues. Assigning a preset total fuel consumption to each unit within the scheduled time horizon is a typical set of fuel restrictions used in modern power systems. These constraints are known as "Take or Pay" contracts. The third category of generators includes wind generators, which produce electricity by harnessing the kinetic energy of moving wind. Utilizing wind resources may result in a number of operational restrictions depending on the rate of the wind intake and environmental restrictions. In power system operation, unit commitment (UC) is a mixed-integer, frequently nonlinear, NP-hard issue. It necessitates determining the binary on/off state of power generation units as well as the real-valued power output of on-line units at the same time, all while adhering to different limitations such as power balance, capacity limit, and minimum up/down time limits. Since global warming has been a major challenge for mankind in the last decade, owing to widespread use of fossil fuels. Fuel releases pollutants including CO2, SO2, and NOx that are harmful to the environment. By enacting emission taxes, many nations are promoting the creation of clean energy in an effort to reduce the impact of global warming. The burning of fossil fuels is the main source of greenhouse gases. Three main forms of conventional fuels—oil, gas, and coal—produce emissions that are represented by greenhouse gases like NOx, CO2, and SO2. The Paris Climate Conference 2015 created a global commitment to keep maximum temperature rises under 2 degrees Celsius by the end of the century [1]. Significant reductions in GHG emissions from both major drivers of emissions, notably power production and transportation, are ultimately a fundamental way to achieving this objective [2]. As a result, both industrialized and developing countries work to utilize abundantly available alternative energy sources that are also environmentally beneficial. As a result, current studies and research have given renewable energy sources a lot of attention. The UC has already been widely investigated for traditional power systems with limited or no participation of variable renewable production. However, in recent years, there has been a significant increase in the use of variable renewable energy sources around the world, posing new issues for power system operators. Modern power systems with a considerable renewable share must be able to respond to fast and large variations in renewable energy output due to the uncertainty and variability of renewable generation, making flexibility a hot topic of research in recent years. A number of studies have been conducted in this area, with the goal of maximizing the use of flexible resources in the operation of highly renewable systems. In renewable energy share wind and generation gain utmost importance and have increased dramatically, allowing them to play a substantial role in wholesale energy markets while lowering fossil fuel consumption. Wind and solar power generation, which are generally classified as negative loads, have substantial swings in their load curves and are likely to induce repeated ramping and start-ups/shut-downs for thermal units [17]. Over the last few decades, extensive research has been performed to answer the problem, leading to the proposal of a variety of traditional approaches, intelligent methods, and hybrid or analytical methods. Generation scheduling problem has long been matter of concern various approaches have been applied to solve this complex problem conventional methods such as exhaustive enumeration [1] in which all possible combinations are tried, this is very complex and tiresome. Priority list [2] is simple sub-optimal approach which sorts units according to their priority, and whose priority is first participate first, followed by next best units. Dynamic programing [3] approach suffers with dimensionality problem. Lagrangian relaxation [4, 5 suffers with its sub-optimal nature as duality gap may not always reduce to zero. Branch and bound [6] Tabu search [7, 8] Unit decommitment method [9].

Performance of conventional approach fails when objective functions are either discontinuous or having multiple optima points. Since almost all classical techniques require objective function and gradient information of objective function since generation scheduling is complex non-convex curve classical method usually stuck in local minima. Soft computation-based solutions primarily make use of theoretical and applied understanding of the inference generated to address UCP. With the use of these techniques, it is possible to qualitatively describe a system's behavior, characteristics, and response without using precise mathematical formulas. Fuzzy logic, artificial neural networks, and evolutionary and GA-based approaches are three main categories in which growth in the field of soft computation techniques can be hypothesized. Hence soft computing-based method is widely adopted to solve generation scheduling problem, such as simulated annealing fuzzy logic [10, 11], genetic algorithm [12], ant colony optimization [13], evolutionary programming [14], harmony search [15], binary gravitational algorithm [16], fire fly algorithm [17], differential evolution [18], teaching–learning-based optimization [19], sine cosine optimizer [20], moth flame optimizer [21]. Various other hybrid optimization approaches owing to their batter balance between exploitation and exploration are quite common nowadays such as BMFO-SIG [22], GWA-SA [23], Hybrid SMA-SCA [24], BGWO [25], Hybrid optimizer [26, 27, 28]

Numerous scientific disciplines have used meta-heuristic optimization techniques. Meta-heuristics are straightforward to comprehend. They draw inspiration from natural phenomena, animal behaviors, and evolutionary theories. Meta-heuristics are easily adaptable to different issues since they typically treat them as "black boxes." Third, the majority of meta-heuristics use processes without derivation. Meta-heuristics stochastically optimize issues as opposed to gradient-based methods. Finally, compared to traditional optimization strategies, meta-heuristics are better able to avoid local optima. Since the search space for real issues is frequently unknown, extremely complicated, and home to a large number of local optima, meta-heuristics are effective tools for optimizing these difficult real-world problems. Applying meta-heuristics to a variety of issues is simple. Finally, meta-heuristics outperform conventional optimization techniques when it comes to avoiding local optima. Meta-heuristics are workable solutions for optimizing these challenging real-world problems since the search space of realistic difficulties is frequently unknown and complex, with a high number of local optima. There isn't a single meta-heuristic that can handle all optimization issues. This outcome was reached according to the "No Free Lunch Theorem" [37]. In other words, a meta-heuristic can produce incredibly promising outcomes for a set of issues. On a different set of tasks, however, the same algorithm might not perform well. We can also suggest new meta-heuristics, integrate multiple meta-heuristics, or enhance existing meta-heuristics thanks to their simplicity. A new hybrid meta-heuristic search algorithm is being created as a result. In order to achieve hybridization, newly developed, well-known metaheuristic algorithms, GTO Hybridizes with chaotic search should integrate both exploration and exploitation search methodologies to address unit commitment and dispatch issues in real-world power systems.

Novel contribution/innovation

Study’s original contribution is the creation of a special hybrid optimizer. By moving the optimizer from the exploration side, GTO enables the optimizer to look for fresh, fruitful search locations. Its exploitation capability which is in charge of hunting for global solutions inside the search space has been boosted by the addition of a chaotic search operator. Its exploration and exploitation capabilities are more stable than its original version. This paper advances the current understanding of hybrid metaheuristic search algorithms. Fig. 1 expresses intent of proposed work.

Fig. 1

Graphical Abstract of the objective of the Study

Reason of choosing CGTO optimizer

It is important to note that one element related to the broader notion of exploration and exploitation is diversity. AGTO has high diversity. Low diversity does not necessarily imply exploitation, while high diversity stimulates exploration because exploitation calls for the utilization of both population and landscape data gathered during the search process. Literature survey further divulges that optimizer has quick convergence rate. However, when they are hybridizes, they tend to show further quick convergence; hence chaotic strategy further improves their shortcoming.

Objective of study

Clear objective of study is to find cost-effective generation schedule which should satisfy the load demand and spinning reserve requirement with lowest fuel cost. This objective has been achieved with the help of proposed hybrid optimizer. The suggested approach achieves the objective of research in three parts. In the first, the CGTO resolves the unit commitment issue without taking into account the various limitations. Second, using a heuristic constraints repair mechanism, prior solutions are made to agree with unavoidable restrictions. Lastly, the suggested method is utilized to determine the best cost-effective solution within a given timeframe. This study looks into wind energy in more detail. The Chaos of Wind pattern is believed to follow the Weibull probability distribution function and be stochastic in nature.

Comprehensive benchmarking against some of the top techniques in the literature and in-depth simulation studies demonstrate that system operators can experience significant cost savings by deploying the CGTO optimizer. Saving in cost further increases if wind power share is taken into consideration along with thermal system.

Organization of the paper

The rest of the paper is organized as follows: The basic survival strategy adopted by the Gorilla Troops and the new improved chaotic Gorilla troop’s optimizer is framed in Sect. 2. Section 3 proposes a mathematical problem formulation of generation scheduling. The characteristic equation in this section shows objective functions with all hard bound constrained. Section 4 for generation of uncertain renewable energy generation scenarios; scenarios of wind power generation in four specific seasons, probabilistic wind power model and estimated wind power for each hour calculation is done in this segments. Section 5 deals with modus operandi of optimizer to solve generation scheduling problem. Result and discussion for large-scale thermal and wind thermal test system constitute Sect. 6. Conclusive result and scope for improvements as well as limit of optimizer is discussed in Sect. 7.

Artificial gorilla troops optimizer

Gorilla troops optimizer is recent metaheuristic optimizer proposed by Benyamin el al [29]. All optimization processes were carried out by five distinct operators. Three distinct operators were utilized throughout the exploration phase, including migration to an uncharted area to advance GTO research. The gorilla movement, the second operator, enhances the harmony between exploration and exploitation. The third machinist in the exploration phase, migration towards a known location, significantly improves the GTO's capacity to explore for various optimization spaces. On the other hand, the use of two operators during the exploitation phase greatly improves search performance.

Artificial gorilla troops: exploration phase

All gorillas in the GTO approach are potential solutions, and the silverback gorilla at each optimization stage is the top candidate solution. The optimizer uses three distinct strategies for the exploration phase: migration to new sites, migration to existing areas, and migration to other gorillas. Each of these three approaches is chosen using a conventional technique. The random parameter determines how migration to an unknown site is decided (p). However, the strategy of approaching other gorillas is chosen if rand is less than 0.5. However, the migration to a known site technique is selected if rand is less than 0.5. Each of the employed techniques offers the GTO algorithm a significant amount of power. While the second mechanism restores algorithm exploration performance and the third technique aids the GTO dropping from local minima, the first mechanism enables the algorithm to keep a close eye on the entire solution universe.

$$GX^{{{\text{iter}} + 1}} = \left\{ \begin{gathered} {\text{Lowerbound}} + r_{1} \times ({\text{Upperbound}} - {\text{Lowerbound}})\begin{array}{*{20}c} ; & {\mu < p} \\ \end{array} \hfill \\ L \times H + (r_{2} - C) \times X_{r} (iter)\begin{array}{*{20}c} ; & {\mu \ge 0.5} \\ \end{array} \hfill \\ X_{i} - L \times [L \times \{ X({\text{iter}}) - {\text{GX}}_{r} ({\text{iter}})\} + r_{3} \times \{ X({\text{iter}}) - {\text{GX}}_{r} ({\text{iter}})\} ]\begin{array}{*{20}c} ; & {\mu < 0.5} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(1)

As described in Eq. (1) in iteration t, ${\text{GX}}^{{{\text{iter}} + 1}}$ is the gorilla candidate position vector. The current vector of the gorilla location is X (t). Furthermore, each iteration updates the values of and random number, which are random numbers ranging from 0 to 1 and play an important part in the optimization process. P is a 0–1 parameter that must be specified prior to the optimization procedure; it affects the possibility of choosing a migration method to an unknown site. The upper and lower limits of the variables are denoted by its original name as “Lower bound”, and “upper bound”, respectively, the places adjusted in each phase were incorporated into one of the gorilla candidate locations vectors, which was chosen at random. Finally, C, L, and H, respectively, are obtained as describe in [29]

Artificial gorilla troops: exploitation phase

The silverback has a long-lasting effect on the other gorillas since he is the strongest and fittest one in the group, drawing them all to him. Additionally, they comply with Silverback's commands to travel to various locations in search of food supplies and to remain with him. Members could also have an effect on how the group moves as a whole. This method is picked when the C & W value is selected. When C ≥ W, Follow the silverback mechanism is selected and when C<W adult females competition is taken. W is variable to be set before optimization operation. Equation is used to model this phenomenon (2).This is dipicted in Fig. 2

Fig. 2

Vector follows the silverback in 2D Search space [29]

$${\text{GX}}^{{{\text{iter}} + 1}} = X^{iter} + L \times M \times [X^{{{\text{iter}}}} - X_{{{\text{silverback}}}} ]$$

(2)

${\text{GX}}^{{{\text{iter}} + 1}}$ In the following t iteration of Equation is the gorilla candidate position vector (1). The gorilla's current location vector is X. (t). Furthermore, the values of r1, r2, r3, and rand, which are random numbers ranging from 0 to 1, are updated with each iteration. p is a 0–1 range parameter that must be specified before to the optimization stage.

$$M\left( {\left| {\frac{1}{N}\sum\limits_{i - 1}^{N} {{\text{GX}}_{i} \left( {{\text{iter}}} \right)} } \right|^{g} } \right)^{\frac{1}{g}}$$

(3)

$$g = 2^{L}$$

(4)

Competition for adult female

In the event where C is less than W, the second mechanism is used. When teenage gorillas reach puberty, they engage in violent competition with other males to choose adult females to add to their group. These fights can go for days and involve numerous participants. Equation is used to model this phenomenon (5)

$${\text{GX}}_{i}^{{{\text{iter}}}} = X({\text{Silverback}}) - A \times [X({\text{silverback}}) \times Q - X({\text{iter}}) \times Q]$$

(5)

$$Q = 2 \times r_{5} - 1$$

(6)

$$A = \beta \times E$$

(7)

$$E = \left\{ \begin{gathered} N_{1} \begin{array}{*{20}c} ; & {{\text{rand}} \ge 0.5} \\ \end{array} \hfill \\ N_{2} \begin{array}{*{20}c} ; & {{\text{rand}} < 0.5} \\ \end{array} \hfill \\ \end{gathered} \right.$$

(8)

In the exploitation phase of the algorithm, two behaviors are used: first is imitation of the silverback and second phase is fatal Competition for fully grown females. The group's leader is a silverback gorilla, who makes all of the decisions, directs the gorillas to food sources, and controls their movements. The silverback is in charge of the gorillas' safety and well-being, and they all obey his commands. The silverback gorilla, on the other hand, may become feeble and elderly, eventually dying, and the group's silverback gorilla may take over as leader, or other male gorillas may challenge the silverback gorilla and seize control of the group. The addition of a chaotic search tool boosts its local search power.

Different chaotic functions

Present manuscripts use a special kind of chaotic function, and hybridize with gorilla troop’s optimizer. The intention behind the same is it improves its exploitation capability since GTO optimizer already proved to be one of best optimizer as far as exploration phase is concerned [29]; hence chaotic gorilla optimizer has batter balance between exploitation and exploration search capability. Many meta-heuristic algorithms employ the notion of probability distribution, which incorporates a degree of randomness in the optimizer, to boost local search capabilities. If unpredictability due to periodicity and idleness is correctly utilized, chaotic maps may be helpful. Equation 21 satisfies all of these chaotic requirements

$$Z(q + 1)\, = \,f(z_{q} )$$

(9)

In Eq. (9) $z_{k + 1} \,\& f(z_{k} )$ are the $(\,k + 1)^{th} \,\,\& \,k^{th}$ chaotic number, respectively. Following popular chaotic functions are usually adopt to improve optimization process. In the propose research singer map function is used to enhance optimizer local search capability. List of chaotic curve has been given in Appendix-A.

Pseudo code of CGTO

Unit commitment problem formulation

The unit commitment problem's goal is to reduce the total operational cost of generating power by as much as possible over a given planning horizon:

$${\text{NOC}} = \sum\limits_{t = 1}^{t = T} {\sum\limits_{i = 1}^{i = N} {[{\text{FC}}_{i} (P_{i}^{t} )*\psi_{i}^{t} + \psi_{i}^{t} (1 - \psi_{i}^{t - 1} ){\text{SUC}}_{i}^{t} + \psi_{i}^{t - 1} (1 - \psi_{i}^{t} ){\text{SD}}_{i}^{t} } }$$

(10)

$${\text{FC}}_{i} (P_{i}^{t} ) = a_{i} *(P_{i}^{t} )^{2} + b_{i} *(P_{i}^{t} ) + c_{i}$$

(11)

$${\text{SUC}}_{i}^{t} = \left\{ {\left. \begin{gathered} {\text{HOTS}}_{i}^{t} ,\begin{array}{*{20}c} {\begin{array}{*{20}c} {} \\ \end{array} {\text{if}}\begin{array}{*{20}c} {{\text{MDT}}_{i} \le T_{{i,{\text{off}}}}^{t} + T_{i}^{{{\text{cold}}}} } \\ \end{array} } \\ \end{array} \hfill \\ {\text{COLDS}}_{i}^{t} ,{\text{if}}\begin{array}{*{20}c} {T_{{i,{\text{off}}}}^{t} > {\text{MDT}}_{i} + T_{i}^{{{\text{cold}}}} } \\ \end{array} \hfill \\ \end{gathered} \right\}} \right.$$

(12)

This objective function is subject to following constraints.

System constraints

Power balance, spinning reserve, and tie line capacity restrictions are all examples of system constraints. These restrictions affect all of the system's generating units, hence they're referred to as coupling constraints. The following is a detailed description of each constrain

Power balance constraint

Power demand plus losses must be met by generation. The load ratio share, or load conferred, in any system is random variable; hence power generation has to be adjusted at every instant of time in accordance with load.

$$\sum\limits_{i = 1}^{{{\text{NG}}}} {P_{i}^{t} *\psi_{i}^{t} = P_{D} } \begin{array}{*{20}c} {} & {(t = 1,2,...,T)} \\ \end{array}$$

(13)

For wind thermal system this constraints amended as follows

$$\sum\limits_{i = 1}^{{{\text{NG}}}} {P_{i}^{t} *\psi_{i}^{t} + P_{{{\text{wind}}}}^{t} = P_{D} } \begin{array}{*{20}c} {} & {(t = 1,2,...,T)} \\ \end{array}$$

(14)

Spinning reserve constraint

On-line units' total maximum capacity must be larger than demand + losses and the spinning reserve required. This spinning reserve is required to meet unexpected increases in demand or forced generating unit outages. The required spinning reserve is normally defined by the maximum capacity of one of the system's two largest generating units or a percentage of predicted peak demand for the time frame in question. In the present manuscript spinning reserve is considered as 10% of peak load demand. The formula for the spinning reserve restriction is provided in

$$\sum\limits_{i = 1}^{{{\text{NG}}}} {P_{i(\max )} *\psi_{i}^{t} \ge P_{D} + {\text{SR}}_{t} } \begin{array}{*{20}c} {} & {(t = 1,2,...,T)} \\ \end{array}$$

(15)

Thermal unit constraints

The beginning condition, minimum and maximum generation output limits, minimum up-time and down-time, and unit status restriction constraints are all unique to each thermal unit. Non-coupling limitations are what they're called. The following is a description of each limitation.

Each constraint is described as follows.

Initial condition

Initial condition of a generating unit includes number of hours that it has consecutively been on-line or off-line and its generation output at an hour before the scheduled time horizon starts. Both initial number of on- line/off-line hours and initial generation output, when associated with the other unit constraints, may limit the on-line/off-line status and the generation output of the unit in the scheduled time horizon.

Minimum and maximum generation output limits

Machine output limits or economic output limits specify a range of unit power outputs. Due to unit ramp rate constraints, the minimum and maximum limits that bound the generation output of each generating unit in a given hour, as stated in Eq. 28, can be adjusted within the range of unit power outputs. The next sections go over these restrictions.

$$P_{i}^{\min } \le P_{i} \le P_{i}^{\max }$$

(16)

Minimum up-time

A unit's minimum up-time is the amount of time it must be on-line once it has been turned on. The number of hours that a unit must be off-line after it has been turned off is known as the minimum down-time

$$\begin{gathered} T_{t,i}^{{{\text{ON}}}} \ge {\text{MUT}}(i)\begin{array}{*{20}c} {(i = 1,2,...,{\text{NG}};t = 1,2,...,T)} \\ \end{array} \hfill \\ \begin{array}{*{20}c} {} & {} \\ \end{array} \hfill \\ \end{gathered}$$

(17)

Minimum down-time constraints

Unit once gets off it should not turned on until it reaches certain minimum time to get completely cooled off. This minimum time is known as minimum down time.

$$T_{t,i}^{{{\text{OFF}}}} \ge {\text{MDT}}(i)\begin{array}{*{20}c} {(i = 1,2,...,{\text{NG}};t = 1,2,...,T)} \\ \end{array}$$

(18)

Wind speed modelling

For the sake of describing stochastic behavior of wind speed in predefined time period, Weibull document has been considered. Weibull division at t th time segment for the wind speed $u^{t}$ (m/s) can be shown as

$$f_{u}^{t} (u) = \frac{{k^{t} }}{{c^{t} }}.\left( {\frac{{v^{t} }}{{c^{t} }}} \right)^{{k^{t} - 1}} .\exp \left( { - \left( {\frac{{u^{t} }}{{c^{t} }}} \right)^{{k^{t} - 1}} } \right)\mathop {}\nolimits^{{}} for\mathop {}\nolimits^{{}} c^{t} > 1;\mathop {}\nolimits^{{}} k^{t} > 0$$

(19)

At the tth time segment, the shape parameter $(k^{t} )$ and scale factor $(c^{t} )$ are expressed as shown.

$$k^{t} = \left( {\frac{{\sigma^{t} }}{{\mu_{u}^{t} }}} \right)^{ - 1.086}$$

(20)

$$c^{t} = \left( {\frac{{\mu_{u}^{t} }}{{\Gamma (1 + 1/k^{t} )}}} \right)^{{}}$$

(21)

Power generation model

Any device that can slow down a mass of moving air, such as a sail or propeller, may extract some of the energy that the wind possesses due to its speed and utilize it to do meaningful work. The wind energy converter's output power is determined by three parameters. Due to chaotic wind dynamics, it is nearly hard to predict wind power reliably and precisely for the following three factors: wind speed, cross-sectional area of wind swept by the rotor, and total rotor conversion efficiency. The hourly output power of wind turbine corresponds to a specific $t^{th}$ time segment

$$P_{{{\text{WT}}}}^{t} = \sum\limits_{g = 1}^{{N_{v} }} {PG_{{{\text{WTg}}}} *P_{u} (u_{g}^{t} )}$$

(22)

The probability of wind speed for each state during any specific time frame is calculated as below.

$$\begin{gathered} P_{u} (u_{g}^{t} ) = \left\{ \begin{gathered} \int\limits_{0}^{{(u_{g}^{t} + u_{g + 1}^{t} )/2}} {f_{v}^{t} (u)dv} \mathop {}\nolimits^{{}} for\mathop {}\nolimits^{{}} g = 1 \hfill \\ \int\limits_{{(s_{g - 1}^{t} + s_{g}^{t} )/2}}^{{(u_{g}^{t} + u_{g + 1}^{t} )/2}} {f_{u}^{t} (u)du\mathop {}\nolimits^{{}} for\mathop {}\nolimits^{{}} g = 2...(N_{s} - 1)} \hfill \\ \int\limits_{{(u_{g - 1}^{t} + u_{g}^{t} )/2}}^{\infty } {f_{u}^{t} (u)du\begin{array}{*{20}c} {} & {for} & {g = N} \\ \end{array} } \hfill \\ \end{gathered} \right\} \hfill \\ \hfill \\ \end{gathered}$$

(23)

Wind speed variability must be taken into account when designing large WEC machines; for each turbine, there is a minimum wind speed called the cut-in wind speed at which rotation can begin, and a rated wind speed at which the generator produces its rated power; the rotation rate is kept constant by varying the generator output. When wind speeds surpass the rated value, the pitch of the blades is automatically adjusted to maintain rotor speed. The blades are feathered, similar to air craft, at very high wind speeds, and spinning stops. Cut off speed is the wind speed at which this occurs, and the electric power production is practically constant.

The variation of generated power with wind speed is approximately shown in Fig. 3

Fig. 3

The variation of generated power with wind speed

The power generation of the wind turbine depends on its performance curve. For nonlinear performance characteristics power generation of wind turbine at average speed for state g is calculated as

$$PG_{{{\text{WT}}}}^{g} = \left\{ \begin{gathered} 0\begin{array}{*{20}c} {} & {} & {} & {\begin{array}{*{20}c} {} & {} & {} & {\begin{array}{*{20}c} {u_{{{\text{ag}}}} < c_{{{\text{in}}}} ,or\begin{array}{*{20}c} {u_{{{\text{ag}}}} > u_{{{\text{count}}}} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} \hfill \\ \left( {a*u_{{{\text{ag}}}}^{3} + b*P_{{{\text{rated}}}} } \right)\begin{array}{*{20}c} {} & {u_{{{\text{cin}}}} \le u_{{{\text{ag}}}} \le u_{N} } & {} \\ \end{array} \hfill \\ P_{{{\text{rated}}}} \begin{array}{*{20}c} {} & {} & {} & {\begin{array}{*{20}c} {} & {} & {u_{N} \le u_{{{\text{ag}}}} \le u_{{{\text{count}}}} } & {} \\ \end{array} } \\ \end{array} \hfill \\ \end{gathered} \right.$$

(24)

where $P_{{{\text{rated}}}}$ is the maximum power that can be generated by Wind turbine, $v_{{{\text{cout}}}}$ is cutout wind speed; constant a and b are function of cut in wind speed $v_{{{\text{cin}}}}$. And Nominal wind speed $v_{N}$, obtained as

$$a = \frac{{P_{{{\text{rated}}}} }}{{\left( {u_{N}^{3} - u_{{{\text{cin}}}}^{3} } \right)}}$$

(25)

$$b = \frac{{u_{{{\text{cin}}}}^{3} }}{{\left( {u_{N}^{3} - u_{{{\text{cin}}}}^{3} } \right)}}$$

(26)

Wind data

Practical wind data was collected from the Kakdwip area, which is located close to the Bay of Bengal at 21.8830 N and 88.183E. A particular characteristic of dramatic weather change is provided by its closeness to the tropic of cancer [26]. In addition, the study time is separated into four seasons: spring, summer (May to July), autumn (August to October), and winter (November to January) (February to April). Each season is represented by 24 segments, which correspond to the season's hourly intervals. 96 time periods make up the year, then (24 for each season). Wind speed in meters per second, mean and standard deviation. Appendix-B provides full data (Mean and Standard deviations) for all the seasons. Each hour is separated into 20 states (scenarios). Practical wind turbine data is given in Appendix-C. In Appendix-D, a wind power given from wind turbine's for all the season’s summer, autumn winter and spring is listed and described. Wind erratic pattern follows Weibull probability function. Probabilistic power in each states has been shown in Fig. 4. Over all power for that hour is sum of all states power generation, and shown in Fig. 5.

Fig. 4

Wind power probability profile with respect to number of state

Number of states for Weibull probability function has been chosen 20. This helps to realize the distribution in discrete form. The discrete probability distribution for all seasons, i.e. summer, spring, and autumn and winter, for four hour has been illustrated in Fig. 4.

Wind power for 24-h

In this work, the wind speed distribution and statistical analysis are determined using the Weibull distribution approach. To derive the Weibull probability density function, you must know the figure and scale parameters. The equation may be used to demonstrate the generic form of the two-parameter Weibull probability density function for wind speed (25). Equation may be used to compute the shape function and scale factor values for each observed time (22–23).

The parameters of the Weibull distribution can be determined using a variety of approaches. The most commonly used strategy is the graphical method, which is a maximum possibility method. The graphical technique was employed in this investigation. The veracity of this technique is based on the minimization of vertical discrepancies between observed and segmented wind speed data. The following equation may be used to represent the average wind speed of the Weibull distribution. In this, the wind power over a 24-h period has been computed and tallied (4). In addition, in Fig. (5), wind power is plotted against wind speed (in m/s2) in each state. Wind power for all the seasons is obtained and tabulated in Appendix-D.

In this manuscripts an only wind data in the seasons of spring has been considered. A composite wind-thermal system has been proposed in witch wind power for 24 h has already calculated. Wind owing to cost free has got precedence over the thermal units hence utilized first in this way effective load on thermal units get reduced. ($P_{D} (t) = P_{{{\text{Load}}}} (t) - P_{{{\text{wind}}}} (t)$These under stressed load is now subject to tack by the thermal units with their all constraints in to considerations.

Fig. 5

Wind power generation in each stat

Wind Power Profile for 24 h for spring seasons has been calculated based on the mean and average speed as depicted in Appendix-B. Using Weibull probability distribution wind power for all the four seasons is calculated and given in Appendix-D. Its profile for 24 h is shown in Figs. 6, 7, 8, 9

Fig. 6

Wind Power Generation Profile in Summer Seasons for 24-Hours

Fig. 7

Wind Power Profile For 24-Hour In the winter seasons for 24-h

Fig. 8

Wind Power Profile For 24-Hour In the autumn seasons for 24-h

Fig. 9

Wind Power Profile For 24-Hour In the spring for 24- hour

Solution approach for generation scheduling problem

This strategy exploits and explores their placement in a suitable search space while taking into account all constraints using a hybrid CGTO method. Finding out the status of committed units is necessary before fixing a scheduling issue. The entire generation schedule issue can be solved using the following steps.

• Step 1: The demonstration of particle is defined in order to tackle the unit commitment problem using CGTO. A particle is a discrete entity that has a binary state of either 1 or 0. These states are similar to the on and off states of a unit. These discrete individuals would exhibit the unit commitment schedule across a time horizon T, and the full set of individuals would be recorded in a binary metrics of order GT*T, the population matrix illustrated below.

  $$u_{NP} = \left[ {\left. \begin{gathered} u_{1}^{1} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {u_{1}^{2} } & {..} & {..} & {u_{1}^{T} } \\ \end{array} \hfill \\ u_{2}^{1} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {u_{2}^{2} } & {..} & {..} & {u_{2}^{T} } \\ \end{array} \hfill \\ . \hfill \\ . \hfill \\ u_{N}^{1} \begin{array}{*{20}c} {} \\ \end{array} \begin{array}{*{20}c} {u_{N}^{2} } & {..} & {..} & {u_{N}^{T} } \\ \end{array} \hfill \\ \end{gathered} \right]} \right._{N \times T}$$

  (27)

  Power generation schedule, P(i:t) for $i^{th}$ generating unit at $t^{th}$ hour within maximum and minimum power generation limits can be found as below

  $$P(i:t) = P(i)_{\min } + rand()*\left\{ {P(i)_{\max } - \left. {P(i)_{\min } } \right\}} \right.$$

  (28)

• Step 2: all the generating units are prioritize according to the average full load cost.

• Step 3: available wind power for each hour has been calculated beforehand with the help of Weibull probability density function. Effective hourly load has now been reduced to $P_{d,t}^{{{\text{new}}}} = P_{d,t}^{{{\text{old}}}} - P_{{{\text{wind}}}}^{t}$. As available wind power for each hour must be utilized and has priority over thermal units, as wind power does not impose any fuel cost on the system. In this study wind power for the spring seasons has been considered for generation scheduling problem.

• Step 4: Because computed priority units may fail to meet spinning reserve restrictions, heuristic methods are used to fix spinning reserve limitations.

• Step 5: For time violation limitations, all individual units of the population matrix are repaired. Because all individual units must have the shortest possible up and down times, if the suggested method violates these restrictions, it should be fixed using an aggressive heuristics method. As a result of this reasoning, the algorithm is compelled to meet these limits at the expense of fitness value.

• Step 6: Repairing the MDT/MUT might result in a larger mismatch in the spinning reserve, resulting in increased operational costs. The program determines the most redundant unit using a heuristic methodology used to repair de-commitment of units. The most redundant units are those with the lowest priority rank, and these units may frequently be turned off without causing further MDT/MUT violations.

• Step 7: When complete patterns of committing and decommitting units within a stipulated time horizon are known, then most economic load problem is solved.

• Step 8: Calculate the fitness value of each particle in entire population using objective function.

• Step 9: Define Artificial Gorilla troops optimizer parameter as described in Sect. (2), obtain position with AGTO and allow chaotic search for local search and obtain new position according to its governing equations, obtained by CGTO Optimizer.

• Step 10: Modify the new updated position given by CGTO by passing through function that handles all vital constraints such as MDT/MUT, Spinning reserve, and de-commitment. This process is known as grey zone modification. Detail of constraints handling strategies and their heuristic code have been presented in Appendix-E.

• Step11: obtain final gbest position through CGTO optimizer.

• Step12: Since after grey zone modification their exist modification in unit schedule, so obtained optimal economic generation schedule.

• Step13: Obtain Startup cost units, startup may be different in all cases when unit is started from cold start, warm start or hot start.

• Step14: If maximum iteration is achieved stop it otherwise increase the iteration number.

• Step 15: Stop the optimal solution for generation scheduling through the particle whose index is g-best.

Entire process has been shown in Fig. 10.

Fig. 10

Flow chart of Generation Scheduling Process by CGTO Optimizer

Result and discussion

For solving UCP, the proposed CGTO is tested on different system sizes based on a basic system of 10 units from literature [30]. The scheduling time horizon T is chosen as one day with 24 intervals of one hour each. The spinning hour requirement is set to 10% of total demand. For the system 20, 40, 60, and 100 unit the basic 10 unit system is duplicated and load demands are adjusted proportionally to the system size. General experimental setup has been carried out with number of trial run = 30, maximum iteration = 100. And matlab-2019a (8.1.0.604) software on windo10, home basic CPU @2.10Ghz, RAM 6 GB, Processor i5, 64 bit operating system is used. Tables 1, 2, 3 show generation scheduling of thermal unit system. Rest of the Table and figures are present in Appendix –F.

Table 1 Generation scheduling of committed units for 10 unit thermal system (with 5%SR)

Table 2 Generation scheduling of committed units for 10 unit thermal system (With 10% SR)

Table 3 Generation scheduling of committed units for 20 unit thermal system

Test system-I (10 -Unit Thermal System With 5% SR)

Test system-II (10 -Unit Thermal System With 10%SR.)

Test system-II (20 -Unit Thermal System)

Comparison of results with other similar state of the art algorithm

The proposed algorithm is tested on medium and large dimensional system 10, 20, 40, 60, 80 and 100 unit system (small test system for 7 unit also tested). The system data and load pattern for basic 10 unit, 24-h test system is adopted from [30]. For 20 unit system the 10 unit system data is taken as a reference, but it is multiplied by two, similarly load demand is duplicated. In all the test system cases cost function is quadratic as shown in equation 11; the spinning reserve is taken 10% of load demand for all the cases. However for 10 unit system 5% SR is also used to evaluate cost solution Wind integrated dynamic generation scheduling and. Convergence curve has been shown in Appendix-F. The amount of dollar saving is depicted in Table 6. Tables 4 and 5 show cost comparison of CGTO algorithm with other state of art optimizer. These tables clearly show the total operational costs achieved by the new improved CGTO algorithm is best for all the test system.

Table 4 Generation scheduling of committed units for 10 and 20 unit thermal system  (Comparison With other optimizer)

Table 5 Generation scheduling of committed units for 40, 60, 80 and 100. Unit thermal system (Comparison with other Optimizer)

Detail discussion in accordance to the main objective

The CGTO technique is used to resolve generation scheduling for various test systems with 10 to 100 units. To successfully use the CGTO for addressing the highly constrained, non-convex generation scheduling problem, the best values for various parameters must be determined. This test system has been used to execute a number of tests. The experiments show that the factors listed below are ideal for the CGTO algorithm's greatest performance: population size is four times larger than the system size (i.e. for 10 unit population size is 40). The Appendix- F description is used to set the CGTO parameters. The highest iterative generations is 100. Under the chosen parameters, we run CGTO 30 times from different initial populations in succession and select the best result as the final optimization solution. In the meantime, for each test system we also performed CGTO 30 trials from different initial populations in succession to examine the variation in their total production costs. Test results are shown in Tables 4 and 5. The best, worst, and average total production costs findings of the CGTO are obtained. From Tables 4 and 5, the average production costs of 30 trials generated variation in a small range and the standard deviation are small and tolerable. Simultaneously, average production costs are near to the middle position between their maximum and minimum values. Therefore, it is evident that solutions are impartial and equally dispersed among the best and worst options. The 10-unit system's optimal solution as determined by the suggested algorithm is displayed in Table 4. While waiting, we look at how the test system's overall fuel cost varies with the number of evolutionary generations. The convergence processes of the top solution after 30 trials are listed in APPENDIX-F for various test systems. It is simple to see from Figs. 15 and 16 that the CGTO has satisfactory convergence and that, at later iterations, the algorithm escaped from the local optima. It demonstrated the effectiveness of the CGTO stochastic searching mechanism, which relies on chaotic interactions between agents. Additionally, the CGTO's performance was enhanced by the suggested mutation procedures.

Interpretation of the results

Tables 4 and 5 demonstrate how the test system's overall fuel cost has changed as a function of the evolutionary generation numbers, same is depicted by Figs. 11 and 12. Wind integrated thermal unit’s solution for generation scheduling further more superior as depicted in Fig. 13 which clearly shows that introducing wind power at available instant reduces the effective load requirement of that hour hence it imposes less unit commitment stress on thermal unit which results in economic saving C solution after 30 trials are shown for several test systems in Figs. 14 and 15

Fig. 11

Variation of the fuel cost as a function of number of generating units for Thermal units

Fig. 12

Variation of the fuel cost as a function of number of generating units for wind-thermal units

Fig. 13

Saving in Fuel cost by considering wind-power share (unit wise)

Fig. 14

Cost as a function of Number of trial runs

Fig. 15

Convergence curve for unit 10–100 for thermal unit system

The distribution of the best solutions is shown in Fig. 11 through 7 to allow for a visual representation of the outcome. The 10 unit system's optimal solution as determined by the suggested algorithm is taken from [28]. In the for the time being, we investigate how the test system's overall fuel cost varies with the number of evolutionary generations. 30 trial runs for various test systems are reported in Fig. 14. It is clear from convergence curve of Figs. 15 and 16 that the recommended CGTO algorithm converges satisfactorily and escapes the local optima thus perform stochastic searching mechanism of CGTO. The proposed chaotic search technique increased the performances of CGTO.

Fig. 16

Convergence curve for unit 10-100 for wind-thermal units

It proved that the stochastic searching mechanism of CGTO, which is conducted by chaotic forces among agents, is efficient and the proposed mutation strategies improved the performance of CGTO. To validate the results obtained with the CGTO, we compare the performance of the CGTO to those of other approaches with respect to the best total production cost. The results were reported in literature when the same problem were solved using EPL [31], ILA [32], QM [32], LR [33], GA [33], EP [34], SA [35], IBPSO[36], DE[18], BFA[37], ICA [38] HSA [39], BGSA [40], Fuzzy SADP [10], GMTLBO-BH [41], BRABC [42] MBABC [43]. Tables 4 and 5 show comparison of cost with other state of the art algorithm. From Table 5, it is clearly shown that the total operation costs achieved by the proposed improved CGTO is the best in terms of the test systems of 10, 40, 60, 80 units. For the test systems with 20 units and 100 units, the result of BGSA is second best. It is obvious that the proposed improved CGTO is superior to the mentioned methods. As the results shown above, the total production costs of the CGTO are demonstrated to be less expensive than those of other methods on generating unit systems. Obviously, the CGTO vastly improves performance than other methods in terms of both solution quality and CPU times especially on the large-scale UCP. The proposed CGTO converges to the solution at a faster rate than the other methods reported in the literature.

From Table 6 it is obvious that for 7 unit wind-integrated thermal unit saving in fuel cost is maximum and is 14.5%, and as the number of units go on increasing percentage cost saving gets reduces and for 100 unit wind-thermal units it is less than 1%.

Table 6 Economic analysis of results while considering effect of wind power share

Conclusions

The chaotic Gorilla troop's optimizer (CGTO), a ground-breaking global optimization method that was developed to address the Generation Scheduling Problem, was inspired by and derived from the multi-phased Social process of gorilla groups. This optimizer presents a new chaotic updating strategy known as singer map based position updating, which logically divides the population into selection of silverback and adopts the strategy used by silverback (CMBCS). A better balance between exploration and exploitation is achieved by using the GTO approach, which looks for global optima, as opposed to chaotic search, which prevents the algorithm from slipping into local minima. The impact of renewable energy on the issue of generation scheduling is further explored in this manuscript. The experiment has used thermal and wind thermal systems with 10 units, 20 units, 40 units, 60 units, and 80 units and 100 unit the extensive computational studies demonstrate that the CGTO strategy suggested in this paper performs better than the majority of the current methods to create high-quality solutions close to global minima. The influence of wind energy on the scheduling of power generation was also emphasized in this research. The precise calculation of wind power using wind data has successfully shown that a significant amount of money may be saved by using a wind integrated thermal system.

The following are potential areas for further research:

• The hydro generating units can be combined along with conventional thermal generating units, renewable and EV based energy sources.

• The maintenance cost and shut down cost of the thermal units can be included in addition to production cost as futuristic overall objective function.

• The generation scheduling problem can be consider with heat as combined cycle consider plants which is the most important concern for the Antarctic countries.

• Hybrid energy sources such as super capacitor based hybrid battery packs and ancillary services based energy sources can be included to fulfill the futuristic power demand in smart grid system.

• The proposed CGTO approach has been tested for single objective function; however same can be used to develop with multi objective function.