^{1}

^{2}

^{2}

^{1}

^{2}

In this paper, a novel approach of hybridization of two efficient metaheuristic algorithms is proposed for energy system analysis and modelling based on a hydro and thermal based power system in both single and multiobjective environment. The scheduling of hydro and thermal power is modelled descriptively including the handling method of various practical nonlinear constraints. The main goal for the proposed modelling is to minimize the total production cost (which is highly nonlinear and nonconvex problem) and emission while satisfying involved hydro and thermal unit commitment limitations. The cascaded hydro reservoirs of hydro subsystem and intertemporal constraints regarding thermal units along with nonlinear nonconvex, mixed-integer mixed-binary objective function make the search space highly complex. To solve such a complicated system, a hybridization of Gray Wolf Optimization and Artificial Bee Colony algorithm, that is,

Hydrothermal unit commitment coordination (HTUC) problem concerns utilizing hydro potential satisfying hydro constraints in such a way that the cost of produced electricity from thermal resources during a scheduling period of time is lowermost and with lowest environmental impact. Many researchers’ have forwarded various solution techniques to solve such problem but due to extreme nonlinearity and compounded structure of hydro and thermal units, the process of finding feasible and optimal solution is quite tough and time-consuming. In addition, nonlinearity of search space increases greatly with increasing complexity of the compounded structure of the system. Most conventional optimization techniques do not work well for problems like HTUC with such nonlinearity and multimodality. The concerned problem involves not only nonlinearity but also nonconvexity. The thermal power generation (due to quadratic cost function), hydro power generation, and emission are nonlinear in nature. In case of thermal power generation the nonconvexity is due to consideration of valve point loading effect for thermal units. In Figure

The waveform of production cost of two thermal units (a) with valve point loading effect and (b) without valve point loading effect where

Several researchers have attempted solving HTUC problem using different solution procedures. Initially most researchers proposed conventional mathematical techniques, such as dynamic programming [

Catalão et al. solved hydro subsystem using a nonlinear optimization method [

Therefore, the objective of this research focuses on developing a suitable algorithm for HTUC problem based on hybridized Artificial Bee Colony (ABC) and Gray Wolf Optimization (GWO) algorithm (

On the other hand, in GWO algorithm alone the global-or-local search operation on individual position reduces the exploration capability of GWO, which makes GWO fail to provide sufficient diversity in the search space for a complex problem like HTUC with a further deficiency regarding local optima entrapment or premature convergence. For this reason in this research the

The performance of the proposed

HTUC involves optimization of nonlinear objective function, with a mixture of linear/nonlinear and dynamic network flow constraints. The problem difficulty is compounded by a number of practical considerations and unless several simplifying assumptions are made, the problem is difficult to solve for practical power systems. The basic optimal HTUC involves minimizing overall thermal cost,

In Figure

The waveform of production cost (see (

In this paper, one day is considered as the scheduling period (

The waveform of hydro power generation cost (see (

The main drawback regarding metaheuristic algorithms is extensive random search where there is a high possibility of searching the same position repeatedly while few places in the vast multidimensional search space did not get any attention at all. Superior nonlinearity and nonconvexity increase the effect of this phenomena even further. In order to search such a complex search space it is necessary to balance between the local and global search capability of the acting algorithm or in other words the exploitation and exploration capability according to the search space behavior. For this purpose a hybridized algorithm, that is, hybrid-Artificial Bee Colony/Grey Wolf Optimization (

The challenge of an algorithm in constrained environment is much more complicated than in an unconstrained environment. Algorithm performance also greatly depends on the technique used to handle the acting constraints. Before explaining the function evaluation method and constraint handling process of HTUC problem, the key arrangement of

The initial population will be created using (

The employed bees will search the neighborhoods of the initial positions, in hopes of better position using (

Consistent individuals that do not improve their position will be converted to scout based on a predefined parameter, limited and forced to search the search space without any guidance using

The best solution so far is taken as the global best for the next iteration. This process continues until the iteration number reaches the maximum cycle number (

The detailed flow chart of hybrid-Artificial Bee Colony (ABC)/Grey Wolf Optimization (GWO) (

The individuals

The individual with minimum

In this section, an algorithm based on

According to the algorithm strategy the hydro discharges (

Constraint handling technique used to satisfy hydro limitations is as follows:

Considering the initial volume as reference, the end volume of

So, the end volume mismatch is

A dependent time variable,

In order to minimize the mismatch, the discharge at

If

This process is applied on every hydro unit present in the system from least depended to most depended units. Normally a piecewise linear function is used along with Lagrangian multiplier to handle the end volume constraint but this process is time-consuming and erroneous as linearization compromises the accuracy of the system. For this reason in this proposed heuristic approach a rectification based method is used to solve the end volume limitation.

Once all the mentioned limitations concerning hydro network are satisfied, the hydro power generation at each hour has to be calculated (see (

The constraint handling technique for power balance is described below:

As the minimum up/down time limitations are considered, the units that are available on a particular time interval have to be found based on previous hours and the value (generation) of unavailable units is set to zero. The required power is distributed among the available units.

A dependent variable,

Power balance mismatch at

The power generation of

If

This process will be implemented for every hour to balance generated power at each hour.

In any metaheuristic algorithm the constraint handling techniques are very crucial factor. But the problem adopted in this research is not only nonlinear but also highly nonconvex in nature as shown in Figures

After creating the subpopulations according to Section

where

In case of multiple objectives, fitness value will be found using (

For each individual according to the grey wolf optimization phase the better individuals will be selected modifying the population.

The new positions will be evaluated using (

Each individual of population will create another position according to (

Based on the fitness values of new found positions and their concerned individual of population found in Step

The best solution of final population will be considered as global best.

Finally, an iteration counter increases.

Steps (b)–(g) will continue until iteration counter reaches

For multiple objectives, instead of best solution best compromised solution needed to be found. In that case the global best compromised solution can be calculated using pareto optimal front given in (

In this section, two test systems are considered for experimentation of the performance of proposed algorithm on HTUC problem. The scheduled time period for both systems is taken as 24 hours. The objective of these studies is to showcase the effectiveness of

The parameters of the

Parameters used for test systems in algorithms.

Test System I | Test System II |
---|---|

Maximum cycle number ( | Maximum cycle number ( |

Limit = 25 | Limit = 35 |

Population size = 40 | Population size = 80 |

The mentioned algorithms are coded in MATLAB (Version: 8.1.0.604 (R2013a)) environment and simulated on Dell XPS15 (2760QM®, Intel configured), 3rd Generation Quad-Core i7 Processor ~3.2 GHz processor speed and 12 GB RAM.

Test System I [

This system is solved in two different scenarios. In first scenario (Case I) all the hydro limitations are considered along with thermal generation limitation (see (

The number of dimensions in the multidimensional search space or decision variables in Case I is

Performance comparison for Test System I (Case I and Case II).

Algorithm | Case I | Case II | ||||
---|---|---|---|---|---|---|

Best | Worst | Time | Best | Worst | Time | |

| 165921.53 | 170201.83 | 202.59 Sec | 271736.07 | 475124.26 | 394.61 Sec |

ABC [ | 168386.23 | 183044.84 | 213.89 Sec | 281645.19 | 489542.39 | 1023.64 Sec |

GWO [ | 180356.68 | 195635.44 | 242.65 Sec | 325689.26 | 524586.69 | 536.98 Sec |

SPPSO [ | 167710.56 | — | — | — | — | — |

IDE [ | 170576.50 | — | — | — | — | — |

MINLP [ | 208706.20 | — | 105.23 Sec | — | — | — |

(a) Convergence characteristic of different algorithms for Test System I, Case I. (b) Convergence characteristic of different algorithms for Test System I, Case II.

From Table

Performance comparison for Test System II for both Cases I and II in single objective environment.

Algorithm | Case I | Case II | ||||
---|---|---|---|---|---|---|

Best | Worst | Time | Best | Worst | Time | |

| 3411361.47 | 3814805.07 | 424.36 Sec | 4604300.83 | 6653645.64 | 394.61 Sec |

ABC [ | 3923430.55 | 4811261.91 | 443.59 Sec | 4828932.22 | 6453793.64 | 1506.64 Sec |

GWO [ | 4259836.27 | 5037176.67 | 797.53 Sec | 5315693.34 | 7037176.67 | 624.64 Sec |

CSA [ | 3503527.75 | — | — | 4823256.68 | — | — |

MINLP [ | 4023565.36 | — | 212.56 Sec | — | — | — |

In Test System II the number of dimensions is even higher than Test System I and contains almost all the features of a practical hydrothermal system. It incorporates 44 hydro units and 54 thermal units. The hydro system [

The thermal system [

This system is solved in two different scenarios like Test System I. In Case I the number of variables is

In order to verify, the results are compared with [

(a) Convergence characteristic of different algorithms for Test System II, Case I. (b) Convergence characteristic of different algorithms for Test System II, Case II.

The convergence characteristics show that the

This problem is solved using the same parameter shown in Section

Performance comparison for Test System I for both Case I and Case II in multiobjective environment.

Algorithm | Best | Worst | ||||
---|---|---|---|---|---|---|

Cost | Emission | Time | Cost | Emission | Time | |

| 113295.63 | 19536.65 | 452.69 | 123569.65 | 23326.65 | 463.59 |

ABC [ | 167386.23 | 21831.56 | 745.36 | 135643.96 | 23549.95 | 745.36 |

GWO [ | 178546.35 | 22469.49 | 721.49 | 154937.19 | 24987.48 | 456.19 |

IDE [ | 134649.96 | 234697.49 | 748.59 | 194853.15 | 31497.19 | 489.46 |

MINLP [ | 123494.46 | 200349.45 | 451.74 | — | — | — |

Convergence characteristic of (a) production cost and (b) emission for Test System I.

From the convergence characteristics, it is clear that the

Like the previous section in this case also Test System II is solved under the influence of the parameter used in Section

Performance comparison for Test System II for both Case I and Case II in multiobjective environment.

Algorithm | Best | Worst | ||||
---|---|---|---|---|---|---|

Cost | Emission | Time | Cost | Emission | Time | |

| 2259863.56 | 438592.35 | 1523.32 | 2456965.58 | 440015.20 | 1589.26 |

ABC [ | 2667386.23 | 438312.56 | 1689.36 | 2785952.63 | 445963.56 | 1702.25 |

GWO [ | 3498830.99 | 671839.44 | 1559.45 | 3925696.54 | 675895.68 | 1517.26 |

MINLP [ | 5939758.65 | 542569.65 | 936.36 | — | — | — |

Convergence characteristic of (a) production cost and (b) emission for Test System II.

In addition to this it is seen that the convergence and time consumption for

The reason for such efficiency of the proposed algorithm is due to using the more sophisticated and efficient exploitation technique of GWO algorithm instead of onlooker bee phase of ABC algorithm. These phenomena give the algorithm necessary attribute to efficiently search the globally best solution from such complicated and vast search space. Also the effect of proposed constraint handling technique mentioned in Section

The hydrothermal unit commitment coordination (HTUC) is modelled having included various limitations of hydro and thermal subsystems. The analysis is performed based on an efficient optimization algorithm for better performance. A hybridized approach of ABC and GWO is presented in this work in both single and multiobjective environment. This proposed method is tested with two different test systems in different scenarios. Though the proposed method is performing better than other mentioned algorithms through thorough analysis, it is found that, in case of Test System I, where the number of variables is comparatively low, the

Unit number

Number of hydro units

Number of thermal units

Scheduled time (1 day (24 hours))

Time delay of

Set of upstream plant of

Inflows to

Spillage of

Reservoir volume or water content of

Discharge from

Load demand at time “

maximum/minimum power limit of

Maximum/minimum discharge limit of

Maximum/minimum volume limit of

Maximum/minimum power limit of

Power from

Power from

ON/OFF status of

Initial status of

Minimum ON/OFF time of

Minimum cold start time of

Time duration for which

Up and down ramp rate limit of

Cold and hot start cost of

There authors declare no conflict of interests.

_{2}emissions constraints