Electricity Generation Scheduling With Integrated Conventional and Non-Conventional Units Using Grey Wolf Optimization Algorithm

1

Electricity Generation Scheduling With Integrated Conventional and Non- Conventional Units Using Grey Wolf Optimization Algorithm

R.Saravanan

, S.Subramanian

, V.Dharmalingam

, S.Ganesan

1Research Scholar, ²Professor, ⁴Assistant Professor

Department of Electrical Engineering, Annamalai University, Chidambaram

3Professor, Department of Electrical Engineering, Pandian Saraswathi Yadav Engineering College, Sivaganagai

1[email protected] , ²[email protected] , ³[email protected] , ⁴[email protected]

Abstract: Integration of wind energy along with conventional energy sources for power production has become a tough task in the problem of generation scheduling. The foremost objective of generation scheduling is to attain better optimal solution with reduced fuel cost. In this paper, a new population based search algorithm named Grey Wolf Optimization algorithm is employed to solve the generation scheduling problem. The performance of the system with proposed GWO method is tested using three different test systems under various constraints. The results obtained by integrating 26 Conventional units and 2 wind farms are compared with the results obtained from integration of 26 conventional, 2 wind farms and 2 hydro units and the results are also compared with the conventional PSO method. It is evident from the comparison of obtained results, that the solution obtained from proposed GWO method reaches the near optimal solution.

Keywords: Generation scheduling, Grey wolf optimization algorithm, Wind power availability, Hydro power availability, Total generation cost.

I.

I

The evolution of new energy sources for electric power generation is to meet the increasing demand or load and the rapid increase in fuel cost. Generally, there are two type of energy sources are available to produce or generate electricity. 1. Conventional energy source like thermal, petroleum, natural gas and nuclear sources of energy. 2. Non-conventional energy sources and they are solar, wind, tidal, hydro, biomass and geothermal energy sources.

The unit commitment and economic dispatch problems are used to calculate the generation cost with respect to the fuel consumption depend upon the demand. The generation cost reduced by maintaining the generation scheduling constraints with varies critical load condition.

The generation scheduling constraints will vary from one utility unit to other unit with respect to the type of combination of energy sources in the particular generating station. When the availability of energy source which is used to produce electricity increases, the fuel consumption decreases and so the generation cost will also decrease with respect to the load.

The impacts of wind, hydro and thermal power generation are modelled by increasing the reserve requirements and the hydro power generation relates to the availability of water level in the reservoir. The

generated power in wind and hydro units are depending on the availability of wind speed and the amount of water. So it is difficult to produce power to meet the demand. Thus integrating conventional and non- conventional unit results in enough power production in all conditions.

In this paper, conventional energy source is integrated with the non-conventional energy sources and the problem of generation scheduling is solved by a newly proposed method called Grey Wolf Optimization method. The main aim of the proposed system is to reduce the generation cost along with the reduction in fuel consumption. The system performance is evaluated by the comparison of results in various test cases with several numbers of trails and iterations and a new swarm intelligence based GWO method is used to determine the feasible optimal solution. The results for different test systems with GWO are also compared with the conventional PSO method to prove that the proposed GWO method is computationally efficient and it attains the best optimal solution and the results of integrated wind, thermal and hydro units led to better results than integrated wind-thermal units.

II.

P

F

(1)

2

The reduction of the generating cost is one of the main focuses when we considering the GS problem, while the operation is go through a various constraints. The objective function entitles the total generation cost;

including the fuel cost, the operation and maintenance costs. Minimization equation is given above.

Where

The unit constraints will be introduced in the following system and the system demand will satisfied in the following equation:

The percentage of total wind power availability is used to determine the reserve requirement of the given system. The above factor (RESW) is also used to the wind power forecast error. Assume RESW is to be 10%

in each wind form.

(3)

The constraints of generation unit must be satisfied and therefore the wind power availability must be satisfied as follows:

(4)

The maximum and minimum generation of the generating units must be satisfied as follows:

(5)

A. Hydro Constraints

The hydro constraints are used to maintain the generation scheduling. The generation scheduling has two types one is long term and another one is long term scheduling. The power generation by hydro turbine is calculated by using the given below equations:

Hydraulic balance constraints

(6) Initial and final reservoir constraints

(7) Hydroelectric power generation

(8) Storage and turbine volume limit

(9) III.

G

M

W

We cannot predict the electricity generation easily in the wind farms due to the atmospheric condition of the site. But it is necessary to accurately calculate the electric power generated by a wind turbine while go to the load forecasting technique. The generated power is related to the speed of the wind. When the speed of thee wind is increased the related power generation also increased similarly the speed of wind is reduced means the power generation also reduced with respect to the speed. The power curve is used to predict the power output of a win turbine. By using the power curve model the demand will be predicted but it is not more accurate compared to the conventional methods. The power curve is plotted between the output powers versus the wind speed.

When the wind speed is between the rated speed and the cut-out speed then only the rated power is generated in the wind turbine. The speed curves are calculated by using power curve. The below following equations are used to find out the power generated:

(10)

Where,

3

The wind power generation unit’s model will be employed to a model a wind turbine

 Rated at 2 MW

 Cut-in, rated and cut-out wind speeds of 2.5,14 and 25m/s.

 Wind speed > 4.3m/s

 Wind power output >0

IV.

I

G

In 2014. GWO is created by based on one type of wolf family is called grey wolves. Mirjalili takes the leadership hierarchy and hunting mechanism of grey wolves to create the new optimization process and it implemented in the GWO optimization algorithm. The four main steps of GWO are given below:

 Hunting

 Searching of prey

 Encircling prey

 Attacking prey

Social hierarchy dominant of grey wolves:

1. Alpha ( ) 2. Beta ( ) 3. Delta ( ) 4. Omega ( )

The main phases of grey wolf hunting according to the social behaviour of grey wolves are as follows:

 Tracking , chasing, and approaching the prey

 Pursuing, encircling, and harassing the prey until it stops moving

 Attack towards the prey A. Mathematical Model

The design and optimization performance of GWO is based on the social hierarchy and hunting techniques of grey wolves. When design a mathematical models the following considerations must be satisfied:

 The fittest solution named as alpha

 The second and third best solutions respectively named as beta and delta

 The rest of the candidate solutions are named as omega

B. Encircling Prey

The encircling prey is mentioned in the optimization during hunting process of grey wolves. The mathematical model equations are given below:

(11)

where

and are calculated as follows:

Linearly decreased from 2- 0 over the course of iterations

and random vectors in {0, 1}

From the above equation,

(x, y) the position of grey wolf Update position of prey

Various places around the best agent can be attained with respect to the current position by adjusting the values

By setting the wolves could reached the instance position .

 The random vectors allow grey wolves to attain any place or position between two particular points.

 So that the wolves can update its position inside the area or space around the prey in any random location by the above equation.

 The search space with ‘n’ dimension can be reached by using the above same concept.

 For attaining the best solution the wolves will move around the prey in hyper-cube.

C. Hunting

These kinds of wolves have the ability to identify the location of prey and it also able to encircle them.

 Alpha - always guide the other wolves during hunting

 Beta & delta – sometimes participated in hunting.

Initially we do not have any idea about the location of prey. The location of prey is calculated by using mathematical calculation. The prey position is determined by the best three solutions and it also used to encircle them. From the wolves behaviour we can able to obtain the best three solutions from the alpha, beta and delta as well as including omega. The position of the best agents is updated depend upon the position of prey by the following equations:

4

(12)

D. Attacking Prey

The grey wolves end the hunts by attacking the prey when it stops moving. The prey value of a decreased by mathematical formulation. The fluctuation value of changed with respect to the value of prey . This approach allows search agents to update their position based on the location on .

 - random value in interval [-2a,2a]

 If - in the interval [-1,1] the search agents next positions can be in anywhere between the current and prey position.

E. Search For Prey

The searching of prey position determined by the position of alpha, beta and delta. To search prey they diverge from one to other and converge of prey. - In the interval [-1, 1] is used for divergence process.

The grey wolves differ from each other to search for prey and converge to attack prey. >1 stimulates the wolves to find the fit prey by diverge process. – Random variables [0, 2] used for exploration.

F. A GWO based algorithm to solve Generation Scheduling problem

In this section, a GWO based optimization algorithm used for solving Generation Scheduling problems. It will be suggested in which the equality and inequality constrains of the Generation Scheduling problems when changing each search agent positions. Here, the search agent positions update their positions depends on the prey position.

 Start the program by initiate the population

 Based on the population identify the position of wolves and position of vectors.

 Identify the search agents based on the fitness values

 Check max number of iteration .

 If the number of iteration less than the k means the process stopped.

 Otherwise the process continued and goes to the next step.

 Depend upon the position of prey the search agents update their positions to find out the prey with high fitness.

 Calculate all the fitness among that identifies the best one.

 Encircle the best one.

 By attacking the prey the process will be stopped.

V.

R

T

S

The proposed GWO optimization algorithm is applied to a model system. By using two approaches its effectiveness was verified. The two methods are one is initialization and another one is simulation parts.

Initially, this optimization approach is applied to the test system 1. And the test results are compared with the other methods to verify the feasibility solution of GS problem while using the proposed GWO optimization method.

A. GWO Initialization Procedure

Test system 1: The test system one has 30 generating units which include 26 thermal units and 2 wind units and 2 hydro units (26C +2W + 2H). The input data for test system 1 is taken from the reference papers [10, 12]. Table 1 denotes the wind power availability with respect to the wind speed. The percentage of annual peak load is calculated. The wind data is mentioned for 12 months. The percentage of annual peak load is calculated by using the reserve requirements and wind speed. The reference values are taken from [12].

Table I. Load Pattern, Reserve Requirements and Wind Farm Output

Perio d

Annual Peak Load (in %)

Reserve Requirements

(MW)

Wind Speed

Wind Power Availability

(MW) Unit 26 Unit 27 Unit 26 Unit 27 1 87.8 120.5483 5.788 8.284 3.576 16.607 2 88 120.5905 5.358 8.149 2.23 15.675 3 75 104.1935 5.829 9.446 3.717 25.718 4 83.7 116.2843 7.193 9.134 9.817 23.076 5 90 124.6211 7.989 8.284 14.604 16.607 6 89.6 123.1303 7.559 7.19 11.905 9.798 7 88 120.6043 7.25 6.826 10.13 7.913 8 80 109.9162 7.063 9.836 9.122 29.202 9 78 108.0626 7.591 8.127 12.097 15.529 10 88.1 121.0406 6.165 8.213 4.937 16.119 11 94 129.6722 6.414 8.966 6.007 21.715 12 100 139.1649 7.035 10.20 8.973 32.676 Test system 2: It includes table 2 and table 3. In table 2 the total cost and accuracy are calculated for various populations using the proposed GWO method. It’s only for 26 conventional and 2 wind units. The total cost which includes minimum cost, average cost and

5

standard deviation represented in M$. Table 3 is also used to calculate total cost and accuracy but it for s26

conventional, 2 wind and 2 hydro units (26C + 2H + 2W).

Table II. The Simulation Results in Different Approaches for 100 Trails and 100 Iterations in Test System 2 (2t + 2w)

Method Population

Total cost(M$)

Standard Deviation Accuracy Minimum cost Average cost

GWO

10 292.065 295.524 1.3034 50.01585

20 293.162 295.707 1.0484 42.52871

30 292.795 296.786 1.1568 49.243

40 293.185 297.904 1.2082 53.697

50 292.108 297.94 1.7087 53.538

Table III. The Simulation Results in Different Approaches for 100 Iterations and 100 Trails in Test System 2 (2T +2W+2H)

Method Population

Total cost(M$)

Standard

Deviation Accuracy Minimum cost Average cost

GWO

10 292.009 295.1 0.71701 62.352

20 293.096 295.825 0.6740 68.718

30 292.729 296.583 0.8424 79.874

40 293.029 297.359 0.9286 67.682

50 292.052 297.884 1.2450 67.398

Table IV. Optimal Results for Supplying Load Contribution in GWO for (26 C + 2 W)

Units m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12

1 11.26 11.26 5.23 3.85 8.265 5.68 5.68 3.85 6.35 8.68 9.365 10.35 2 5.58 8.58 10.23 8.69 11.68 11.684 11.684 8.69 9.65 9.25 11.265 11.35 3 10.59 10.59 11.65 11.243 10.68 11.356 11.356 11.243 10.68 11.35 10.35 9.36 4 3.89 5.89 9.25 10.265 8.94 10.68 10.68 10.265 11.95 5.32 8.69 11.894 5 9.54 11.54 8.25 6.25 3.298 8.69 8.69 6.25 5.68 10.65 3.25 10.358 6 18.68 18.68 6.18 8.265 19.268 17.268 17.268 8.265 19.368 9.35 19.865 19.25 7 5.265 13.265 12.35 6.842 15.36 19.684 19.684 6.842 5.98 19.65 9.23 18.59 8 9.584 9.584 18.25 13.268 18.95 14.865 14.865 13.268 4.97 12.35 15.365 15.26 9 15.84 19.84 19.025 17.268 15.69 12.987 12.987 17.268 16.98 5.39 18.35 9.68 10 25.68 25.68 16.028 29.487 75.268 74.856 74.856 29.487 75.98 16.25 17.36 75.68 11 75.356 75.356 57.86 53.598 69.68 19.684 59.684 53.598 59.687 72.65 74.25 72.68 12 56.298 56.298 66.87 67.268 74.68 39.258 69.258 67.268 69.68 32.68 75.98 74.258 13 71.021 75.021 75.016 75.169 69.268 48.587 74.587 75.169 74.95 55.62 69.32 69.25 14 90.265 96.265 99.016 99.04 96.268 93.102 93.102 99.04 98.674 59.32 99.35 96.58 15 98.265 98.265 75.013 91.468 98.674 92.125 92.125 91.468 32.957 78.96 89.25 78.265

6

16 92.154 92.154 88 97.468 92.157 88.145 98.145 97.468 31.57 96.35 59.35 89.36 17 55.265 59.684 115.13 147.54 149.59 151.98 151.98 146.54 145.98 145.3 145.37 154.35 18 59.357 57.684 55.025 154.27 154.27 145.7 154.7 134.27 96.25 132.7 125.35 148.96 19 125.33 96.358 56.021 149.89 153.27 55.025 55.013 128.89 100.65 55.36 108.69 56.258 20 110.6 102.66 57.036 55.032 55.024 59.025 59.025 55.203 86.32 121.4 119.35 146.4 21 198.25 198.25 68.648 77.059 69.035 68.068 69.235 56.098 119.7 152.4 186.35 196.38 22 178.86 161.88 75.321 78.365 190.27 196.03 116.73 129.26 109.68 71.68 142.37 123.35 23 179.66 187.21 178.27 189.92 196.28 169.36 144.19 70.251 169.61 107.1 99.35 120.35 24 240.21 340.21 286.6 248.86 256.22 245.87 345.87 248.86 298.32 320.2 348.65 310.35 25 303.27 283.27 280.25 168.46 252.68 321.35 285.35 337.65 151 320.7 190.35 350.38 26 220.27 160.27 190.35 312.51 166.62 328.35 189.35 150.99 254.35 368.3 366.25 320.35 27 75.26 75.26 65.26 65.878 19.68 20.53 40.65 36.298 12.68 12.65 78.36 25.36 28 25.015 25.015 19.26 12.68 78.94 89.26 89.26 66.258 26.35 67.35 36.98 75.35 Total

Cost (M$/yr)

341.9 342.7 292 325.9 350.4 348.9 342.7 311.5 303.7 343 366 389.4

Table V. Optimal Results for Supplying Load Contribution in GWO for (26 C + 2 W + 2 H)

Units m1 m2 m3 m4 m5 m6 m7 m8 m9 m10 m11 m12

1 9.544 9.544 5.245 11.245 10.265 10.265 3.265 11.245 3.245 3.265 10.27 10.265 2 7.567 7.567 3.825 3.825 9.257 9.257 10.257 3.825 5.825 10.257 9.257 9.257 3 4.245 4.245 8.154 8.154 8.278 8.278 8.278 8.154 8.154 8.278 8.278 8.278 4 5.245 5.245 6.326 3.326 11.587 11.587 5.587 3.326 3.326 5.587 11.59 11.587 5 3.145 3.145 8.245 6.245 3.258 3.258 3.258 6.245 6.245 3.258 3.258 3.258 6 6.256 6.256 5.215 5.2148 18.278 18.278 8.278 5.2148 5.2148 8.278 18.28 18.278 7 12.245 12.245 10.24 10.239 12.478 12.478 5.478 10.239 10.239 5.478 12.48 12.478 8 9.245 9.245 12.59 12.589 8.268 8.268 10.268 12.589 12.589 10.268 8.268 8.268 9 5.235 5.235 14.27 8.268 5.287 5.287 5.287 8.268 8.268 5.287 5.287 5.287 10 35.265 35.265 19.26 12.257 38.577 38.577 38.577 12.257 12.257 38.577 38.58 38.577 11 25.145 25.145 41.24 31.238 18.251 18.251 18.251 31.238 31.238 18.251 18.25 18.251 12 55.145 55.145 54.27 41.267 32.449 32.449 32.449 41.267 21.267 33.149 32.45 32.449 13 26.746 25.746 45.26 45.257 36.214 36.214 36.214 45.257 35.257 36.214 36.21 36.214 14 42.256 42.256 31.49 31.487 95.012 95.012 95.012 31.187 31.187 95.012 95.01 77.012 15 79.265 79.265 41.24 41.235 50.215 50.215 50.215 41.235 41.235 51.215 50.22 50.215 16 30.314 36.314 26.25 26.248 38.245 38.245 38.245 26.248 26.248 38.145 38.25 38.245 17 70.284 70.284 56.25 56.248 62.124 62.124 62.124 56.248 46.248 53.124 62.12 62.124 18 104.95 64.954 90.6 70.598 100.59 87.587 84.587 70.198 70.198 95.387 130.6 120.59 19 110.95 120.95 91.58 91.578 58.145 58.145 58.145 91.378 92.378 58.145 58.15 58.145 20 55.989 55.989 85.02 85.024 75.874 75.874 75.874 85.024 86.024 75.174 73.87 73.874

7

21 106.25 116.25 70.25 70.248 94.268 94.268 94.268 70.248 70.248 94.268 94.27 94.268 22 95.215 115.22 89.16 79.157 83.478 83.478 73.478 79.157 79.157 73.478 83.48 73.478 23 70.519 70.125 95.27 95.268 99.271 99.271 99.271 95.268 95.268 99.271 99.27 99.271 24 150.22 155.22 141.6 141.6 156.3 156.3 156.3 141.6 141.6 156.3 156.3 156.3 25 108.25 108.25 119.3 119.25 109.38 109.38 109.38 120.25 110.25 109.38 119.4 119.38 26 260.97 261.97 110 110.9 110.65 112.95 102.95 110.9 110.9 102.95 110.7 110.65 27 12.838 12.235 39.22 39.215 36.254 36.254 36.254 39.215 39.215 36.254 36.25 36.254 28 60.214 60.214 52.25 52.245 69.245 69.245 69.145 52.245 52.245 69.145 69.25 69.245 29 446.83 446.24 375.2 500.22 552.33 552.23 567.13 400.22 400.22 567.13 622.3 722.33 30 360.28 356.26 276.3 450.26 426.18 426.18 418.18 450.26 450.26 418.18 426.2 526.18 Total

Cost (M$/Yr)

340.9 341.6 291.5 323.9 349.4 348.2 342.3 310.5 302.7 342.8 365 387.3

Test system 3: it includes table 4 and table 5. In table 4 the load in month wise data (12 months) is calculated for conventional and wind units (26+2). By using the month load values we can calculate the annual load. In table 5 also we calculate the load in moth wise including the hydro units (2C + 2W+ 2H). In table 6 the comparison is given between the conventional with wind (2C+2W) and conventional with wind and hydro unit (2C + 2W+ 2H). The proposed GWO performances are compared with the PSO method. The PSO approach values are taken from [12]. In annexure 1 the table 4 and 5 is mentioned.

Table VI. Best Result in Different Approaches of GWO for 100 Trails and 100 Iterations in Different Test Data Test

system Methods Population Size

Total cost(M$)

Accuracy

Min Avg. Std.dev

26 C +

2W PSO 10 293.6009 297.9482 2.0575 46.177

26C+ 2W Proposed

GWO 10 292.065 295.1544 1.202046 50.01585

26C+

2W+ 2H

Proposed

GWO 10 292.009 295.1 0.717013 66.3516

B. Simulation Result of Gwo

The simulation procedure is based on the acceleration values (appropriate values). The minimum, average and standard deviation of the objective function of GS was obtained by those acceleration values. Table 2 & 3 denotes the best result of this GS problem employing 100 iterations and 100 trails. Table shows the best result of this GS problem employing 100 iterations and 100 trails. However, by comparison of proposed GWO approaches, it can be concluded that the proposed GWO with 2 hydro units + 2 wind + 26 conventional units which has a total cost which is less than the cost related to the proposed GWO with 26 conventional + 2 wind units.

Figure 1 shows the 100 trails values for wind and thermal unit with 5 populations 10, 20, 30, 40 and respectively.

Fig.1. Sensitivity Analysis of Parameter’s Selection for Proposed GWO (2W+26C)

10 Pop 20Pop 30 Pop 40 Pop

50 Pop

8

Figure 2 shows the 100 trail values for including 2 hydro unit with 5 populations 10, 20, 30, 45 and 50 respectively. The above colour and populations are similar for this figure also.

10

Pop 20Pop 30 Pop

40 Pop

50 Pop

Fig.2. Sensitivity Analysis of Parameter’s Selection for Proposed GWO (2wind + 26 conventional+ 2 hydro) Figure 3 shows the 100 iteration values for 2 wind + 26 conventional units which indicates the total generation cost with respect to the iterations.

Fig. 3. Convergence Characteristics of Proposed GWO with 100 Iterations (26 conventional + 2 wind units Figure 4 shows the 100 iteration values for 2 wind + 26 conventional + 2 hydro units which indicates the total generation cost with respect to the iterations. It shows the total generation cost of wind + hydro + thermal / conventional unit is low compared to wind + hydro units.

Fig. 4. Convergence Characteristics of Proposed GWO with 100 Iterations (26 Conventional+ 2 wind + 2

Hydro Units)

VI.

C

This paper presents a new method for solving GS problems based on the GWO algorithm. The result or solution of this problem is used to satisfy the constraints. When the position of search agents is changed from one position to another one. By using the principle of ‘sharing method’ the conventional and non conventional energy sources sharing the load depends on the requirement. By using the load sharing principle the fuel cost is reduced in wind, thermal and system (26+2+2) compared to the wind and thermal units (26+2). This method is implemented to 3 test systems.

It shown when we combine the hydro unit with thermal and wind unit the fuel cost surely reduced compared to the power generated by wind and thermal units.

VII.

R

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0 10 20 30 40 50 60 70 80 90 100

292 293 294 295 296 297 298 299 300 301 302

Trail

Total Generation Cost(M$/yr)

x10⁸

0 10 20 30 40 50 60 70 80 90 100

2.95 3 3.05 3.1 3.15 3.2 3.25

iteration

Total Generation Cost(M$/yr)

x10⁸

0 10 20 30 40 50 60 70 80 90 100

2.8 2.85 2.9 2.95 3 3.05 3.1 3.15 3.2 3.25 3.3

iteration

total Generation Cost(M$/yr)

X10⁸

9

Technology, Azadi Ave.,P.O. Box 11365-8639, Tehran, Iran.

[11] Siyu Lu, Suhua Lou, Yaowu Wu, Xianggen Yin.

“Power System economic dispatch under Low – carbon economy with carbon capture plants considered”. DOI: 1049/iet – gtd. 2012.0590.

2013.

[12] Anup Shukla, Student Member, IEEE and S. N.

Singh, Senior Member, IEEE, “Cluster based Wind-Hydro-Thermal Unit Commitment Using GSA Algorithm”, Department of Electrical

Appendix

The coefficients of generating unit g

Weighting factors called acceleration constants D dimension of the particle

g index for thermal generator unit n(t) number of hours in time t

Number of thermal generator units Number of wind units

OMFCT(g) operation and maintenance fixed cost of thermal unit g ($/MWyr)

OMFCW(w) operation and maintenance fixed cost of wind unit w ($/MWyr)

OMVCW(w) operation and maintenance variable cost of wind unit w ($/MWyr)

System demand at time t (MW)

Load contribution of thermal unit g at time t (MW)

Reserve contribution of thermal unit g at time t (MW)

A fraction of total system load for

system reserve requirements (first part) at time t (MW)

Generation of wind unit w at time t (MW)

RESW a fraction of total wind power employed to compensate wind power prediction errors (%)

T number of periods under study (12 months) T index for time

W index for wind unit

Commitment state of unit g at time t (on = 1, off=0)

Commitment state of wind unit w at time t (on = 1, off=0)

Maximum available wind power of wind unit w at time t (MW)

Water inflow to the reservoir k during hour Generation of hydro unit at hour t

Spilled outflow to the reservoir k during hour t

Turbine outflow for reservoir k during hour t Max turbine outflow for reservoir k during hour t

Min turbine outflow for reservoir k during hour t

Volume of water for reservoir k during hour t Max volume of water for reservoir k Min volume of water for reservoir k

Input / output characteristics of hydro unit