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The solar and wind renewable energy is developing very rapidly to fulfill the energy gap. This specific increasing share of renewable energy is a reaction to the ecological trepidations to conciliate economics with security due to the new challenges in power system supply. In solar and wind renewable energy, the only partially predictable is the output with very low controllability which creates unit commitment problems in thermal units. In this research paper, a different linear formulation via mixed integer is presented that only requires “binary variables” and restraints concerning earlier stated models. The framework of this model allows precisely the costs of time-dependent startup & intertemporal limitations, for example, minimum up & down times and a ramping limit. To solve the unit commitment problem efficiently, a commercially available linear programming of mixed-integer is applied for sizeable practical scale. The results of the simulation are shown in conclusions.

The new era of technology created a competitive environment in power system which needs perfect and useful tools to support the resources scheduling decisions. In a centralized power system, the unit commitment problem has been solved traditionally by determination of time when to start up or shut down the thermal generating units to fulfill the demands while dispatching online. Keeping in view of the spinning reserve necessities by satiating all generation constraints such as ramping limits, production and up & down minimum times, this operation is made over an explicit short-term time duration, to minimize inclusive process cost [

The independent system operator (ISO) solves the generating scheduling problems in present electricity market [

In the field of potential saving for several decades the combinatorial of mixed-integer on a large scale and the nonlinear programming problem is a vigorous topic for researchers. As a significance numerous elucidations methods have been offered such as dynamic programming [

The Dynamic programming is a way to solve problems of different variety of sizes, which can be easily changed to the model features of explicit utilities. The disadvantage of dynamic programming is that, it needs to limit the commitments measured at any time period and the suboptimal treatment of constraints (minimum up/down time) along with startup costs (time dependent). The heuristic algorithm is used when exact solutions are ineludibly computationally costly but imprecise solutions are adequate and produce solution independently. The hindrance of heuristic algorithm produces near optimal outcomes very rapidly. So there is always doubt of correctness. The Lagrangian Relaxation is utilized in manufacture UC programs more than Dynamic. The sub-optimality moves to zero when the number of units rises. One of its characteristic is that, it easily accommodates to the model characteristic of specific utilities. The key drawback of lagrangain relaxation is its intrinsic sub-optimality. The simulated annealing is the analogy among annealing procedure and optimization problem. Its strong feature is being autonomous to the preliminary solution and mathematical intricacy.

The unique advantages of mixed integer linear programming are as follows: 1) the parameter and structural can be achieved at the same time. 2) The discrete and logical constraints can be fingered plainly with binary numbers. 3) With mathematical tools a general systematic framework can be expressed for different problems. 4) It can deal with binary variables (linear) and separable from continuous variable (nonlinear). 5) Its efficient representation leads to superstructure which can deal with problems of any size in convenient time. 6) The efficient optimization can also exploit the structure of problems. In a finite number of steps, the MILP warranties the convergence to the optimal solution [

The MILP was applied in [

In the realistic scenario, the power system consists of several generators, and the model in [

The linear expression, which requires a single binary variable, was formulated in [

A linear formulation is presented in this paper for the thermal unit commitment with an alternative mixed-integer which is symbolized by MILP-UC which needs a binary variable with a single set of one per unit as per period. In contrast to different MILP previous methods [

The proposed model is also capable of solving the problems of scheduling which arises in markets for electricity furthermore, Market clearing processes (solved by ISO’s), self-scheduling problems (solved by generating companies) to originate bidding tactics which are used to analyze the market participant’s behavior by using market simulators. Hence this MILP-UC can also benefit the market agents.

This paper contributes to a new formulation, of MILP-UC which needs less binary variables and constraints to model the thermal unit commitment problem which can accurately reduce the burden of computational on existing MILP approaches. Furthermore, the scientific experience is presented by resolving a real application.

The proposed MILP-UC detailed description is provided in this section. Furthermore, it also includes the intertemporal and physical constraints as a precise model for power generators. While in section III, the numerical outcomes are discussed and presented. In IV Section, the appropriate and relevant conclusion is presented. Lastly, in the appendix, the statistics cast-off for the simulation is given.

The assumptions playing in the proposed model are out listed below in numerical order for the purpose of clarity.

1) A Pool-Based energy-only power wholesale marketplace is considered where a market agent Clears the market one time a day with day-ahead and hourly-horizon. The market operator seeks to minimalize the total generation cost considering the generation price generation limits submitted by the generation unit, as well as the demand and reserve existing in this market. The market clearing outcomes are hourly unit state, production, start-up and shunt down cost.

2) Reserve in the market is set to be 10% of overall system demand, which reflects the GB market. However, the reserve can be adjusted accordingly, which will be considered in future work.

In the model, the most critical task is to minimize and control the Total Operation Cost (TOC). The TOC is directly proportional to three main components the overall cost of production and the cost of startup and shutdown which can be formulated as [

min ∑ j , k ( c p j k + c d j k + c u j k ) (1)

V = { p j k , v j k , c u j t , c d j k } (2)

In scheduling Problems usually, the quadratic production cost function is used which can be written as (3). As presented in

c p j k = A j k + B j k ( p j k − P min j ) , ∀ j , k (3)

The stair wise start-up cost is proposed with the help of a mixed integer linear formulation. The stair wise function asymptotically approximates the discrete start-up cost which is a more accurate due number of intervals as increased. As the time span, the start-up cost is decentralized in hourly periods, so it can be written as a discrete function of start-up cost as shown in Equation (4).

A j k = a j v j k + b j P min j + c j P min j 2 , ∀ j , k (4)

In [

B j k = b j + 2 c j P min j , ∀ j , k (5)

With the span of time for every generating unit set of constraints are formulated as thermal constraints. The Power security constraints as shown in (6) provide spinning reserve margins, where the reserve in the market is set to be 10% of overall system demand, which has been assumed in Section III-A.

c u j t ≥ C U j t ( v j k − ∑ n = 1 t v j k − n ) , ∀ j , k , t ∈ { 1 , ⋯ , k − 1 } (6)

c d j k ≥ C D j ( v j k − 1 − v j k ) , ∀ j , k ∈ { 1 , ⋯ , T } (7)

Power Security: constraints (8) provide spinning reserve margins, where the reserve in the market is set to be 10% of overall system demand, which has been assumed in Section III-A.

∑ j ∈ J p j k ≥ D K + R K , ∀ k (8)

The (9) constraints confine the minimum power output of the generation and with an all-out existing power output of unit-j in k period, p ¯ j , It is a non-negative bounded variables of the above unit capacity (10).

P _ j v j k ≤ p j k ≤ p ¯ j v j k , ∀ j , k (9)

Note here that If in the period of k, the unit j is offline then v j k = 0 , so both p j k and p ¯ j will be equal to 0.

0 ≤ p ¯ j ≤ P ¯ j v j k , ∀ j , k (10)

In (11) the rates of ramp-up and start-up and in (13) the shutdown ramp rates both constrained the p ¯ j k while (12) represents the constraints in which ramp-down limits are enforced on the power output. In [

p ¯ j k ≤ p j k − 1 + R U j v j k − 1 + S U j ( v j k − v j k − 1 ) − P ¯ j ( 1 − v j k ) , ∀ j , k ∈ { 2 , ⋯ , T } (11)

p j k ≥ p j k − 1 − R D j v j k + S D j ( v j k − 1 − v j k ) + P ¯ j ( 1 − v j k − 1 ) , ∀ j , k ∈ { 2 , ⋯ , T } (12)

p ¯ j k ≤ P ¯ j v j k + 1 + S D j ( v j k − v j k + 1 ) , ∀ j , k ∈ { 1 , ⋯ , T − 1 } (13)

The minimum up and down constraints was initially expressed in [

∑ k = 1 G j [ 1 − v j k ] = 0 , ∀ j (14)

In (15) the G j is expressed mathematically. As well-defined by G j in (15) is subject to the preliminary position of the units which are related to constraints (14)

G j = min { T , ( U T j − U 0 j ) V 0 j } , ∀ j (15)

To satisfy the minimum time up constraint, the (16) constraints are used for the subsequent periods during all consecutive period of size U T j for all possible sets.

∑ n = k k + U T j − 1 v j n ≥ U T j ( v j k − v j k − 1 ) , ∀ j , k ∈ { G j + 1 , ⋯ , T − U T j + 1 } (16)

The Final U T j − 1 period in constraints (17) model where if unit j will be started up, and continue online till the end of spam time (minimal up and downtime)

∑ n = k T { v j k − ( v j k − v j k − 1 ) } ≥ 0 , ∀ j , k ∈ { G j + 1 , ⋯ , T − U T j + 1 } (17)

Similarly, the constraints for minimum downtime are formulated as follow (18)-(21)

∑ k = 1 L j v j k = 0 , ∀ j (18)

L j = min { T , ( D T j − S 0 j ) ( 1 − V 0 j ) } , ∀ j (19)

∑ n = k k + D T j − 1 ( 1 − v j n ) ≥ D T j ( v j k − 1 − v j k ) , ∀ j , k ∈ { G j + 1 , ⋯ , T − U T j + 1 } (20)

∑ n = k T { 1 − v j n − ( v j k − 1 − v j k ) } ≥ 0 , ∀ j , k ∈ { G j + 1 , ⋯ , T − U T j + 1 } (21)

In short, from the problem (1)-(21) establishes the proposed MILP-UC, that only needs binary variables of single types, namely, v j k , a contrast to the formulation of other MILP [

In test market, a case study is carried out with hourly day ahead resolution. The examined wholesale market reflects the properties of the GB power system and includes 7 electricity producers, with their cost features, and all-out yield limits are resulting from [

A case study with a real size is based on a ten-unit system of [

the load demand has also been multiplied by 10 accordingly. The time span is divided to meet 10% of the spinning reserve requirement each of the 24 hourly periods. The inventive ten-unit structure can be seen in the appendix [

The recommended model has been executed with CPLEX 9.0 [

Name | MILP-UC | MILP-2 | MILP-2R |
---|---|---|---|

Binary Variables (No’s) | 2100 | 6900 | 4500 |

Continuous Variables (No’s) | 18,901 | 18,901 | 21,301 |

Constraints (No’s) | 69,189 | 74,049 | 74,049 |

Non-zero Elements in constraints Matrix (No’s) | 388,161 | 411,741 | 411,741 |

Computing Times in seconds | 113 | 314 | 399 |

Total Operating Cost ($) | 564,889 | 5,611,828 | 5,606,577 |

Unit J | Period K | |||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | |

1 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 |

2 | 150 | 150 | 195 | 305 | 360 | 450 | 455 | 455 | 455 | 455 | 455 | 455 |

3 | 20 | 65 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 |

4 | 120 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 |

5 | 25 | 25 | 25 | 25 | 25 | 25 | 75 | 130 | 162 | 162 | 162 | 162 |

6 | 0 | 0 | 0 | 0 | 0 | 20 | 20 | 20 | 78 | 80 | 80 | 80 |

7 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 25 | 63 | 85 |

8 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 55 | 55 | 55 |

9 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 38 | 55 | 55 |

10 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 10 | 10 | 43 |

Sum | 770 | 825 | 935 | 1045 | 1100 | 1210 | 1265 | 1320 | 1430 | 1540 | 1595 | 1650 |

D | 700 | 750 | 850 | 950 | 1000 | 1100 | 1150 | 1200 | 1300 | 1400 | 1450 | 1500 |

R | 70 | 75 | 85 | 95 | 100 | 110 | 115 | 120 | 130 | 140 | 145 | 150 |

Unit J | Period K | |||||||||||

13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | |

1 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 | 455 |

2 | 455 | 455 | 455 | 415 | 360 | 455 | 455 | 455 | 455 | 455 | 275 | 165 |

3 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 |

4 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 | 130 |

5 | 162 | 162 | 130 | 25 | 25 | 40 | 100 | 162 | 162 | 25 | 0 | 0 |

6 | 80 | 78 | 20 | 0 | 0 | 0 | 20 | 80 | 68 | 20 | 0 | 0 |

7 | 25 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

8 | 55 | 10 | 0 | 0 | 0 | 0 | 10 | 55 | 10 | 0 | 0 | 0 |

9 | 55 | 10 | 0 | 0 | 0 | 0 | 10 | 55 | 10 | 0 | 0 | 0 |

10 | 10 | 0 | 0 | 0 | 0 | 0 | 10 | 18 | 10 | 0 | 0 | 0 |

Sum | 1540 | 1430 | 1320 | 1155 | 1100 | 1210 | 1320 | 1540 | 1430 | 1210 | 990 | 880 |

D | 1400 | 1300 | 1200 | 1050 | 1000 | 1100 | 1200 | 1400 | 1300 | 1100 | 900 | 800 |

R | 140 | 130 | 120 | 105 | 100 | 110 | 120 | 140 | 130 | 110 | 90 | 80 |

Number of units | MILP-UC | MILP-2 | MILP-2R |
---|---|---|---|

1 - 10 | 0.88 | 2.05 | 2.51 |

11 - 20 | 3.45 | 15.12 | 14.99 |

21 - 30 | 10.72 | 62.78 | 43.06 |

31 - 40 | 18.48 | 75.18 | 75.04 |

41 - 50 | 24.60 | 118.63 | 117.90 |

51 - 60 | 36.78 | 170.15 | 160.80 |

61 - 70 | 49.66 | 170.19 | 16.80 |

71 - 80 | 132.83 | 215.33 | 202.92 |

81 - 90 | 166.69 | 230.92 | 348.38 |

91 - 100 | 172.45 | 322.95 | 408.95 |

Ramping limit, 455 MW. However, with the operation of the system, the units start to change their states while meeting the constraints and system demand and spinning reserve requirements.

The Purple (D) line and dark green (R) line in

The production costs of 10 units in 24 hours are also presented in

In this paper, the linear formulation of thermal unit commitment problem with uncertainties through a computationally efficient mixed integer is presented. The significant features of the proposed method are the single type binary numbers requirement, which precisely model the startup costs (time dependent), along with the intertemporal constraints and, especially individual contributions to spinning reserve. The problem of reduction in computation time which is normally a huge issue in unit commitment is decreased by the accessible commercial software. The recommended model has been effectively tested on a genuine situation study for wind and PV uncertainties. The numerical results show the efficient and correct computational performance. Finally, it is also applicable to the new scheduling problem arising in electricity market although the formulation has been used in traditional centralized power system for the unit commitment problem.

The authors would like to thank the financial support provided by the National Science Foundation under Grant 61273142, Foundation for Six Talents by Jiangsu Province (2012-DZXX-045), Priority Academic Program Development of Jiangsu Higher Education Institutions (PADP), and Technology research project of Ministry of public security of China (grant number 2015JSYJC029).

Ahsan, M.K., Pan, T.H. and Li, Z.M. (2018) The Linear Formulation of Thermal Unit Commitment Problem with Uncertainties through a Computational Mixed Integer. Journal of Power and Energy Engineering, 6, 1-15. https://doi.org/10.4236/jpee.2018.66001

In

Unit J | p ¯ j (MW) | P _ j (MW) | UT (h) | DT (h) | V_{0} (h) |
---|---|---|---|---|---|

1 | 455 | 150 | 8 | 8 | 8 |

2 | 455 | 150 | 8 | 8 | 8 |

3 | 130 | 20 | 5 | 5 | -5 |

4 | 130 | 20 | 5 | 5 | -5 |

5 | 162 | 25 | 6 | 6 | -6 |

6 | 80 | 20 | 3 | 3 | -3 |

7 | 55 | 25 | 3 | 3 | -3 |

8 | 55 | 10 | 1 | 1 | -1 |

9 | 55 | 10 | 1 | 1 | -1 |

10 | 55 | 10 | 1 | 1 | -1 |

Unit J | a (£/h) | b (£/MWh) | C (£/MW2h) |
---|---|---|---|

1 | 1000 | 16.19 | 0.00048 |

2 | 970 | 17.26 | 0.00031 |

3 | 700 | 16.60 | 0.00200 |

4 | 680 | 16.65 | 0.00211 |

5 | 450 | 19.70 | 0.00398 |

6 | 370 | 22.26 | 0.00712 |

7 | 480 | 27.74 | 0.00079 |

8 | 660 | 25.92 | 0.00413 |

9 | 665 | 27.27 | 0.00222 |

10 | 670 | 27.79 | 0.00173 |

j ∈ J Set of generation units

k ∈ K Set of periods

ParametersP _ j The Minimum generation limit of generation unit-j (MW)

P ¯ j The Maximum generation limit of generation unit-j (MW)

U T j The Minimum start-up time of generation unit-j (s)

D T j The Minimum shunt down time of generation unit-j (s)

U 0 j The time of Initial start-up to generation unit j (s)

S 0 j The time of Initial shunt-down to generation unit-j (s)

V 0 j Initial commitment state of generation unit-j (online: 1, offline: 0)

a j The coefficient of Fixed quadratic production cost function to generation unit j (£/h)

b j The Linear coefficient for the quadratic production cost function of generation unit-j (£/MWh)

c j Quadratic coefficient for the quadratic production cost function of generation unit-j (£/MWh^2)

A j k Cost of generation unit j at k (£) period

B j k The Cost of generation unit-j at k (£) period

C U j t The cost of Maximum start-up for generation unit-j at period k (£/MWh)

C D j The cost of Maximum shut down for generation unit-j at period k (£/MWh)

S U j Start-up ramping of generation unit-j (MW)

S D j Shunt down ramping of generation unit -j (MW)

D K Demand at period K (MW)

R K Reverse at period K (MW)

Variablesv j k On-Off state of generation unit-j at period t [0/1]

p j k Generation output of generation unit-j at period t (MW)

c u j t Start-up cost of generation unit-j at period k (£/MWh)

c d j k The cost of Shunt down for generation unit-j at the k period (£/MWh)