Alternative Formulations for Unit Commitment Xitan Qiang, Boran Zhou, Quanyuan Jiang

School of Electrical Engineering Zhejiang University

Hangzhou, China qiangxitan@yahoo.cn

Abstract—The short-term unit commitment (UC) problem in

thermal power generation is a large-scale, mixed-integer nonli-near programming problem, which is difficult to be solved effi-ciently. It is possible to approximate the non-linear objective by means of piece-wise linear functions, so that UC can be approx-imated by a mixed-integer linear programming (MILP). This pa-per presents tighter constraints for the formulation to solve the thermal unit commitment (UC) problems, which is more compu-tationally efficient. Results based on realistic studies support this conclusion.

Index Terms—Unit commitment, mixed-integer linear pro-gramming (MLIP), optimization

NOTATION Constants

jA Coefficient of the piecewise linear production cost function of unit j .

ja , jb , jc Coefficients of the quadratic pro-duction cost function of unit j .

, , coldj j jcc hc t Coefficients of the startup cost

function of unit j .

jC Shutdown cost of unit j .

( )D k Load demand in period k .

jDT Minimum down time of unit j .

tjK Cost of the interval t of the stairwise

startup cost function.

jP Maximum power output of unit j .

jP Minimum power output of unit j .

( )R k Spinning reserve requirement in pe-riod k .

jRD Ramp-down limit of unit j .

jRU Ramp-up limit of unit j .

jSD Shutdown ramp limit of unit j .

jSU Startup ramp limit of unit j .

T Number of periods of the time span.

jUT Minimum up time of unit j .

(0)jV Initial commitment state of unit j = 1 if it is online, 0 otherwise).

Variables( )d

jc k Shutdown cost of unit j in period k .

( )pjc k Production cost of unit j in period

k .

( )ujc k Startup cost of unit j in period k .

( )jp k Power output of unit j in period k .

( )jp k Maximum available power output of unit j in period k .

( )jv k Binary variable that is equal to 1 if unit j is online in period k and 0 otherwise.

SetsJ Set of indices of the generating

units. K Set of indices of the time periods.

I. INTRODUCTION Unit commitment (UC) in electric power systems is to mi-

nimize system-wide operational costs of power generators by optimizing generating resources to supply system load while satisfying prevailing constraints, such as minimum on/off time, ramping up/down, minimum/maximum generating ca-pacity, and fuel and emission limit.

The UC problem is used to be modeled as a large-scale non-convex problem. For several decades, this large-scale, mixed-integer, combinatorial and nonlinear programming problem has been an active research topic because of potential savings in operation costs. As a consequence, several solution techniques have been proposed such as heuristics [1]-[3], dy-namic programming [4]-[6], mixed-integer linear program-ming (MILP) [7],[8], Lagrangian relaxation [9]-[15], and evo-lution-inspired approaches [16]-[17]. Among these approach-es, mixed-integer linear programming has become attractive owing to the availability of commercially MILP algorithm and modern software, such as Cplex [18].

MILP UC formulation has been described and extended in [4]-[7]. In [5], 3-binary variable format formulations with ri-gorous constraints was provided while in [21], more compact formulations were presented by reducing two sets of binary variables. In addition, computational results were provided in [21] to prove that formulations with one set of binary variable performed better than those of 3 sets of binary variables in terms of total cost and time needed.

This work was supported by National Natural Science Foundation of China (50977082), and National High Technology Research and De-velopment Program of China (2011AA05A118).

978-1-4577-1600-3/12/$26.00 © 2012 IEEE

For realistic large-scale power systems, the formulations reported might be computationally intensive for state-of-the-art implementations of branch-and-cut algorithms [20] and current computing capabilities, which require large sums of time to get the best optimization result below a certain gap to-lerance. Therefore, the objective of this paper is to present al-ternative ramp constraints based on MILP formulations with one set of binary variable to reduce the overall time needed in solving unit commitment problems, in short, to obtain a faster convergence rate. The new constraints would make the com-putation more efficient and requires less time in obtaining the expected cost.

The remaining sections are outlined as follows. Section Ⅱ provides a detailed description of the alternative MILP unit commitment formulation. In section Ⅲ, we provide computa-tional results regarding the efficiency of the past formulation and the alternative formulation we proposed. Section Ⅳ dis-cuss one practical parameter which is needed for more precise result. Conclusions will be given in section Ⅴ.

II. IMPROVED 1-BINARY VARIABLE FORMULATION The unit commitment problem can be formulated as [19]

minimize ( ) ( ) ( )p u d

j j jk K j J

c k c k c k∈ ∈

+ +∑∑ (1)

subject to ( ) ( ),j

j Jp k D k k K

∈

= ∀ ∈∑ , (2)

( ) ( ) ( ),jj J

p k D k R k k K∈

≥ + ∀ ∈∑ , (3)

( ) ( ), ,j jp k k j J k K∈ ∀ ∈ ∀ ∈∏ , (4)

where ( )j

k∏ represents the region of feasible production of

generating unit j in time period k [21]. The goal of the unit commitment problem is to minimize

the total operation cost, which is defined as the sum of the production cost, the startup cost, and the shutdown cost (1). The production cost is typically expressed as a quadratic func-tion of the power output, while the startup cost is usually modeled as an exponential function of the offline time prior to the startup [19]. Constraints (3) provide spinning reserve mar-gins. The operational constraints defining ( )

jk∏ contain

generation limits, ramp rate limits, and minimum up and down times [21]. The binary variables are used to model on/off deci-sions for each generator in every time period including the startup status and shutdown status. Problem described in (1)-(4) is mixed-integer and nonlinear optimization problem that is difficult to solve.

In [21], a piecewise linear function is described and we use the same method to approximate the production cost.

A. Objective Function 1) Production Cost: The quadratic production cost func-

tion can be formulated as 2( ) ( ) ( ) ( ), ,p

j j j j j j jc k a v k b p k c p k j J k K= + + ∀ ∈ ∀ ∈ (5)

Detailed piecewise linear process can be found in [21] 2) Startup Cost: A mix-integer linear formulation for the

stairwise startup cost was proposed in [22].

1( ) ( ) ( ) ,

, , 1

tu tj j j j

n

j

c k K v k v k n

j J k K t ND=

⎡ ⎤≥ − −⎢ ⎥⎣ ⎦

∀ ∈ ∀ ∈ ∀ =

∑ (6)

( ) 0, ,ujc k j J k K≥ ∀ ∈ ∀ ∈ (7)

in which the coefficients , 1 ,

, 1 .

coldj j jt

j coldj j j j

hc t t DT j JK

cc t t DT ND

⎧ = + ∀ ∈⎪= ⎨= + +⎪⎩

(8)

3)Shutdown Cost:

( ) [ ( 1) ( ),

,

dj j j jc k C v k v k

j J k K

≥ − −

∀ ∈ ∀ ∈ (9)

( ) 0, , .djc k j J k K≥ ∀ ∈ ∀ ∈ (10)

B. Thermal Constraints 1)Generation Limits: If a generator is turned on, the power

output from that generator must be within certain operational limits:

( ) ( ) ( ), , ,j j j jP v k p k p k j J k K≤ ≤ ∀ ∈ ∀ ∈ (11)

0 ( ) ( ), ,j jjp k P v k j J k K≤ ≤ ∀ ∈ ∀ ∈ (12) Note that if unit j is offline in period k , both ( )jp k and

( )jp k are equal to 0. 2)Minimum Up and Down Time Constraints:

1

[1 ( )] 0,jG

jk

v k j J=

− = ∀ ∈∑ (13)

1

( ) [ ( ) ( 1)],

, ( 1) ( 1)

jk UT

j j j jn k

j j

v n UT v k v k

j J k G T UT

+ −

=

≥ − −

∀ ∈ ∀ = + − +

∑ (14)

{ }( ) [ ( ) ( 1)] 0,

, ( 2)

T

j j jn k

j

v n v k v k

j J k T UT T=

− − − ≥

∀ ∈ ∀ = − +

∑ (15)

where jG is the number of initial periods during which unit j must be online. jG is mathematically expressed as

{ }0, (0) .j j j jG Min T UT U V⎡ ⎤= −⎣ ⎦ Constraints (13) are related to the initial status of the units

as defined by jG . Constraints (14) are used for the subsequent periods to satisfy the minimum up time constraint during all the possible sets of consecutive periods of size jUT . Con-straints (15) model the final 1jUT − periods in which if unit j is started up, it remains online until the end of the time span [21].

Analogously, minimum down time constraints are formu-lated as follows:

1

( ) 0,jL

jk

v k j J=

= ∀ ∈∑ (16)

1

[1 ( )] [ ( 1) ( )],

, ( 1) ( 1)

jk DT

j j j jn k

j j

v n DT v k v k

j J k L T DT

+ −

=

− ≥ − −

∀ ∈ ∀ = + − +

∑ (17)

{ }1 ( ) [ ( 1) ( )] 0,

, ( 2)

T

j j jn k

j

v n v k v k

j J k T DT T=

− − − − ≥

∀ ∈ ∀ = − +

∑ (18)

where jL is the number of initial periods during which unit j must be offline. In [21], jL is mathematically expressed as

{ }, (0) 1 (0)j j j jL Min T DT S V⎡ ⎤ ⎡ ⎤= − −⎣ ⎦ ⎣ ⎦ . 3) Ramping Constraints: The original ramp constraints were described in [21] as

follows: ( ) ( 1) ( 1)

( ) ( 1) 1 ( )j j jj

jj j j j

p k p k RU v k

SU v k v k P v k

≤ − + − +

⎡ ⎤ ⎡ ⎤− − + −⎣ ⎦ ⎣ ⎦ (19)

( ) ( 1)

( ) ( 1) ,

, 1 1.

j jj

j j j

p k P v k

SD v k v k

j J k T

≤ + +

⎡ ⎤− +⎣ ⎦∀ ∈ ∀ ∈ −

(20)

( 1) ( ) ( )

( 1) ( ) 1 ( 1)j j j j

jj j j j

p k p k RD v k

SD v k v k P v k

− − ≤ +

⎡ ⎤ ⎡ ⎤− − + − −⎣ ⎦ ⎣ ⎦ (21)

The above formulation is not tight enough when consider-ing generator status in the following cases.

In (19), for each generator and every time period, this in-equality is satisfied under different conditions of status change. We classify the status change into four categories:

a. from ( 1) 1jv k − = to ( ) 1jv k = . b. from ( 1) 0jv k − = to ( ) 1jv k = . c. from ( 1) 0jv k − = to ( ) 0jv k = . d. from ( 1) 1jv k − = to ( ) 0jv k = . Next we analyze these conditions separately. a. At time period k and 1k − (suppose 1 0k − > ), the ge-

nerator j was on, where ( 1) 1jv k − = and ( ) 1jv k = . Con-strain (19) transformed into the following inequality:

( ) ( 1) , ,j jjp k p k RU j J k K≤ − + ∀ ∈ ∀ ∈ (22) which means the maximum available power output of unit j in period k is no more than the sum of power output

( 1)jp k − of last period and ramp-up rate from period 1k − to period k . This is mainly how this constraint makes sense.

b. At time period 1k − , the generator was off whereas at k , the generator was on, where ( 1) 0jv k − = and ( ) 1jv k = . Constraint (19) transformed into the following inequality:

( ) , ,jjp k SU j J k K≤ ∀ ∈ ∀ ∈ (23)

which means the maximum available power output of unit j in period k is no more than the startup ramp rate from 1k − to k . Note that when ( 1) 0jv k − = , ( 1) 0jp k − = .

c. At time period 1k − , the generator was off whereas at k , the generator was off as well, where ( 1) 0jv k − = and

( ) 0jv k = . Constraint (19) transformed into the following in-equality:

0 , ,jP j J k K≤ ∀ ∈ ∀ ∈ . (24) which is obviously satisfied.

d. At time period 1k − , the generator was on whereas at k , the generator was off, where ( 1) 1jv k − = and ( ) 0jv k = . Constraint (19) transformed into the following inequality:

0 ( 1) , ,jj j jp k RU SU P j J k K≤ − + − + ∀ ∈ ∀ ∈ (25) Practically and physically, owing to the negative sign in

front of jSU , this inequality does not reflect any physical process. Nevertheless, let us consider whether the inequality is satisfied and how well it is satisfied. Firstly, is the sum of the power output of last period, ramp up rate and the capacity of the generator bigger than startup ramp rate? For ordinary ge-nerators, this inequality is well satisfied. However, extreme cases are that the startup ramp rate is too big to satisfy this in-equality. In addition, the inequality is too loose for those gene-rators comprising of large ramp up rate and capacity but rela-tively small startup rate. In these cases, however, the inequali-ty is not “just” satisfied, but “easily” satisfied, which generates a bigger but useless feasible region of variable ( 1)jp k − , leading to a negative computing efficiency when using the branch-and-cut algorithm in commercially available software, such as Cplex.

In fact, the loose inequality leaves Cplex more choices when using the branch-and-cut algorithm to optimize the best solution. There would be more branches on the objective func-tion tree thus requiring more time to choose and cut. When UC problems are of large size, complex dimensions, numerous variables and even more constraints, cutting these branches which are brought in by loose constraints would largely de-crease the efficiency. To make matters worse, computing me-thod with low efficiency is practically unavailable, in that the UC problems usually cannot make a long-time plan due to the difficulty in predicting the demand of daily use. That is, opti-mizing tomorrow’s startup and shutdown plan is most likely precise. Therefore, a computationally efficient formulation is highly needed and the following improved formulation we presented could solve the above-mentioned problems:

( ) ( 1) ( 1) ( ) ( 1)

1 ( ) , ,

j j j j j jj

jj j j

p k p k RU v k SU v k v k

SU RU P v k j J k K

⎡ ⎤≤ − + − + − −⎣ ⎦⎡ ⎤ ⎡ ⎤+ − − − ∀ ∈ ∀ ∈⎣ ⎦ ⎣ ⎦

(26)

Now let’s reconsider conditions d. At time period 1k − , the generator was on whereas at k , the generator was off, where ( 1) 1jv k − = and ( ) 0jv k = . Constraint (19) trans-formed into the following inequality:

0 ( 1) , ,jjp k P j J k K≤ − − ∀ ∈ ∀ ∈ (27) This inequality does not contain ramp up rate, startup ramp

rate or maximum capacity. Firstly, any generator of extreme parameters we have mentioned above would be satisfied in

this inequality. Furthermore, for ordinary generators, it de-creases the feasible region of variables ( )jp k , giving Cplex less choices when optimizing the best solution thus requiring less time and improving the efficiency. In fact, inequality (27) is a redundant and repetitive constraint since it has been ite-rated in (11). Note that we have restricted our discussion under the conditions of ( 1) 1jv k − = and ( ) 0jv k = . Thus, constraint (27) is in congruence with (11).

In (21), ramp-down constraint has the same problem as well. For the reason of simplicity, we only analyze the condi-tions of ( 1) 0jv k − = and ( ) 1jv k = . Under these conditions, the inequality of (21) transformed into the following:

0 ( ) , ,jj j jp k RD SD P j J k K≤ + − + ∀ ∈ ∀ ∈ (28) Again, it would not apply to generators of extreme para-

meters and it offers a big but unnecessary feasible region of variables ( )jp k , making it computationally inefficient. We present an alternative formulation:

( 1) ( ) ( ) ( 1) ( )

1 ( 1) , ,

j j j j j j j

jj j j

p k p k RD v k SD v k v k

SD RD P v k j J k K

⎡ ⎤− − ≤ + − − +⎣ ⎦⎡ ⎤ ⎡ ⎤− − − − ∀ ∈ ∀ ∈⎣ ⎦ ⎣ ⎦

(29)

This formulation would transform into the following in-equality when ( 1) 0jv k − = and ( ) 1jv k = :

( ), ,j jP p k j J k K≤ ∀ ∈ ∀ ∈ (30) which is a tighter constraint and in congruence with (11).

Next, several realistic cases are studied and computing re-sults are presented to back our analysis.

III. NEMERICAL RESULTS We have studied the realistic cases provided in [21], in

which the time span is divided into 24 hourly periods. Howev-er, for the sake of generosity and solidarity, we have studied generating units from 10 to 100, with 10 units’ interval. We have also linearized the quadratic production costs through a piecewise linear approximation with four segments.

Detailed system data could be found in Appendix. Howev-er, in [21], no ramping data is provided. Therefore, we use da-ta provided by [21] adapted to include ramping data in Appen-dix Table A. We compare the time requested to solve this UC problem for models provided by [21] and our paper, using the same set of data, with only models differing.

Two models have been implemented on a Lenovo Y460 with Intel i3 processor at 2.26GHz and 2GB of RAM memory using Cplex 12.1 to solve MILP unit commitment problem de-scribed above.

Note that the time required to solve a problem is the most important measure of the quality of the formulation. However, looking only at time required to solve problems does not pro-vide an indication of why one formulation is better than another. In addition to time, we also use the integrality gap to set a same criterion for the two formulations. The integrality gap is defined to be ( ) /MILP LP MILPgap z z z= − , where LPz is the optimal value of the LP relaxation and MILPz the optimal value of the MILP. TABLE Ⅰ compares the time required in both formulations using 0.5% integrality gap as a stopping cri-terion.

In TABLE Ⅰ, O-time (original) equals the time required of original formulation while A-time (alternative) equals the alternative formulation we proposed. As the result suggests, the time required for the formulation we proposed is either faster or similar to the original formulation with one exception, 40 units.

TABLE Ⅰ Results with 0.5% integrality gap

Num Units O-time O-node A-time A-node1 10 9.37 1600 9.69 18002 20 25.44 1599 13.37 5503 30 17.61 600 16.21 6004 40 27.32 562 49.58 20005 50 88.92 2338 40.28 5006 60 45.87 600 30.12 5057 70 69.72 554 69.18 5368 80 173.9 4023 168.2 34009 90 260.5 3804 32.65 53210 100 123.3 551 21.21 517

TABLE Ⅱ Results with 0.4% integrality gap Num Units O-time O-node A-time A-node

1 10 13.01 2400 12.2 31002 20 50.46 4552 40.6 3900 3 30 39.33 3334 31.58 13004 40 65.35 2376 53.6 22135 50 110.9 3057 41.46 5956 60 105.4 2782 61.23 5117 70 73.66 566 70.72 5548 80 200.5 4472 169.2 3500 9 90 281.9 3804 100.4 53410 100 148.6 580 24.71 519 Next, as has been shown in TABLE Ⅱ, when using 0.4%

integrality gap as a stopping criterion for both formulations, the time required for the formulation we proposed is either faster or similar to the original formulation with no exception. That being said, the formulation we proposed has shown a faster convergence rate in most cases. Nevertheless, for those cases that required similar time in both formulations also con-form to our theory. In fact, in cases like 30 or 70 units, the feasible region of variables ( )jp k in ramping constraints is limited first by other constraints, leaving an already tight re-gion for ramping constraints, thus making the ramping con-straints less effective. Therefore, it makes no big difference for our improved ramping constraints and thus requiring similar time for both formulations. This kind of situation could take place only when the original generator data itself is loose, leading to the less effectiveness of many constraints. When we

change the original data, specifically, ramp up and ramp down rate to make it tight, cases like 30 or 70 units requires less time to solve the UC problem using our formulation than the original formulation, which backed our analysis.

Note that although time required for the alternative formu-lation is sometimes a little faster than the original ones, it is because the model and analysis are both restricted to a one-bus system for the sake of simplicity. For instance, network con-straints and losses as well as other security constraints were not included into our analysis. Due to the mechanism of branch-and-cut algorithm in Cplex for MILP unit commitment problems, the formulation we proposed would have a signifi-cant reduction in time requirement as the complexity of the whole model increases.

We have also given the performance of the two formula-tions in terms of how objective function value changes with time going by. We have analyzed the total costs of the two formulations with 100 units by allowing the branch-and-cut algorithm to run for a maximum of 900s. Figure 1 shows the evolution with computing time of the best solution found by each MILP formulation.

5655000

5660000

5665000

5670000

5675000

5680000

5685000

5690000

0 100 200 300 400 500 600

t (s)

cost

($)

AlternativeOriginal

Figure 1. Evolution of the best solution of two formulations Obviously, the alternative formulation we proposed has a

faster convergence rate than the original one, especially in the first 20 seconds, which on the other hand proves the superiori-ty of the alternative formulation.

Furthermore, the evolution with computing time of inte-grality gap of 100 units is shown in figure 2.

0.00%0.10%

0.20%0.30%

0.40%0.50%

0.60%0.70%

0.80%0.90%

1.00%

0 100 200 300 400 500 600

t (s)

gap

AlternativeOriginal

Figure 2. Evolution of integrality gap of the two formulations

Recall that ( ) /MILP LP MILPgap z z z= − . The integrality gap is a major factor in the time expected to solve the MILP prob-lem. Gap tolerance is the compromise and balance between the best solution and a suitable time required to solve the problem. Therefore, given the same period of time, the smaller the gap is, the better the objective function. The rate of the gap in the first several seconds is also a significant criterion in determin-ing how well a formulation performs. Again, this important criterion has also backed our analysis.

IV. DISCUSSION The time required to compute for the best solution in this

paper is reported by Cplex. However, Cplex starts to count time only after all the constraints have been added to the ob-jective function, follow by the function cplex.solve(). That be-ing said, to the best of our knowledge, the paper we have used as our references as well as this paper only provide data of the optimizing time, not including the time spent for importing da-ta, establishing objective function and adding constraints, in short, running the program before optimizing. However, for larger and more complex cases, running the program before optimizing would also consume large sums of time. From the perspective of practical use, this time consumption should also be included into the total required time in assessing the effi-ciency so that the performance of program as a whole could be reflected.

As far as we know, two types of functions could be im-plemented on MILP unit commitment problems, namely, mat-lab class function and matlab toolbox function. We used class function to translate mathematical formulations into pro-gramming language and so far we found it efficient. More complete computing results containing total running time are expected to be given in future studies and the program itself could also be optimized by choosing suitable algorithm and techniques, such as sparse matrix.

V. CONCLUSION In this paper, we presented an alternative formulation

based on single type of binary variables to solve MILP unit commitment problems. The improved constraints in ramping limits decrease the possible but unavailable region of va-riables, making the contemporary commercial solver more ef-ficient to implement the branch-and-cut algorithm. As a result, the time required to solve the same size problem has been re-duced and the convergence rate has increased.

APPENDIX

TABLE A SYSTEM DATA Ⅰ

Units

P (MW)

P (MW)

UT DT (h)

SU SD

(MWh)

RU RD

(MWh)

inistate (h)

1 455 150 8 150 225 8 2 455 150 8 150 225 8 3 130 20 5 20 50 -5 4 130 20 5 20 50 -5 5 162 25 6 25 60 -6 6 80 20 3 20 60 -3 7 85 25 3 25 60 -3 8 55 10 1 20 135 -1 9 55 10 1 20 135 -1

10 55 10 1 20 135 -1

TABLE B SYSTEM DATA Ⅱ Units

a

($/h) b

($/MWh) c

($/M2h) hc

($/h) cc

($/h) tcold (h)

1 1000 16.19 0.00048 4500 9000 5 2 970 17.26 0.00031 5000 10000 5 3 700 16.6 0.002 550 1100 4 4 680 16.5 0.00211 560 1120 4 5 450 19.7 0.00398 900 1800 4 6 370 22.26 0.00712 170 340 2 7 480 27.74 0.00079 260 520 2 8 660 25.92 0.00413 30 60 0 9 665 27.27 0.00222 30 60 0 10 670 27.79 0.00173 30 60 0

ACKNOWLEDGMENT The authors would like to thank Professor Quanyuan Jiang

for the assistance to complete this paper. This work was sup-ported by National Natural Science Foundation of China (50977082), and National High Technology Research and De-velopment Program of China (2011AA05A118).

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