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Accelerating the Benders decomposition for network-constrained unit commitment problems

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Abstract

This paper presents an optimization method by generating multiple strong Benders cuts for accelerating the convergence of Benders Decomposition (BD) when solving the network-constrained generation unit commitment (NCUC) problem. In NCUC, dc transmission network evaluation subproblems are highly degenerate, which would lead to many dual optimal solutions. Furthermore, the classical BD cuts are often low-density which involve only a limited number of decision variables in the master problem. Therefore, the dual optimal solutions and the corresponding Benders cuts are of crucial importance for improving the efficiency of the BD algorithm. The proposed method would generate multiple strong Benders cuts, which are pareto optimal, among candidates from multiple dual optimal solutions. Such cuts would be high-density in comparison with low-density cuts produced by the classical BD. The proposed multiple strong Benders cuts are efficient in terms of reducing the total iteration number and the overall computing time. The high-density cuts may restrict the feasible region of the master unit commitment (UC) problem in each iteration as they incorporate more decision variables in each Benders cut. The multiple strong Benders cuts would accordingly reduce the iteration number and overall computing time. Numerical tests demonstrate the efficiency of the proposed multiple strong Benders cuts method in comparison with the classical BD algorithm and the linear sensitivity factors (LSF) method. The proposed method can be extended to other applications of BD for solving the large-scale optimization problems in power systems operation, maintenance, and planning.

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Abbreviations

a,b,m,n:

Denote a bus

d,d′:

Denote a BD iteration

i,j:

Denote a unit

k :

Denote a segment of a cost curve function

l,l′:

Denote a transmission line

o,o′:

Denote a subset

p :

Denote a phase shifter

t,τ:

Hourly index

B,B o ,Bo′:

Set/subsets of buses

f(b),t(b):

Set of transmission lines starting from bus b/ending at bus b

PL t :

Vector of power flow variables at hour t

PL max :

Vector of power flow upper limit

U(b):

Set of generators located at bus b

Z,Zc,SZ o ,SZo′:

Sets of buses

γ(m):

Denote a set of buses, for each bus n in the set, there is a phase shifter at line m to n, and m is the tap side and n is the non-tap side

γ t :

Vector for phase shifter at hour t

γmin,γmax:

Vector of phase shifter lower/upper limits

DT i ,UT i :

Number of hours unit i must be initially offline/online due to its minimum off/on time limits

G :

Objective value of the global optimal solution to the original problem

I it ,P it :

Unit commitment decision/generation dispatch of unit i at hour t

LB d ,UB d :

Lower/upper bound of the original problem objective value obtained at BD iteration d

LB best ,UB best :

Best obtained lower/upper bounds for the objective value to the original problem

offset :

Offset term in the linear expression of the power loss

OR it ,SR it :

Non-spinning/spinning reserve provided by unit i at hour t

P ikt :

Generation dispatch of unit i at hour t at segment k

P Losst :

System loss at hour t

PL lt :

Power flow of transmission line l at hour t

q,q1,q2,Q:

Dual variables for the dual problem

s :

Slack variable

SD it ,SU it :

Shutdown/startup cost of unit i at hour t

θ at ,θ bt :

Phase angle of bus a and bus b at hour t

θ ref :

Phase angle of the reference bus

γ abt :

Phase shifter value on the line from bus a to bus b at hour t

λ,κ,π:

Dual variables for the primal problem

|Z|:

Number of buses that belong to set Z

\(\hat{\cdot},\tilde{\cdot}\) :

Value for variables calculated at previous BD iterations

c :

A constant integer number determined during the algorithm

C b :

Loss distribution factor corresponding to bus b

c ik ,N i :

Incremental cost for segment k and no-load cost of unit i

D bt :

System load at bus b hour t

DP i ,UP i :

Shutdown/startup ramp limits of unit i

DR i ,UR i :

Ramping down/ramping up limits of unit i

LF b :

Loss factor related to bus b

\(\mathit{LSF}_{ab}^{m}\) :

The sensitivity of the power flow on line l (from bus a to bus b) to power injection at bus m

M,ε:

Very large/small positive number

MSR i :

Spinning reserve that can be provided by unit i in one minute

NB :

Number of buses

NT :

Number of hours under study

P Dt ,P Lt :

System load/loss at hour t

\(P_{i}^{\mathrm{min}},P_{i}^{\mathrm{max}}\) :

Minimum/maximum capacity of unit i

\(P_{ik}^{\mathrm{max}}\) :

Maximum capacity of unit i of segment k

QSC i :

Quick start capacity of unit i

R Ot ,R St :

System non-spinning/spinning reserve requirements at hour t

\(T_{i}^{\mathit{on}},T_{i}^{\mathit{off}}\) :

Minimum on/off time limits of unit i

x ab :

Line reactance between bus a and bus b

\(X_{i0}^{\mathit{on}},X_{i0}^{\mathit{off}}\) :

On/off time counter of unit i at the initial status

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Authors and Affiliations

  1. Electrical and Computer Engineering Department, Illinois Institute of Technology, Chicago, IL, 60616, USA

    Lei Wu & Mohammad Shahidehpour

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  1. Lei Wu
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  2. Mohammad Shahidehpour
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Correspondence to Mohammad Shahidehpour.

Additional information

This work is supported in part by the National Science Foundation grants ECCS-0725666 and ECCS-0801853.

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Wu, L., Shahidehpour, M. Accelerating the Benders decomposition for network-constrained unit commitment problems. Energy Syst 1, 339–376 (2010). https://doi.org/10.1007/s12667-010-0015-4

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