two-stage robust power grid optimization problem · 2011-01-03 · two-stage robust power grid...

Two-Stage Robust Power Grid Optimization Problem

Ruiwei Jiang†, Muhong Zhang‡, Guang Li§, and Yongpei Guan†

†Department of Industrial and Systems Engineering

University of Florida, Gainesville, FL 32611, USA‡School of Computing, Informatics, and Decision Systems Engineering

Arizona State University, Tempe, AZ 85281, USA§Electric Reliability Council of Texas (ERCOT), Austin, TX 78744, USA

Abstract

For both regulated and deregulated electric power markets, due to the integration of renew-

able energy generation and uncertain demands, both supply and demand sides of an electric

power grid are volatile and under uncertainty. Accordingly, a large amount of spinning re-

serve is required to maintain the reliability of the power grid in traditional approaches. In this

paper, we propose a novel two-stage robust integer programming model to address the power

grid optimization problem under supply and demand uncertainty. In our approach, uncertain

problem parameters are assumed to be within a given cardinality or polyhedral uncertainty set.

We study cases with and without transmission capacity and ramp-rate limits. We also analyze

solution schemes to solve each problem that include an exact solution approach, and an efficient

heuristic approach that provides a tight lower bound for the general robust power grid opti-

mization problem. The final computational experiments on a modified IEEE 118-bus system

verify the effectiveness of our approaches, as compared to the worst-case scenario generated by

the nominal model without considering the uncertainty.

Key words: unit commitment, transmission capacity limits, mixed integer programming, sepa-

ration, robust optimization

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1 Introduction

Current US electric power markets include both regulated and deregulated markets (e.g., see [16]).

For the regulated market (e.g., the state of Florida and the state of Arizona), utility companies own

power plants and clients. A utility company solves the traditional unit commitment problem plus

transmission capacity limits for a given planning horizon (e.g., one week) based on the forecasted

demand, with the objective of maximizing the social welfare or minimizing the total power gener-

ation cost to satisfy the given demand. In real time at each operating hour, the utility company

adjusts the economic dispatch amount of each online power generation unit, in order to satisfy

the real time demand. The utility company may need to turn on pickers (e.g., gas generators) or

purchase through bilateral contracts to cover the extra demand when the demand is significantly

larger than forecasted, and salvage its extra power if the demand is smaller than the online gener-

ators’ minimum output. The deregulated market contains day ahead and real time markets. For

the day ahead market, three part offers that specify start up cost, minimum energy, and an energy

offer curve are submitted to the Independent System Operator (ISO). An ISO optimizes the power

grid operations based on the submitted offers to decide the unit commitment status (e.g., online

or off-line) of each generator. In the real time market, an ISO determines the most economical

dispatch of individual resources across the grid, based on the real time demand information. Sev-

eral electric power markets (e.g., ERCOT) execute reliability unit commitment studies after the

close of the day ahead market, and evaluates the generation needs for the next operating day (e.g.,

see [27]). Reliability unit commitment ensures that there is sufficient generation capacity in the

proper locations to reliably serve the forecasted demand and forecasted transmission congestion by

leveraging offline resources when necessary.

In practice, the demands for a power grid are highly uncertain (e.g., see [16]) due to weather

and other conditions. Meanwhile, high penetration of renewable generation, such as wind, triggers

volatile supply (e.g., see [3]). All these factors bring great challenges in grid management and

generation scheduling to power system operators. The inherent intermittency and variability of

renewable resources, such as wind, plus uncertain demands require both utility companies under

the regulated market and ISOs under the deregulated market to refine their unit commitment and

economic dispatch policies to accommodate these uncertainties. While a large amount of research

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exists on how to formulate and improve the general unit commitment algorithm [20], the unit

commitment research that considers uncertain wind power and demand is limited.

To address demand uncertainty, previous research progress has been made on stochastic pro-

gramming approaches to solve the problem with the objective of minimizing the total expected

cost. A multistage stochastic programming formulation was developed early in [23]. At the same

time, an augmented Lagrangian decomposition framework was studied in [11]. Other relevant La-

grangian decomposition literature includes [12] and [15], among others. Recently, the stochastic

programming approach for unit commitment to generate supply curves in electric power markets

was studied in [19], to serve as a decision aid for scheduling and hedging in the wholesale electric

power market was studied in [22], and to estimate the potential contribution of demand flexibility

in replacing operating reserves was studied in [18].

To address wind supply uncertainty, Barth et al. [3] presented the early stage of the Wind Power

Integration in the Liberalised Electricity Markets (WILMAR) model [1]. Tuohy et al. [24] examined

the effects of stochastic wind and demand on the unit commitment and dispatch of power systems

with high levels of wind power by using the WILMAR model. Ummels et al. [25] analyzed the

impacts of wind power on thermal generation unit commitment and dispatch in the Dutch system,

which has a significant share of combined heat and power units. A rolling commitment method is

used to schedule the thermal units where the common constraints (e.g., ramp-rate limit constraints

and minimum on/off time constraints) are considered. The wind power forecasting errors are

captured by an autoregressive moving average process. Bouffard and Galiana [10] proposed a

stochastic unit commitment model to integrate significant wind power generation while maintaining

the security of the system. Rather than being pre-defined, the reserve requirements are determined

by simulating the wind power realization in the scenarios. Ruiz et al. [21] proposed a stochastic

formulation to manage uncertainty in the unit commitment problem. The formulation captures

several sources of uncertainty and defines the system reserve requirement for each scenario. Wang

et al. [26] presented a security-constrained unit commitment algorithm that takes into account the

intermittency and variability of wind power generation. The uncertainty in wind power output is

captured in a number of scenarios, and a Benders’ decomposition approach is applied to reduce the

computational efforts.

3

In the stochastic optimization approaches, scenarios are generated based on the forecasted

demand or wind output distribution with the objective of minimizing total expected cost. This

approach may cause two potential issues: 1) the problem size increases dramatically due to a large

amount of samples to generate scenarios, and 2) it is very challenging to provide a distribution of

the wind output amount due to its intermittent nature. Besides, due to the objective of minimizing

the total expected cost, a significant portion of wind energy may be curtailed to achieve the optimal

objective value or facilitate the scheduling of other generation units. For instance, a large upward

ramp in wind power may be unfavorable in a system in which sufficient downward reserves from

other resources are not present. In this case, wind power may have to be curtailed, which leads

to a waste of available resources and violates the policy of utilizing renewable energy as much as

possible.

In this research, we study a novel two-stage robust optimization version of the problem to

address the demand and wind supply uncertainties. Since wind supply can be considered as a

negative demand, we combine demand and wind supply uncertainties, and regard it as demand

uncertainty. Instead of providing a detailed description of the demand distribution, the demand

is assumed to be within an uncertainty set. In addition, the error of the wind supply is set much

larger than that of the demand, because wind output is much harder to forecast, as compared to the

demand. Our proposed robust optimization approach will enhance the reliability unit commitment

for both utility companies under the regulated market and ISOs under the deregulated market.

Accordingly, our approach will benefit the current practice, because it is crucial to provide a robust

unit commitment solution that keeps the power grid system away from blackout under both markets.

The robust optimization approach has received rich research attention in recent decades. With

the parametric uncertainty sets, the robust optimization approach provides tradeoffs between the

optimality and the robustness of the solutions. Moreover, it becomes an attractive approach because

of its computational advantages without enumerating a large number of scenarios. Ben-Tal and

Nemirovski [5, 4] and El Ghaoui et al. [13, 14] are among the first to study the robust optimization

approach with ellipsoidal uncertainty sets. In this research, we adopt cardinality and polyhedral

uncertainty sets which were first proposed by Bertsimas and Sim [6, 7] and studied in various

literature, such as inventory theory [8, 9], network design [2], and lot-sizing [2]. However, to

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the best of our knowledge, there is no previous research on robust optimization for power grid

optimization problems. In this paper, we study the two-stage robust power grid optimization

problem. We provide unit commitment decisions in the first stage with the objective of minimizing

the system-wide power generation cost including the unit commitment cost and dispatch cost under

the worst-case scenario.

The remaining part of this paper is organized as follows. Section 2 describes the mathematical

formulation of the robust power grid optimization problem. Section 3 studies the solution approach

to solve a simplified version of the power grid optimization problem, in which the transmission and

ramp-rate limit constraints are relaxed. This problem can be solved efficiently and can be used to

approximate the general power grid optimization problem. In Section 4, we explore the solution

schemes to solve the general robust power grid optimization problem. Finally, Section 5 reports

the computational results, and Section 6 concludes our study.

2 Notation and Mathematical Formulation

For a T -period power grid optimization problem, we let E = {1, 2, · · · ,M} represent the set of buses

and A represent the set of transmission lines linking two buses. For each bus m ∈ E, we let Nm be

the set of generators in this bus. Accordingly, for each generator i ∈ Nm, we let Smi represent the

start-up cost, Wmi represent the shut-down cost, Gm

i represent the minimum-up time, Hmi represent

the minimum-down time, Lmi represent the minimum output of electricity if the generator is on,

Umi represent the maximum output of electricity if the generator is on, V m

i represent the ramp-up

rate limit, and Bmi represent the ramp-down rate limit. For each transmission line (i, j) ∈ A, we

let Cij represent the capacity of the transmission line, and Kmij represent the line flow distribution

factor for the transmission line, due to the net injection at bus m, ∀m ∈ E. To describe the demand

uncertainty set, we let Dℓmt and Du

mt represent the lower and upper bounds of the demand at bus

m in time period t. For notation brevity, we define Drmt := Du

mt −Dℓmt, ∀m ∈ E, t = 1, 2, · · · , T .

For our two-stage robust power grid optimization problem, in the first stage we provide the unit

commitment decisions (ymit , umit , v

mit ) for each generator that include: 1) if generator i at bus m is

on or not in time period t (i.e., ymit = 1 if yes; ymit = 0 o.w.), 2) if generator i at bus m is started

up or not in time period t (i.e., umit = 1 if yes; umit = 0 o.w.), and 3) if generator i at bus m is shut

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down or not in time period t (i.e., vmit = 1 if yes; vmit = 0 o.w.). In the second stage, we let random

parameter dmt represent the demand at bus m in time period t, and decision variable xmit represent

the amount of electricity generated by generator i at bus m in time period t. First, the nominal

model can be described as follows:

Nominal model

zPO = miny,u,v,x

T∑t=1

M∑m=1

∑i∈Nm

(Smi umit +Wm

i vmit + fmit (x

mit ))

s.t. −ymi(t−1) + ymit − ymik ≤ 0, (1)

1 ≤ k − (t− 1) ≤ Gmi ,∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T

(PO) ymi(t−1) − ymit + ymik ≤ 1, (2)

1 ≤ k − (t− 1) ≤ Hmi , ∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T

−ymi(t−1) + ymit − umit ≤ 0, ∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T (3)

ymi(t−1) − ymit − vmit ≤ 0, ∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T (4)

Lmi ymit ≤ xmit ≤ Um

i ymit , ∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T (5)

xmit − xmi(t−1) ≤ (2− ymi(t−1) − ymit )Lmi + (1 + ymi(t−1) − ymit )V

mi , (6)

∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T

xmi(t−1) − xmit ≤ (2− ymi(t−1) − ymit )Lmi + (1− ymi(t−1) + ymit )B

mi , (7)

∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , TM∑

m=1

∑i∈Nm

xmit =

M∑m=1

dmt, t = 1, 2, · · · , T (8)

−Cij ≤M∑

m=1

Kmij

( ∑n∈Nm

xmnt − dmt

)≤ Cij , ∀(i, j) ∈ A, t = 1, 2, · · · , T (9)

ymit , umit , v

mit ∈ {0, 1}, and ymi0 = 0, ∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T.

In the above nominal model, the objective is to minimize the total cost, including start-up, shut-

down, and fuel costs. Constraints (1) describe the minimum-up time required for generator i once it

is started up. Accordingly, constraints (2) describe the minimum-down time required for generator

i once it is shut down. Constraints (3) and (4) indicate the start-up and shut-down operations

for each generator i. Constraints (5) describe the upper and lower bounds of power output of

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generator i if it is on in period t. Constraints (6) and (7) describe the ramp-up and ramp-down

rate limit restrictions, respectively. Constraints (8) ensure the power grid flow balance. Finally,

constraints (9) describe the transmission capacity limits which restrict the flow passing through

each transmission line (i, j).

Two-stage robust optimization model

When the demand for the second-stage is unknown and is within an uncertainty set, i.e., D, the

two-stage robust counterpart of the nominal model (PO) is to minimize the total cost under the

worst-case scenario. Before further characterizing the model, we first use a P -piece piecewise linear

function to approximate the non-decreasing convex cost function fmit (x

mit ) = cmit (x

mit )

2+ bmit xmit +amit .

After choosing P breakpoints

Lmi = qm1

i < qm2i < · · · < qmP

i = Umi , ∀m ∈ E, ∀i ∈ Nm,

we can represent the function with

fmit (x

mit ) ≥ ymit (α

mpit + βmp

it xmit ) = αmpit ymit + βmp

it xmit , ∀m ∈ E, ∀i ∈ Nm, p = 1, 2, · · · , P,

where the equation follows from (5), and αmpit , βmp

it for p = 1, 2, · · · , P are given by{αmpit = amit − cmit (q

mpi )2

βmpit = 2cmit q

mpi + bmit

, ∀p = 1, 2, · · · , P.

Now the two-stage robust power grid optimization problem can be described as follows:

zR = miny,u,v

T∑t=1

M∑m=1

∑i∈Nm

(Smi umit +Wm

i vmit ) + maxd∈D

min(x,ϑ)∈X (y,d)

T∑t=1

M∑m=1

∑i∈Nm

ϑmit

(RPO) s.t. (1), (2), (3), (4),

ymit , umit , v

mit ∈ {0, 1}, and ymi0 = 0, ∀m ∈ E, ∀i ∈ Nm, t = 1, 2, · · · , T,

where X (y, d) ={(x, ϑ) : (5), (6), (7), (8), (9),

ϑmit ≥ αmp

it ymit + βmpit xmit , ∀m ∈ E, ∀i ∈ Nm, p = 1, 2, · · · , P, t = 1, 2, · · · , T

}(10)

and D is the uncertainty set that describes the range of demand.

In this paper, we study the cases with and without transmission capacity and ramp-rate limits.

For each case, we consider both cardinality and polyhedral uncertainty sets [2].

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1. The model without transmission capacity and ramp-rate limits: This is an aggregated for-

mulation. For the polyhedral uncertainty set (e.g., budget uncertainty set), we consider that

the demand for each operating hour t at each bus m is between a lower bound Dℓmt and an

upper bound Dumt. In addition, the summation of the weighted demands within the planning

horizon is bounded above. The uncertainty set can be described as follows:

DB ={d : Dℓ

mt ≤ dmt ≤ Dumt, ∀m ∈ E, ∀t,

T∑t=1

M∑m=1

πtdmt ≤ π0

}, (11)

where π0 is the given upper bound for the planning horizon.

For the cardinality uncertainty set [7, 8], we can describe the set as follows:

DC ={d : Dℓ


T∑t=1

⌈ ∣∣∣∣∣∑M

m=1 dmt − Dt

Dt

∣∣∣∣∣⌉≤ Γ

}, (12)

where Dt =∑M

m=1(Dumt + Dℓ

mt)/2 and Dt =∑M

m=1Drmt/2 are the nominal value and the

maximum variation of∑M

m=1 dmt.

2. The model with transmission capacity and ramp-rate limits: For this case, we consider that

the demand for each operating hour t at each bus m is between a lower bound Dℓmt and

an upper bound Dumt. Besides the overall budget constraint to make the summation of the

weighted demands within the planning horizon bounded above, for each given time period t,

the summation of the weighted demands at all buses is bounded above.

DB ={d : Dℓ

mt ≤ dmt ≤ Dumt,∀m ∈ E, ∀t,

M∑m=1

πmtdmt ≤ πt, ∀t,

M∑m=1

T∑t=1

πmtdmt ≤ π0

}, (13)

where πt is the given upper bound for time period t. Again, the corresponding cardinality

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uncertainty set is described as follows:

DC ={d : Dℓ


M∑m=1

⌈ ∣∣∣∣∣dmt − Dmt

Dmt

∣∣∣∣∣⌉≤ Γt, ∀t,

M∑m=1

T∑t=1

⌈ ∣∣∣∣∣dmt − Dmt

Dmt

∣∣∣∣∣⌉≤ Γ

},

(14)

where Dmt = (Dumt+Dℓ

mt)/2 and Dmt = Drmt/2 are the nominal value and maximum variation

of dmt.

3 Two-Stage Robust Unit Commitment Problem

We first analyze the power grid optimization problem in which transmission capacity and ramp-

rate limits are relaxed. This is the robust optimization version of the traditional unit commitment

problem. It will help us understand the insights of the basic problem structure and provide an

approximation model for the general robust power grid optimization problem.

In this case, for notation brevity, we omit the index m in the formulation. In addition, we

assume there will be penalty and salvage costs generated if the electricity output does not match

the demand. We let zt and wt represent the surplus and shortage amount of electricity at time

period t, and accordingly, let Rt and Pt represent the unit penalty and salvage cost. To illustrate

the main results, we assume the fuel cost is linear (e.g., fit(xit) = αit+ βitxit) in this section. Note

that our conclusion for the multiple-piece approximation case also holds. Then, the corresponding

robust optimization model can be simplified as follows:

miny,u,v

T∑t=1

∑i∈N

(Siuit +Wivit+αityit) + maxd∈D

min(x,z,w)∈X (y,d)

T∑t=1

(∑i∈N

βitxit +Rtwt − Ptzt

)(RUC) s.t. −yi(t−1) + yit − yik ≤ 0, 1 ≤ k − (t− 1) ≤ Gi, ∀i ∈ N, ∀t (15)

yi(t−1) − yit + yik ≤ 1, 1 ≤ k − (t− 1) ≤ Hi, ∀i ∈ N, ∀t (16)

−yi(t−1) + yit − uit ≤ 0, ∀i ∈ N, ∀t (17)

yi(t−1) − yit − vit ≤ 0, ∀i ∈ N, ∀t (18)

yit, uit, vit ∈ {0, 1}, and yi0 = 0, i ∈ N, ∀t,

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where X (y, d) ={(x, z, w) : Liyit ≤ xit ≤ Uiyit, i ∈ N, ∀t∑

i∈Nxit + wt − zt = dt, ∀t

xit, zt, wt ≥ 0, i ∈ N, ∀t}.

The uncertainty sets described in (11) and (12) are simplified as follows:

DB ={d :

T∑t=1

πtdt ≤ π0, Dℓt ≤ dt ≤ Du

t , ∀t}

and (19)

DC ={d :

T∑t=1

⌈ ∣∣∣∣∣dt − Dt

Dt

∣∣∣∣∣⌉≤ Γ, Dℓ

t ≤ dt ≤ Dut , ∀t

}, (20)

where Dℓt and Du

t represent the lower and upper bounds for the demand in time period t, Dt =

(Dut +Dℓ

t)/2, and Dt = (Dut −Dℓ

t)/2.

To solve (RUC), we first analyze the optimal value function gt(yt, dt) for the subproblem in

the second stage once the first stage decision variable (y, u, v) is fixed. In this case, (RUC) is

decomposed into T subproblems. Corresponding to each time period t, we have

gt(yt, dt) = min∑i∈N

βitxit +Rtwt − Ptzt

s.t. Liyit ≤ xit ≤ Uiyit, i ∈ N, (21)∑i∈N

xit + wt − zt = dt, (22)

xit, zt, wt ≥ 0, i ∈ N.

In the optimal solution for the subproblem corresponding to each time period t, we have z∗t =

(∑i∈N

Liyit − dt)+ and w∗

t = (dt −∑i∈N

Uiyit)+, where x+ = max{x, 0}.

Proposition 1 The value function gt(yt, dt) is piecewise linear, nondecreasing, and convex in dt.

Proof: Without loss of generality, we assume N = {1, 2, ..., N} and

0 <Pt < β1t ≤ β2t ≤ . . . ≤ βNt < Rt. (23)

From constraints (21) and (22), based on the cost relationship shown in (23), we can observe

the following:

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• If dt ≤N∑i=1

Liyit, then in the optimal solution, each generator generates at its lower bound

and salvages the over generated power in the amount ofN∑i=1

Liyit − dt. The total cost is

gt(yt, dt) = ϕt0(dt) =

N∑i=1

βitLiyit − Pt(

N∑i=1

Liyit − dt) = φt0(y) + Ptdt, (24)

where φt0(y) =N∑i=1

βitLiyit − Pt

N∑i=1

Liyit.

• If dt ≥N∑i=1

Uiyit, then in the optimal solution, each generator generates at its upper bound.

The shortage part is in the amount of dt −N∑i=1

Uiyit. The total cost is

gt(yt, dt) = ϕt(N+1)(dt) =N∑i=1

βitUiyit +Rt(dt −N∑i=1

Uiyit) = φt(N+1)(y) +Rtdt, (25)

where φt(N+1)(y) =N∑i=1

βitUiyit −Rt

N∑i=1

Uiyit.

• For the general case, assuming (N∑i=θ

Liyit +θ−1∑i=1

Uiyit) ≤ dt ≤ (N∑

i=θ+1

Liyit +θ∑

i=1Uiyit), θ =

1, . . . , N , the total cost is

gt(yt, dt) = ϕtθ(dt) =N∑

i=θ+1

βitLiyit+θ−1∑i=1

βitUiyit+βθt(dt−N∑

i=θ+1

Liyit−θ−1∑i=1

Uiyit) = φtθ(y)+βθtdt,

(26)

where φtθ(y) =N∑

i=θ+1

βitLiyit +θ−1∑i=1

βitUiyit − βθtN∑

i=θ+1

Liyit − βθtθ−1∑i=1

Uiyit.

From (24), (25), and (26), we can observe that ϕtθ(dt), θ = 0, . . . , N + 1, is a linear func-

tion of dt. Based on (23) and ϕtθ(N∑

i=θ+1

Liyit +θ∑

i=1Uiyit) = ϕt(θ+1)(

N∑i=θ+1

Liyit +θ∑

i=1Uiyit) =

N∑i=θ+1

βitLiyit +θ∑

i=1βitUiyit, we have that the value function gt(yt, dt) is piecewise linear, nonde-

creasing, and convex in dt. Therefore, the conclusion holds.

Since the value function gt(yt, dt) is piecewise linear and convex, the following corollary holds.

Corollary 1 The value function gt(yt, dt) = maxθt=0,...,N+1

ϕtθt(dt), where θt represents the value of θ

at time period t.

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Based on the conclusion obtained in Proposition 1 and Corollary 1, (RUC) can be reformulated

as follows:

miny,u,v

T∑t=1

N∑i=1

(αityit + Siuit +Wivit) + maxd∈D

T∑t=1

gt(yt, dt)

=T∑t=1

N∑i=1

(αityit + Siuit +Wivit) + maxd∈D

T∑t=1

( maxθt=0,...,N+1

ϕtθt(dt))

s.t. (15), (16), (17), (18),

yit, uit, vit ∈ {0, 1}, i ∈ N, ∀t.

Now we can introduce a new continuous decision variable µ for the second stage as follows:

miny,u,v

T∑t=1

N∑i=1

(αityit + Siuit +Wivit) + µ

s.t. µ ≥T∑t=1

maxθt=0,...,N+1

ϕtθt(dt) for all d ∈ D,

(15), (16), (17), (18),

yit, uit, vit ∈ {0, 1}, i ∈ N, ∀t, µ ∈ ℜ.

In the optimal solution, for a given (y, u, v), since ϕtθt(dt) and ϕt′θt′(dt′) are mutually indepen-

dent, we have

µ = maxθt,1≤t≤T

{max

T∑t=1

ϕtθt(dt) : d ∈ D

}. (27)

Separation: The separation problem of (27) can be stated as:

(SP): Given a solution (y, u, v, µ), does there exist (θ1, ..., θT ) and d ∈ D, such that

µ <T∑t=1

ϕtθt(dt)?

In the remaining part of this section, we analyze the separation problem with the uncertainty

sets (19) and (20), respectively.

If (19) is given, for a given θt, the dual of (27) can be described as follows:

min π0ζ +

T∑t=1

(Dut ηt −Dℓ

tρt) +

T∑t=1

φtθt(y)

(Dual) s.t. πtζ + ηt − ρt ≥ ϑtθt , ∀t,

ζ, η, ρ ≥ 0,

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where ϑtθt is the coefficient of dt in function ϕtθt(dt). That is, ϑtθt = Pt if θt = 0, ϑtθt = Rt if

θt = N + 1, and ϑtθt = βθtt if 1 ≤ θt ≤ N. We also notice that φtθt(y) is a constant number here

for the dual problem once t, θt and y are given.

Therefore, if θt for each t is given, we can add the dual constraints to the first stage master

problem. Note here the dual constraints are linear. However, there are exponential number of such

combinations of θt, ∀t. We can add these constraints gradually by running a separation algorithm.

Next, we describe a separation algorithm to discover θt, given a solution of the master problem.

Theorem 1 The separation problem (SP) is NP-hard under the budget uncertainty set DB as

described in (19).

Proof: We prove the claim by reducing a 0 − 1 knapsack problem to (SP). Given an instance of

a 0− 1 knapsack problem (e.g., max{cx : ux ≤ b, x ∈ {0, 1}T , b ∈ Z+}), we construct a separation

problem (SP) as follows:

There are T time periods and 2T generators. At each time period t, only two generators 2t− 1

and 2t are turned on. Assume that the minimum-up and minimum-down times for both generators

are one time period. In addition, the capacities of generators 2t − 1 and 2t are ut − 1 and 1, and

the corresponding unit production costs are 0 and ct, respectively. The budget uncertainty set is

DB = {d :T∑t=1

dt ≤ b, 0 ≤ dt ≤ ut, b ∈ Z+}.

From the above constructed separation problem (SP), we can observe that for each time period,

we use generator 2t − 1 to satisfy the demand if dt ≤ ut − 1, because the unit production cost is

zero. If dt = ut, then we have to use generator 2t to produce one unit at the cost ct. Therefore,

it can be observed that there exists an optimal (SP) solution d∗, such that for any t, d∗t is either

0 or ut. It is equivalent to the knapsack solution with the same objective value. Therefore, the

separation problem with uncertainty set DB is NP-hard.

Since the separation problem under the budget uncertainty set is closely related to the knapsack

problem, we propose a dynamic programming algorithm to solve it. Given a first stage solution y,

we will determine the extreme scenario d which gives the most violated constraint, or determine

that the current solution is feasible. The algorithm is based on the following observation.

13

Observation 1 For any extreme scenario d ∈ DB, there exists a time period t′, such that for any

t = t′, dt = Dℓt or Du

t .

This observation is obtained by the convexity of function gt(yt, dt) and the extreme points of

uncertainty set DB. Moreover, we can derive the following lemma for the extreme scenario d.

Lemma 1 For any extreme scenario d ∈ DB with respect to the first stage solution (y, µ), if

dt ∈ (N∑i=θ

Liyit +θ−1∑i=1

Uiyit,N∑

i=θ+1

Liyit +θ∑

i=1Uiyit) for some θ, then d corresponds to the optimal

solution of the following knapsack problem

max ztθ =∑j∈J

(gj(yj , Duj )− gj(yj , D

ℓj))s

tθj

(SPNtθ) s.t.∑j∈J

πj(Duj −Dℓ

j)stθj ≤ π0,

stθj ∈ {0, 1}, j ∈ J,

where J = {j : βθt/πt ≤ βNj/πj , j = t}, π0 = π0−∑j =t

πjDℓj −πt(

N∑i=θ

Liyit+θ−1∑i=1

Uiyit), and stθj is the

indicator of dj(i.e., stθj = 0 if dj = Dℓ

j, and stθj = 1 if dj = Duj ).

Proof: From the definition of J , if j ∈ J , then we can observe that dj = Dℓj in the extreme scenario.

If not, then dj = Duj based on Observation 1. Then, decreasing dj by a small δ/πj > 0 while

increasing dt by δ/πt will increase the second stage cost by δ(βθt/πt − βNj/πj), which contradicts

with the claim that d is the extreme scenario. Thus, the optimal objective value z∗tθ for (SPNtθ)

corresponds to the worst second-stage cost increment by picking periods j in which dj = Duj .

With Lemma 1, we only need to consider T (N + 2) knapsack problems to solve (SP), because

there are T (N+2) combinations of t and θ. The dynamic programming algorithm for the knapsack

problem can be adopted directly to solve each (SPNtθ).

Now we consider obtaining the extreme demand d after solving the T (N+2) knapsack problems.

For each combination of t and θ, let the optimal solution of (SPNtθ) be (stθ)∗ with the corresponding

optimal objective value z∗tθ, and

λtθ = (π0 −∑j∈J

πj(Duj −Dℓ

j)(stθj )

∗)/πt.

If λtθ ≤ (Uθ − Lθ)yθt, let ztθ = z∗tθ and (t, θ) be a candidate to generate the extreme demand d.

Otherwise, let ztθ = 0, which implies that in the extreme scenario, dt is not in the θth interval. Let

(t∗, θ∗) = argmaxt=1,··· ,T,θ=0,1,··· ,N,N+1{ztθ}.

14

Then, based on (SPNtθ), we can observe that if µ ≥ z∗t∗θ∗ +∑t =t∗

gt(yt, Dℓt) + gt∗(yt∗ , λt∗θ∗ +

N∑i=θ∗

Liyit +θ∗−1∑i=1

Uiyit), the solution is feasible. Otherwise, we can construct the extreme demand d

as follows:

dt∗ = λt∗θ∗ +N∑

i=θ∗

Liyit +θ∗−1∑i=1

Uiyit;

dt = Dℓt + (Du

t −Dℓt)s

t∗θ∗t , t = t∗.

From the above analysis, it is easy to observe the following proposition holds, based on the fact

that the knapsack problem can be solved in O(T π0) time (without loss of generality, we assume

that π0 and Dut −Dℓ

t , ∀t, are integers).

Proposition 1 The separation problem (SP ) can be solved in pseudo-polynomial time O(T 2π0N).

If (20) is given, the separation problem is easy and can be described as follows:

Theorem 2 The separation problem (SP) is polynomial time solvable under the cardinality uncer-

tainty set DC as described in (20).

Proof: For the cardinality uncertainty set, the (SP) problem is equivalent to finding the set of

time periods T ′ ⊆ T corresponding to the extreme scenario d, where for any t ∈ T ′, dt = Dut and

for any t ∈ T \ T ′, dt = Dt. The detailed steps are shown in Algorithm 1.

Algorithm 1: Separation algorithm under uncertainty set DC

Data: The first stage solution y, µ.Result: Current solution is feasible or return (θ1, ..., θT )

∗ and d∗ maximizing the secondstage cost.

forall the t = 1, 2, · · · , T do

Set gt = gt(yt, Dut )− gt(yt, Dt);

Set θt and θt such that gt(yt, Dut ) = ϕtθt

(Dut ) and gt(yt, Dt) = ϕtθt

(Dt).

Let g[1], g[2], · · · , g[T ] be a sorted nonincreasing order of g1, g2, · · · , gT .

Set µ =Γ∑

t=1

g[t](y[t], Du[t]) +

T∑t=Γ+1

g[t](y[t], D[t]).

if µ ≥ µ thenthe current solution is a feasible solution;

else

return (θ[1], · · · , θ[Γ], θ[Γ+1], · · · , θ[T ]) and (Du[1], · · · , D

u[Γ], D[Γ+1], · · · , D[T ]).

15

The algorithm returns the extreme demand that corresponds to the maximum second stage

cost. Note that finding the value of gt(yt, dt) is equivalent to a linear program, which is polynomial

time solvable. Thus, this is a polynomial time algorithm and the conclusion holds.

4 Two-Stage Robust Power Grid Optimization Problem

In this section, we develop solution methods to solve the general two-stage power grid optimization

problem (RPO). We develop solution approaches that can provide exact and near-optimal solutions,

respectively. In terms of tractability, similar as shown in the previous section, the cardinality

uncertainty set case is relatively easier than the polyhedral uncertainty set case. We focus on the

polyhedral uncertainty set case in this section, and describe that the cardinality uncertainty set

case can be solved by a similar approach.

Given a first-stage decision variable y, let ω(y) represent the optimal value function for the

second-stage problem of (RPO). By dualizing the constraints in X (y, d), we have

ω(y) = maxγ,η,τ,ρ,d,ξ,ζ

T∑t=1

(M∑

m=1

∑i∈Nm

(Lmi ymit γ

m+it − Um

i ymit γm−it ) +

M∑m=1

dmtηt

−M∑

m=1

∑i∈Nm

(ξmit (2− ymi(t−1) − ymit )Lmi + ξmit (1 + ymi(t−1) − ymit )V

mi )

−M∑

m=1

∑i∈Nm

(ζmit (2− ymi(t−1) − ymit )Lmi + ζmit (1− ymi(t−1) + ymit )B

mi )

+∑

(i,j)∈A

M∑m=1

(Kmij dmtτ

+ij,t −Km

ij dmtτ−ij,t)−

∑(i,j)∈A

Cij(τ+ij,t + τ−ij,t)

+M∑

m=1

∑i∈Nm

P∑p=1

(ymit αmpit ρmp

it )

(28)

(SUB) s.t. γm+it − γm−

it + ηt + ξmi(t+1) − ξmi(t) + ζmit − ζmi(t+1) −P∑

p=1

(βmpit ρmp

it )

+∑

(i,j)∈A

(Kmij τ

+ij,t −Km

ij τ−ij,t) = 0, ∀m ∈ E, ∀i ∈ Nm, ∀t (29)

P∑p=1

ρmpit = 1 ∀m ∈ E,∀i ∈ Nm, ∀t (30)

16

Dℓmt ≤ dmt ≤ Du

mt ∀m ∈ E, ∀t (31)M∑

m=1

πmtdmt ≤ πt, ∀t (32)

M∑m=1

T∑t=1

πmtdmt ≤ π0 (33)

γm+it , γm−

it , τ+ij,t, τ−ij,t, ξ

mit , ζ

mit , ρ

mpit ≥ 0,

∀m ∈ E, ∀i ∈ Nm, ∀p = 1, 2, · · · , P, ∀(i, j) ∈ A, ∀t,

where the decision variables γm±it , ξmit , ζ

mit , ηt, τ

±ij,t and ρmp

it are the dual variables for constraints

(5)-(10). We observe that (SUB) is a bilinear program with products of (η, τ) and d in the objective

function, and thus NP-hard under general data settings (e.g.,∑M

m=1 πmtDumt > πt).

In this section, we provide separation schemes embedded in a Benders’ decomposition framework

to solve (RPO). We develop both exact and heuristic approaches. First, we can solve the following

master program iteratively by adding new constraints to cut off infeasible or non-optimal solutions

in the Benders’ decomposition framework:

zM = miny,u,v

T∑t=1

M∑m=1

∑i∈Nm

(Smi umit +Wm

i vmit ) + ω(y)

(Master) s.t.

T∑t=1

M∑m=1

∑i∈Nm

σmsit ymit ≤ κs, s = 1, · · · , S, (34)

ω(y)−T∑t=1

M∑m=1

∑i∈Nm

σmrit ymit ≥ κr, r = 1, · · · , R, (35)

(1), (2), (3), (4),

ymit , umit , v

mit ∈ {0, 1}, and ymi0 = 0, ∀m ∈ E, ∀i ∈ Nm,∀t,

where constraints (34) represent the selected feasibility cuts added to the master problem, while

constraints (35) represent the optimality cuts.

4.1 Exact separation approach

To obtain an exact separation, we need to solve the subproblem (SUB) into optimality. In the

following, we show how we can solve the bilinear subproblem by solving one mixed-integer linear

program. First, we can observe that the optimal solution for (RPO) satisfies the following condition.

17

Proposition 2 There exists an optimal solution (γ∗, η∗, τ∗, ξ∗, ζ∗, ρ∗, d∗) to (SUB) that satisfies

the following condition: For each time period t, d∗mt = Dumt or Dℓ

mt, except at most one bus in

which d∗mt ∈ [Dℓmt, D

umt] to make the constraint

∑Mm=1 πmtdmt ≤ πt tight, or to make the constraint∑M

m=1

∑Tt=1 πmtdmt ≤ π0 tight.

Proof: We note that (SUB) maximizes a bilinear function over a polyhedron, where the linear

constraints of (γ, η, τ, ξ, ζ, ρ) and those of d are disjoint. Hence, there exists an optimal solution

(γ∗, η∗, τ∗, ξ∗, ζ∗, ρ∗, d∗) to (SUB) such that d∗ is an extreme point of the polyhedron represented

by the constraints of d, i.e., D. If this is not the case, we can fix (γ∗, η∗, τ∗, ξ∗, ζ∗, ρ∗) and solve

(SUB), which, after optimized, gives us an extreme point d∗ of D. Besides, based on the problem

structure, it is easy to check that d∗ is an extreme point of D if and only if d∗ satisfies the condition

in the above claim.

Based on Proposition 2, as an extreme point, the demand in each bus at each time period will

make: 1) lower or upper bound constraint tight, or 2) inequalities (32) tight, or 3) inequality (33)

tight. Then, we let binary decision variable zumt = 1 represent that d∗mt = Dumt, binary decision

variable zfmt = 1 if d∗mt ∈ [Dℓmt, D

umt] and it makes the corresponding inequality (32) tight, and

binary decision variable zgmt = 1 if d∗mt ∈ [Dℓmt, D

umt] and it makes the corresponding inequality (33)

tight. In addition, we let binary decision variable zt = 1 if the corresponding inequality (32) is

tight. We can observe the following constraints hold:

zumt + zfmt + zgmt ≤ 1, ∀m ∈ E, ∀t, (36)M∑

m=1

zfmt ≤ 1, ∀t, (37)

M∑m=1

T∑t=1

zgmt ≤ 1, (38)

zt =

M∑m=1

zfmt, ∀t, (39)

zgmt + zt ≤ 1, ∀m ∈ E, ∀t, (40)

where (36) indicates that the demand in each bus at each time period will play at most one role,

(37) indicates that, at each time period t, at most one bus in which the demand plays the role to

make the corresponding inequality (32) tight, and similarly (38) indicates that at most one bus at a

18

particular time period in which the demand plays the role to make (33) tight. To avoid duplicated

calculation, (40) indicates that if the demand for a bus in time period t is used to make (33) tight,

then no bus in time period t will be considered to make (32) tight.

Now, we reformulate the subproblem to be a mixed-integer linear program in the following way.

We first can write down the subproblem as follows:


T∑t=1

(M∑

m=1

∑i∈Nm

(Lmi ymit γ

m+it − Um

i ymit γm−it ) +

M∑m=1

∑i∈Nm

f(ymit , ξmit , ζ

mit )

+

M∑m=1

dmtηmt −∑

(i,j)∈A

Cij(τ+ij,t + τ−ij,t) +

M∑m=1

∑i∈Nm

P∑p=1

(ymit αmpit ρmp

it )

s.t. (29), (30), (36)− (40),

ηmt = ηt +∑

(i,j)∈A

Kmij τ

+ij,t −

∑(i,j)∈A

Kmij τ

−ij,t, ∀m ∈ E, ∀t, (41)

ηmt/πmt − ηnt/πnt ≤ M(zumt − zunt + 1), ∀m,n ∈ E, ∀t, (42)

ηmt/πmt − ηnt/πnt ≤ M(zumt − zfnt + 1), ∀m,n ∈ E, ∀t, (43)

ηmt/πmt − ηnt/πnt ≤ M(zumt − zgnt + 1), ∀m,n ∈ E, ∀t, (44)

πt −M∑n=1

πntDℓnt −

M∑n=1

πntDrntz

unt ≤ πmtD

rmt + M(1− zfmt), ∀m ∈ E, ∀t, (45)

(SUBR) πt −M∑n=1

πntDℓnt −

M∑n=1

πntDrntz

unt ≥ 0, ∀t, (46)

π0 −∑t

πt ≥ 0, (47)

π0 −∑t

πt ≤ πmtDrmt + M(1− zgmt),∀m ∈ E, ∀t, (48)

π0 −∑t

πt ≤ (πt −M∑n=1

πntDℓnt −

M∑n=1

πntDrntz

unt) + M(1− zgmt), ∀m ∈ E, ∀t,(49)

πt ≤ πt, ∀t, (50)

πt ≥ πt − M(1− zt), ∀t, (51)

πt ≤M∑

m=1

πmtDℓmt +

M∑m=1

πmtDrmtz

umt + Mzt, ∀t, (52)

πt ≥M∑

m=1

πmtDℓmt +

M∑m=1

πmtDrmtz

umt − Mzt, ∀t, (53)

γm+it , γm−

it , ξmit , ζmit , τ

+ij,t, τ

−ij,t, ρ

mpit ≥ 0, zumt, z

fmt, z

gmt ∈ {0, 1}, (54)

∀m ∈ E, ∀i ∈ Nm, p = 1, 2, · · · , P, ∀(i, j) ∈ A, ∀t, (55)

19

where

f(ymit , ξmit , ζ

mit ) = −ξmit (2− ymi(t−1) − ymit )L

mi − ξmit (1 + ymi(t−1) − ymit )V

mi

−ζmit (2− ymi(t−1) − ymit )Lmi − ζmit (1− ymi(t−1) + ymit )B

mi . (56)

Constraints (42)-(44) ensure that if the weighted cost coefficient of dmt (i.e., ηmt/πmt) is larger than

that of dnt, then zumt ≥ zunt, zumt ≥ zgnt, and zumt ≥ zfnt. Constraints (45) and (46) ensure a demand

dmt in the interval [Dℓmt, D

umt] if z

fmt = 1. Note that here it is allowed this demand dmt = Dℓ

mt or

dmt = Dumt, and constraint (32) is automatically satisfied. Similarly, constraints (47)-(49) ensure

a demand dmt in the interval [Dℓmt,min{Du

mt, Dℓmt + (πt −

∑Mn=1 πntD

ℓnt −

∑Mn=1 πntD

rntz

unt)/πmt}]

to make constraint (33) automatically satisfied. Constraints (50) to (53) indicate that πt = πt if

zt = 1 and πt =∑M

m=1 πmtDℓmt +

∑Mm=1 πmtD

rmtz

umt otherwise.

In the following part, we show how we linearize the bilinear term dmtηmt in the objective function

in (SUBR). First, we can write

dmt = Dℓmt + zumtD

rmt + zfmt(πt −

M∑n=1

πntDℓnt −

M∑n=1

πntDrntz

unt)/πmt

+zgmt

(π0 −

∑s

(πszs + (1− zs)(

M∑n=1

πnsDℓns +

M∑n=1

πnsDrnsz

uns)

))/πmt.

Then, we have

dmtηmt = Dℓmtηmt (57)

+

(Dr

mtzumtηmt + zfmtηmt(πt −

M∑n=1

πntDℓnt)/πmt

)(58)

+

(zgmtηmt

(π0/πmt −

∑s

M∑n=1

(πnsDℓns/πmt)

))(59)

−M∑n=1

zuntzfmtηmtπntD

rnt/πmt (60)

+∑s

zszgmtηmt

(M∑n=1

πnsDℓns/πmt − πs/πmt

)(61)

−∑s

M∑n=1

zunszgmtηmtπnsD

rns/πmt (62)

+∑s

M∑n=1

zgmtzszunsηmtπnsD

rns/πmt. (63)

20

Constraints (58)-(63) can be linearized as follows:

+

(Dr

mtzumt + zfmt(πt −

M∑n=1

πntDℓnt)/πmt

)(64)

+ zgmt

(π0/πmt −

∑s

M∑n=1

(πnsDℓns/πmt)

)(65)

+

M∑n=1

zufmntπntDrnt/πmt (66)

+∑s

zgmst

(πs/πmt −

M∑n=1

πnsDℓns/πmt

)(67)

+∑s

M∑n=1

zugmnstπnsDrns/πmt (68)

+∑s

M∑n=1

zugmnsstπnsDrns/πmt (69)

s.t. zumt ≤ ηmt + M(1− zumt), zumt ≤ Mzumt, (70)

zfmt ≤ ηmt + M(1− zfmt), zfmt ≤ Mzfmt, (71)

zgmt ≤ ηmt + M(1− zgmt), zgmt ≤ Mzgmt, (72)

zufmnt ≤ −ηmt + M(2− zfmt − zunt), zufmnt ≤ Mzfmt, zufmnt ≤ Mzunt, ∀n, (73)

zgmst ≤ −ηmt + M(2− zgmt − zs), zgmst ≤ Mzs, zgmst ≤ Mzgmt, ∀s, (74)

zugmnst ≤ −ηmt + M(2− zgmt − zuns), zugmnst ≤ Mzuns, zugmnst ≤ Mzgmt, ∀s, ∀n, (75)

zugmnsst ≤ ηmt + M(3− zgmt − zs − zuns), ∀s,∀n, (76)

zugmnsst ≤ Mzuns, zugmnsst ≤ Mzgmt, zugmnsst ≤ Mzs, ∀s,∀n. (77)

Therefore, the final subproblem in the form of mixed-integer linear problem is


T∑t=1

(M∑

m=1

∑i∈Nm

(Lmi ymit γ

m+it − Um

i ymit γm−it ) +

M∑m=1

∑i∈Nm

f(ymit , ξmit , ζ

mit )

+

M∑m=1

((57) + (64) + (65) + (66) + (67) + (68) + (69))

(SUBM) −∑

(i,j)∈A


M∑m=1

∑i∈Nm

P∑p=1

(ymit αmpit ρmp

it )

(78)

s.t. (29), (30), (36)− (56), (70)− (77).

21

4.2 Heuristic separation approach

It is challenging to use the exact separation algorithm to solve large size instances, due to a large

amount of integer decision variables. To solve large-scale problems, we propose a heuristic algorithm

that can solve the problem quickly. We first write down the following two linear programs, generated

by fixing γ, η, τ, ρ, ξ, ζ and d in (SUB), respectively.

maxγ,η,τ,ρ,ξ,ζ

T∑t=1

(M∑

m=1

∑i∈Nm

(Lmi ymit γ

m+it − Um

i ymit γm−it ) +

M∑m=1

dmtηt

+∑

(i,j)∈A

((

M∑m=1

Kmij dmt − Cij)τ

+ij,t − (

M∑m=1

Kmij dmt + Cij)τ

−ij,t

)

(SUB1) +

M∑m=1

∑i∈Nm

P∑p=1

(ymit αmpit ρmp

it ) +

M∑m=1

∑i∈Nm

f(ymit , ξmit , ζ

mit )

s.t. (29), (30), (56),

γm+it , γm−

it , τ+ij,t, τ−ij,t, ξ

mit , ζ

mit , ρ

mpit ≥ 0 ∀m ∈ E,∀i ∈ Nm, ∀p, ∀(i, j) ∈ A, ∀t;

maxd

T∑t=1

M∑m=1

(ηt +

∑(i,j)∈A

Kmij (τ

+ij,t − τ−ij,t)

)dmt +

M∑m=1

∑i∈Nm

(Lmi ymit γ

m+it − Um

i ymit γm−it )

(SUB2) −∑

(i,j)∈A

(Cijτ−ij,t + Cijτ

+ij,t) +

M∑m=1

∑i∈Nm

P∑p=1

(ymit αmpit ρmp

it ) +M∑

m=1

∑i∈Nm

f(ymit , ξmit , ζ

mit )

s.t. (31), (32), (33).

4.3 Feasibility cuts

We say a first-stage solution y is infeasible if ω(y) = +∞. In other words, the problem (SUB)

is unbounded. Since the constraints of d in (SUB2) construct a compact set, and the objective

function is continuous, we know that ω(y) < +∞ with γ, η, τ, ξ, ζ and ρ fixed. Hence, we have the

following claim.

Proposition 3 ω(y) = +∞ if and only if (SUB1) is unbounded.

Remark 1 From duality theory, the unboundedness of (SUB1) implies the infeasibility of the sub-

problem min(x,ϑ)∈X (y,d)

T∑t=1

M∑m=1

∑i∈Nm

ϑmit , and thus the set X (y, d) is empty. Intuitively, it means that

given a unit commitment decision y, there is a time period in which the demand d in the power

22

grid cannot be satisfied under the restriction of ramp-rate limit, and transmission and generator

capacity constraints.

Applying the Benders’ method as described in [17], we can construct the following program

ωf (y) = maxγ,η,τ,ξ,ζ

T∑t=1

(M∑

m=1

∑i∈Nm

(Lmi ymit γ

m+it − Um

i ymit γm−it + f(ymit , ξ

mit , ζ

mit )) +

M∑m=1

dmtηt

+∑

(i,j)∈A

((M∑

m=1

Kmij dmt − Cij)τ

+ij,t − (

M∑m=1

Kmij dmt + Cij)τ

−ij,t)

(79)

(FEA) s.t. γm+it − γm−

it + ηt + ξmi(t+1) − ξmit + ζmit − ζmi(t+1)

+∑

(i,j)∈A

(Kmij τ

+ij,t −Km

ij τ−ij,t) = 0, ∀m ∈ E,∀i ∈ Nm, ∀t,

0 ≤ γm+it , γm−

it ≤ 1 ∀m ∈ E, ∀i ∈ Nm, ∀t,

−1 ≤ ηt ≤ 1, ∀t,

0 ≤ τ+ij,t, τ−ij,t ≤ 1 ∀(i, j) ∈ A, ∀t,

0 ≤ ξmit , ζmit ≤ 1 ∀m ∈ E, ∀i ∈ Nm, ∀t,

and test the feasibility of a given y as follows:

• If ωf (y) = 0, y is feasible. Otherwise, if ωf (y) > 0, we generate a feasibility cut in the form

(79) ≤ 0, where γm±it , ηt, ξ

mit , ζ

mit , and τ±ij,t are the optimal solution to (FEA) and dmt is given

for solving (FEA), and we add the cut to (Master).

4.4 Optimality cuts

From (SUB), we observe that for a given first-stage decision variable y,

ω(y) ≥T∑t=1

(M∑

m=1

∑i∈Nm

(Lmi ymit γ

m+it − Um

i ymit γm−it ) +

M∑m=1

∑i∈Nm

f(ymit , ξmit , ζ

mit )

+M∑

m=1

dmtηmt −∑

(i,j)∈A


M∑m=1

∑i∈Nm

P∑p=1

(ymit αmpit ρmp

it )

(80)

for any feasible (γ, η, τ, ξ, ζ, ρ, d). Thus, cut (80) is valid for (Master). Meanwhile, cut (80) does

not necessarily support the epigraph of ω(y) at the boundary point (y, ω(y)), unless the exact

separation as shown in (78) is applied.

For large size problems, we use the following heuristic to generate the cut for a given feasible

solution y and ω(y) to (Master) as follows:

23

Step 1: Pick an extreme point d of D;

Step 2: Solve (SUB1), and store the optimal objective value ω1(y, d);

Step 3: Solve (SUB2), and store the optimal objective value ω2(y, γ, η, τ, ρ, ξ, ζ);

Step 4: If ω2(y, γ, η, τ, ρ, ξ, ζ) > ω1(y, d), go to Step 2, otherwise go to Step 5;

Step 5: If ω2(y, γ, η, τ, ρ, ξ, ζ) > ω(y), generate the corresponding optimality cut (80), and

add the cut to (Master).

4.5 Lower and upper bounds

We construct lower and upper bounds of the optimal value of (RPO), i.e., zR. In the separation

scheme, we track these bounds to estimate the solution quality. First, it is easy to observe that

if we fix d to be any point in D and solve (PO), then zPO ≤ zR, providing a lower bound for zR.

Second, let yPO be an optimal solution of (PO). Then, we will obtain an upper bound for the total

cost zR if we fix the variable y to yPO when solving (RPO).

Proposition 4 The objective value obtained by our heuristic separation approach provides a lower

bound for the optimal objective value of (RPO).

Proof: The conclusion immediately follows from the fact that heuristic separation algorithm cannot

guarantee to obtain an optimal solution for the subproblem.

Remark 2 If the budget uncertainty set in (RPO) is replaced by the cardinality uncertainty set as

shown in (14), then we can replace constraints (32) and (33) by the cardinality constraints

M∑m=1

|zmt| ≤ Γt, ∀t, andT∑t=1

M∑m=1

|zmt| ≤ Γ,

where zmt ∈ {−1, 0, 1} represents the demand hits the lower bound, the nominal value, and the

upper bound respectively. It can be represented as zmt = z+mt − z−mt, where z±mt ∈ {0, 1}. Then,

following the similar argument as shown in constraints (70) to (72), to linearize the bilinear terms

for the dual of the subproblem, we can obtain the optimality cuts by solving a mixed integer program,

and provide an exact separation approach.

24

Remark 3 If ramp-rate limit constraints are ignored, then X (y, d) =

T∩t=1

Xt(y, d), where

Xt(y, d) ={(x, ϑ) : Lm

i ymit ≤ xmit ≤ Umi ymit , ∀m ∈ E, ∀i ∈ Nm,

M∑m=1

∑i∈Nm

xmit =M∑

m=1

dmt,

−Cij ≤M∑

m=1

Kmij

( ∑n∈Nm

xmnt − dmt

)≤ Cij , ∀(i, j) ∈ A,

ϑmit ≥ αmp

it ymit + βmpit xmit , ∀m ∈ E,∀i ∈ Nm, ∀p

}.

Then, the second-stage problem can be decomposed into several subprograms in which each corre-

sponds to a specific time period, if the overall budget constraint is relaxed. Therefore, the computa-

tional time will be decreased significantly.

5 Computational Experiments

We present numerical experiments of the proposed algorithms in Sections 3 and 4. All the experi-

ments are performed by CPLEX 12.1, at Intel Quad Core 2.40GHz with 8GB memory.

5.1 Computational results for the robust unit commitment problem

In this subsection, we report the case described in Section 3. In the experiments, we assume there

are 30 generators and 24 time periods. The upper and lower bounds, Dut and Dℓ

t , of the demand

in each time period are generated by first setting Dt and Dt in the intervals [0, 40] and [0, 20]

respectively, and then letting Dℓt = (Dt − Dt)

+ and Dut = Dt + Dt. The budget restriction of the

uncertainty set DB isT∑t=1

(dt − Dt)/Dt ≤ π0.We control the conservatism of the robust optimization

approach by controlling π0. Note that π0 is between −T and T . When π0 = −T and Dℓt > 0 for

each t, the only possible scenario is that all demands are at the lower bounds. When π0 = T ,

the demand of each time period can take any value within the interval of the lower and upper

bounds. The computational results are summarized in Table 1. All the results are the average

of 10 random instances. We report the optimal objective value, the number of iterations of the

Benders’ approach, and the average computational time.

From this experiment, we first observe that as π0 increases, the uncertainty set becomes larger

and more scenarios are taken into consideration. The corresponding objective value increases as

25

π0 Objective value Number of iterations CPU time (sec)6 8920 27 40.19 8996 25 21.512 9303 24 28.015 9436 24 34.918 9509 28 45.6

Table 1: Computational results for the robust unit commitment problem

the problem becomes more conservative. Second, in these experiments, the size of the scenarios is

2432. However, in our algorithm, the optimal solution is achieved within 30 iterations, which shows

the effectiveness of our proposed approach.

5.2 Computational results for the power grid optimization problem

In this subsection, we present the numerical experiments for the general power grid optimization

problem. We first report the computational results for the case without ramp-rate limit constraints

and the overall budget constraint is relaxed. In this case, as shown in Remark 3, the subproblem can

be decomposed into T problems, which will reduce the computational time. We test a small size data

set that satisfies this condition to compare our bilinear heuristic approach with the exact separation

algorithm, to verify the effectiveness of our algorithm by its solution quality and computational

time. This one allows us to evaluate the optimality gap of our bilinear heuristic approach. Then,

we test a large size data set problem, where the solution quality is provided by the lower and

upper bounds we discussed in Section 4.5. The large size data set (containing 118 buses and 186

transmission lines) is a modified IEEE 118-bus system. For convenience, we normalize the weight

parameter πmt = 1, ∀m, ∀t. In addition, in the experiment, we use a four-piece piecewise linear

function to approximate the non-decreasing convex cost function.

The computational results on the small size data set are summarized in Table 2. In this ex-

periment, there are 16 buses, 10 generators, and 20 transmission lines in the power grid and the time

horizon is 24 hours. We letDℓmt = 0.9Dmt,D

umt = (1+UB%)Dmt, and πt = (1+Budget%)

∑Mm=1 Dmt.

We allow UB% and Budget% to vary from 0 to 20% in this experiment. Note that when Budget%

> UB%, the restriction of the total demand is actually relaxed, and thus the computational results

are the same as the case Budget% = UB%. Besides, since the demand changes as time goes by

within 24 hours, we assume that for each bus m its nominal demand Dmt changes in accordance

26

with the trend shown in Figure 1, which is given by the statistics from a US deregulated energy

market. As shown in Table 2, our bilinear approach converges to optimality with a small gap (less

than 0.05% in all the settings) in a reasonable and much shorter time than the exact separation

algorithm does. With the size of power grid increasing, the computational time of the exact sep-

aration algorithm grows exponentially. Hence, it becomes intractable for us to employ it in larger

power grids. On the other hand, our bilinear separation approach still performs very well with the

power grid swelling, and we use the lower and upper bounds discussed in Section 4.5 to verify its

solution quality.

UB%Budget%

5 10 15 20Opt. Val. Time (s) Opt. Val. Time (s) Opt. Val. Time (s) Opt. Val. Time (s)

5Exact 311635 297 311635 297 311635 297 311635 297Bilinear 311501 143 311501 143 311501 143 311501 143Gap (%) 0.04 0.04 0.04 0.04




Table 2: Computational results for the small size data set

The computational results on the large size data set are summarized in Table 3. In this ex-

periment, there are 118 buses, 33 generators, and 186 transmission lines in the power grid and

the time horizon is 24 hours. We assume the overall budget amount π0 = 90%∑T

t=1

∑Mm=1D

rmt +∑T

t=1

∑Mm=1D

ℓmt, the ramp-up and ramp-down rate limits are 50% of the maximum electricity

output of each generator i at each bus m, and the nominal demand follows the pattern illustrated

in Figure 1. In Table 3, we provide the following lower and upper bounds:

(1) Lower bound L1 is obtained by relaxing the transmission capacity and ramp-rate limits (i.e.,

solving RUC), in which case we can employ the algorithm presented in Section 3.

(2) Lower bound L2 is obtained by using our bilinear heuristic separation algorithm to solve (RPO).

(3) Upper bound U is a statistical upper bound. To obtain U , we first fix the first-stage decision to

27

0.2

0.4

0.6

0.8

1

1.2

1.4

0 5 10 15 20 25

Nom

inal

dem

and

Time period

Nominal demand pattern

Figure 1: Nominal demand pattern, with the average nomalized to 1

be the robust optimal solution in (RPO) obtained by our bilinear heuristic separation approach,

and then generate random large size demands to evaluate the performance of the robust optimal

solution and record the corresponding cost. Finally, we assign the largest cost to U . In the

experiment, the random demand at each bus m in time period t is generated from a truncated

normal distribution with mean (Dumt +Dℓ

mt)/2 and standard deviation (Dumt −Dℓ

mt)/4 in the

interval [Dℓmt, D

umt]. Additionally, the size of samples is 200, 000.

(4) Upper bound WC represents the worst-case value, obtained by solving (RPO) with the first-

stage decision fixed to be the optimal solution of the nominal problem. In other words, we first

solve (PO) with demands chosen to be their nominal values, and then solve (RPO) with its first-

stage decision fixed at the nominal optimal solution. Thus, WC estimates how the nominal

optimal solution performs under the worst-case scenario. In the experiments we found that

the nominal optimal solutions are infeasible to (RPO) for all the instances. This observation

indicates that it would be risky to make unit commitment decisions based only on the nominal

demand information. To compare the performance between these solutions, we introduce a

linear penalty cost function for any unsatisfied demands or transmission capacity/ramp-rate

28

limit violations. The unit penalty cost ϱt is set 20% higher than the sum of the maximal unit

fuel cost and the maximal start-up cost, i.e.,

ϱt = 1.2×(

maxm=1,2,··· ,M

maxi∈Nm

Smi + max

m=1,2,··· ,Mmaxi∈Nm

βmPit

), t = 1, 2, · · · , T. (81)

Correspondingly, we calculate the following gaps based on the lower and upper bounds:

(1) TC Gap = (L2 − L1)/L1×100%. It estimates the difference between the objective value of the

power grid optimization problem without considering the transmission capacity and ramp-rate

limit constraints (i.e., the case studied in Section 3) and the one of the general power grid

optimization problem (RPO) (i.e., the case studied in Section 4).

(2) Opt. Gap = (U − L2)/L2 × 100%. It estimates the optimality gap of our bilinear heuristic

approach.

(3) WC Gap = (WC − U)/U × 100%. It estimates the difference between the performance of the

robust optimal solution and the nominal optimal solution, when the linear penalty cost function

is introduced in calculating WC.

In the experiment, we first observe that the given power grid can tolerate less than a 25%

increase of demand in the worst case. When UB% = 25%, we have (RUC) feasible and (RPO)

infeasible. Thus, the failure of the power grid stems from the transmission line overload or ramp-

rate limit violation, although the power grid has sufficient generation capacity to satisfy the large

demand.

Then, we observe that statistically our bilinear heuristic approach provides a feasible solution

for all the instances with UB% ∈ [5, 20], and the computational results are shown in Table 3. We

can observe from the table that the optimality gap of our bilinear heuristic algorithm is less than

0.2% for all the instances. Meanwhile, we can observe that the computational time for the algorithm

is less than 300 seconds for any instance. This result shows that our bilinear heuristic algorithm

can provide an optimal solution that is very close to the optimal solution, within a short time.

Accordingly, it is sufficient to apply our bilinear heuristic approach to solve large size problems, in

order to obtain near-optimal solutions.

Third, from the WC Gap information, we can observe that when the demand fluctuation is

very restrictive (e.g., UB% ≤ 5% and Budget% ≤ 5%), the WC Gap is small; and with UB%

29

UB%Budget%

5 10 15 20

5

L1 1254672 1266449 1266449 1266449L2 1271248 1282667 1282667 1282667U 1272361 1283950 1283950 1283950WC 1309893 1321585 1321585 1321585Time (s) 159 90 90 90TC Gap (%) 1.32 1.28 1.28 1.28Opt. Gap (%) 0.09 0.10 0.10 0.10WC Gap (%) 2.95 2.93 2.93 2.93

10


15


20


Table 3: Computational results for the large size data set

30

and Budget% increasing, the WC gap grows fast. Note that the nominal optimal solutions are

infeasible for all the instances, and we have introduced a linear penalty cost function, with the unit

penalty cost ϱ defined in (81), to make up for any unsatisfied demand, transmission line overload,

or ramp-rate limit violation. In this sense, the WC gaps measure the relative infeasibility of the

nominal optimal solution in different scenarios. In view of that, we can claim that the extent

of infeasibility (and hence the corresponding making up costs) of the nominal optimal solution

grows fast as demand fluctuation increases. Further, we observe that apart from the case when

UB% = 25% discussed above, we can also see the importance of the transmission capacity and

ramp-rate limits in the analysis on the feasibility of the nominal optimal solution. If we depict

the evolution of total demand under different Budget% scenarios, versus the power grid generation

capacity in 24 hours as shown in Figure 2, we observe that the power grid generation capacity is able

to cover at least a 5% increase in total demand, while is not after that. This observation implies

that the nominal optimal solution is infeasible when Budget% ≤ 5% because of transmission line

overload or ramp-rate limit violation, and it is infeasible when Budget% > 5% because of demand

unfullfillment, as well as transmission and ramp-rate limit violations.

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

0 5 10 15 20 25

Dem

and

Time period

Demand fluctuation

CapacityNominal

5% increase10% increase15% increase20% increase25% increase

Figure 2: Demand fluctuation versus the power grid capacity

31

Fourth, we can observe from Table 3 that the TC Gaps are smaller than 4% for all the instances.

It indicates that we can utilize the solution approach for (RUC) described in Section 3 to provide

a good approximation on the total cost for the general robust power grid optimization (RPO)

problem.

Finally, our computational results indicate that as the demand fluctuation increases, the cor-

responding objective value increases accordingly. For instance, the objective value increases by

around 18% as the uncertainty set chances from UB% = 5% and Budget% = 5% (the correspond-

ing objective value is $1,271,248) to UB% = 20% and Budget% = 20% (the corresponding objective

value is $1,500,437).

6 Conclusion

In this paper, we provided one of the first studies on the robust power grid optimization problem.

In our approach, we addressed the case in which the demand or supply at each bus in each oper-

ating hour may be uncertain. Instead, they are within an uncertainty set in two different forms:

cardinality uncertainty set and polyhedral uncertainty set. We developed solution approaches to

address each type of uncertainty set. Our computational results indicate that the robust optimiza-

tion approach can provide a much better solution as compared to the nominal model approach,

in terms of total cost for the worst-case scenario. More importantly, our proposed approach will

keep the power grid much more reliable than the traditional nominal model approach. In general,

this approach will provide an ISO or utility company (for the regulated market) an alternative ap-

proach to address demand and, especially, wind farm output uncertainty. Finally, as a byproduct,

our approach addressed a set of budget constraints to describe an uncertainty set, instead of just

a single budget constraint in previous studies. Our linearization technique gives insights of this

general polyhedral uncertainty set, and our approach provides the first study on deriving an exact

separation algorithm for this type of uncertain set.

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