A Chance-Constrained Unit Commitment With an $n-K$ Security Criterion and Significant Wind Generation

This paper presents a new approach for the joint energy and reserves scheduling and unit commitment with n-K reliability constraints for the day-ahead market. The proposed method includes a novel n-K criterion where demand must be met with a specified probability under any simultaneous loss of K generating units. A chance-constrained method is proposed with an α-quantile measure to determine the confidence level to meet the demand under K simultaneous contingencies. The chance-constrained optimization problem is recast as a mixed integer linear programming optimization problem. Wind and demand uncertainty are included into the model. The methodology proposed is illustrated with several case studies where the effect of increasing wind power penetration is analyzed showing the performance of our model.

I. Nomenclature

The mathematical symbols used throughout this paper are described below. The operators and are used as expectation and probability of a random function, respectively. The bold-typed symbol is defined as a random function, and represents a particular realization or scenario. Index

Generating unit.

Time period scheduling interval.

Index for the scenario realization .

Random Functions

Demand in period depending on the random set .

Wind power generation in period depending on the random set .

Net load in period depending on the random set .

Parameters

Number of unavailable generator units.

Total number of generator units.

Confidence level of the ELNS/EWS probabilistic constraint in period . Typically takes values between 0.95 and 1.

ELNS/EWS limits for period .

Parameters related with for the whole time horizon.

Ramp-down/ramp-up limit for unit .

Shutdown/start-up ramp limit for unit .

Minimum/maximum power output for unit .

Scheduled down/up reserve upper bound for unit .

Demand/wind production/net load expected value in period .

Demand/wind production/net load standard deviation in period .

Fixed/linear cost for unit .

Linear down/up reserve cost for unit .

Unit cost for the load energy not served in $/MWh.

Unit cost for the wind energy spilled in $/MWh.

Outage probability for unit .

Probability of simultaneous generating units outages.

Binary Variables

On/off status of unit in period .

Equal to 0 if unit has a contingency in period , 1 otherwise.

Positive Variables

Scheduled energy for unit in period .

Scheduled down reserve for unit in period .

Scheduled up reserve for unit in period .

Auxiliary variable.

Auxiliary variable.

Lagrange multipliers associated with the worst-contingency problem.

ELNS

Expected load not served.

EWS

Expected wind spillage.

Table I Energy and Reserve Dispatching with Security Criteria Models

Author and Year	Reference	Wind Power	Multiperiod	Network	Outages criteria	Security variables	Stochastic
Bouffard et al., 2004	[17]	no	no	no	any generating unit can fail		contingencies
Arroyo et al., 2005	[8]	no	no	yes	umbrella of units and lines		contingencies
Bouffard et al., 2005	[9]	no	no	yes	umbrella of units		contingencies
Bouffard et al., 2005	[10], [11]	no	yes	yes	umbrella of units and lines		contingencies
Bouffard et al., 2008	[14]	yes	yes	no	umbrella of units		contingencies, demand and wind
Street et al., 2011	[15]	no	no	no	any generating unit can fail		none
Street et al., 2011	[16]	no	yes	no	any generating unit can fail		none
Restrepo et al., 2011	[12]	yes	yes	no	any generating unit can fail		demand and wind
Karangelos et al., 2012	[13]	no	yes	yes	umbrella of units and lines		contingencies
Current paper		yes	yes	no	any generating unit can fail	*	demand and wind

*

Independent and joint security criteria with an confidence level.