Solving Nonlinear Single-Unit Commitment Problems with Ramping Constraints

Abstract

We present a dynamic programming algorithm for solving the single-unit commitment (1UC) problem with ramping constraints and arbitrary convex cost functions. The algorithm is based on a new approach for efficiently solving the single-unit economic dispatch (ED) problem with ramping constraints and arbitrary convex cost functions, improving on previously known ones that were limited to piecewise-linear functions. For simple convex functions, such as the quadratic ones typically used in applications, the solution cost of all the involved (ED) problems, consisting of finding an optimal primal and dual solution, is O n 3 . Coupled with a special visit of the state-space graph in the dynamic programming algorithm, this approach enables one to solve (1UC) with simple convex functions in O n 3 overall.

Robust unit commitment problem with demand response and wind energy

“…. , T is known, we present next Wind-UC Model by integrating wind energy into classical day-ahead UC-model (Cerisola et al (2009), Frangioni and Gentile (2006), Takriti et al (2000)), which also serves as the nominal model to its robust counterpart.…”

“…Constraints (4) stands for start up operation (Cerisola et al (2009)), that is, unit i is started up at the beginning of period t if its status is off at time t − 1 and is on at time t. Constraint (5) illustrates the generation capacity of each unit (Frangioni and Gentile (2006)), where l i and u i stand for the minimum and maximum output of unit i respectively. Constraint (6) ensures that the customer demand d t should be satisfied.…”

“…Constraint (6) ensures that the customer demand d t should be satisfied. Constraints (7) and (8) are ramping up/down limits in unit commitment system (Frangioni and Gentile (2006)). …”

“…A set of I = 36 thermal units from an IEEE 118 system (Ma and Shahidehpour (1999)) are used in all experiments and some parameters are adjusted for our models: (i) the fuel cost is linear in stead of quadratic form, and (ii) the ramping up/down parameters are adjusted according to the rule in Frangioni and Gentile (2006). The fixed price before response is 15 and the same L = 10 levels price and demand increase/decrease are pre-defined at each time (the discrete increase/decrease levels are sampled from experiment results in Faruqui and Sergici (2009)).…”

“…However, the scheduling operation which is known as the classical unit commitment problem (UC) is hard to solve because complex and critical technical constraints are to be subjected such as (i) generation capacity limit of each unit, (ii) minimum time period a unit should be on(off) once it is start up(shut down), (iii) maximum rate that the generation level can ramp up or down from this time period to the next. Many deterministic and stochastic UC problems as the extension of the classical UC problem have been solved by various approaches during the past two decades: Frangioni and Gentile (2006) solved the deterministic single unit commitment problem using dynamic programming approach; While Takriti et al (2000) presented a stochastic model that incorporates fuel constraints and electricity spot prices and used Lagrangian relaxation and benders decomposition to solve it; More previous research about UC problems could be referred to a survey by Padhy (2004).…”