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Combined cycle plants (CCs) are broadly used all over the world. The inclusion of CCs into the optimal resource scheduling causes difficulties because they can be operated in different operating configuration modes based on the number of combustion and steam turbines. In this paper a model CCs based on a mixed integer linear programming approach to be included into an optimal short term resource optimization problem is presented. The proposed method allows modeling of CCs in different modes of operation taking into account the non convex operating costs for the different combined cycle mode of operation.

The gas-turbine combined cycle plant has been used extensively in power generation, representing the great majority of new generating unit installations across the globe. Combined cycle (CC) technology is now well established, becoming one of the most effective energy conversion technology at present [1,2].

The progress on CC generation technology allowed improving their thermal efficiency up to 50% approximatelly.

In addition, CCs present other advantages such as better environmental performance, reducing greenhouse gases, short construction lead time and low capital cost to power ratio. Moreover, the price of natural gas, which is the primary fuel used for combined cycle plants, dropped.

As an example of CCs use evolution,

Despite of their benefits, the utilization of CCs created new challenges. One of these challenges is the inclusion of combined cycle plants into the unit commitment problem (UC). Modeling of CCs for UC studies is quite difficult due to the tight iteraction between the gas turbine and the steam turbine generating units. These units have different operating modes; each of the operating modes has parameters such as limits or incremental heat rate that can differ considerably from each other depending on which mode is operating at the time. Therefore, the problem needs to be expanded to determine which operating mode the combined cycle units have to be in operation at each time.

Several researches have been conducted to include detailed CC model into the UC related problems. A flexible modeling approach in order to take the multiple possible configurations into account is described in [

Reference [

In [

Combined cycle models were also included into economic dispatch problems based on Genetic Algorithms, Evolutionary Programming, and Particle Swarm techniques [

Recently, advances on mixed integer programing techniques (MIP) allow applying this technique to very largescale, time-varying, non-convex, mixed-integer modeling and optimization, such as unit commitment problems.

A mixed integer programing approach that allows a rigorous modeling to obtain the optimal response of thermal unit to an electricity spot market is proposed in [

An extension of this work is presented in [

Furthermore, a MIP solution for solving the PJM unit commitment problem is described in [

The first formulation of a combined cycle unit model using MIP is presented in [

Another alternative approach that model combustion turbines and steam turbines individually is presented in [

According to what has been said above, it is evident that CC has acquired great relevance, particularly the development of models used for UC problems.

The aim of this paper is to develop a general and accurate CC model purposely designed with the idea that it could be easily included into any optimal short-term resource optimization problem based on a mixed integer linear programing approach. The proposed model, which has taken into account studies previously done by different researchers, includes non-convex cost curves and exponential start up cost curves for each operational CC mode, keeping the number of constraints to the minimum.

The paper is organized as follows; first the combined cycle model based on different modes of operation is presented, then the unit commitment problem formulation based on mixed integer linear programming approach including CC units is described. After that, a numerical example is given, finally presents the most important conclusions of the paper.

Typically, a combined cycle unit comprises of several combustion turbines (CT) and several steam turbines (ST), the waste heat from the CTs is used to produce the steam to generate additional power using STs, and this process enhances the efficiency of electricity generation.

Depending on the number of CTs and STs, combined cycle units can operate in different configurations, also known as modes. Each configuration is determined based on a possible combination of CTs and STs, having a determined generating region and an incremental heat rate curve. Configurations or modes with STs are more efficients; however, since the modes are restricted due to the generation region may not be more efficient for a particular load and a particular simulation period.

As an illustration, considering a combined cycle unit with two CTs and one STs, the related possible configurations are shown in

In addition, combined cycle units have other constraints such as transition between modes and minimum time on and off for each of the modes. Furthermore, there are modes that, for a particular period, can not be eligible, for example, a CT may need to be in service for several hours prior to turn on an associated ST.

Considering all these issues, a mixed integer formulation for combined cycle plants compatible with general MIP software is described next.

t: Index for simulation hours.

b: Index for cost curve segments.

n: Index for startup cost curve.

cc: Index for combined cycle units.

m: Index for combined cycle modes.

T: Total number of simulation hours.

G: Total number of thermal units.

B: Total number of segments for production cost curve.

N: Total number of startup cost curve steps.

CC: Total number of combined cycle units.

M: Total number of combined cycle modes.

Cp_{g}_{,t}: Production cost for unit g at hour t [$/h].

Cp_{m}_{,t}: Production cost for mode m at hour t [$/h].

Cup_{g}_{,t}: Startup cost for unit g at hour t [$].

Cup_{m}_{,t}: Startup cost for mode m at hour t [$].

p_{g}_{,t}: Active generation for unit g at hour t [MW].

p_{m}_{,t}: Active generation for mode m at hour t [MW].

r_{g}_{,t}: Active reserve contribution of unit g, hour t [MW].

r_{m}_{,t}: Active reserve contribution of mode m, hour t [MW].

δ_{b}_{,m,t}: Active generation for segment b, mode m, hour t [MW].

: Counter of hours off for mode m, hour t.

dv_{m}_{,t}: Slack variable for the discretization of the startup cost function of mode m, hour t.

u_{g}_{,t}: Binary state variable for unit g, hour t.

u_{m}_{,t}: Binary state variable for mode m, hour t.

s_{m}_{,t}: Startup variable for mode m, hour t.

z_{m}_{,t}: Shutdown variable for mode m, hour t.

j_{b}_{,m,t}: Activation variable for segment b, mode m, hour t.

w_{n}_{,m,t}: Binary variable which activates the step n of the stepwise startup cost of mode m at hour t.

y_{m}_{,t}: Startup variable for transitions to mode m, hour t.

D_{t}: System demand at time t [MW].

R_{t}: System spinning reserve requirement at time t [MW].

c_{m}: Fixed cost for mode m [$/h].

F_{b}_{,m}: Slope for segment b, mode m [$/MWh].

: Maximum capacity for mode m [MW].

P_{m}: Minimum capacity for mode m [MW].

P_{b}_{,m}: Upper limit for segment b, mode m [MW].

K_{n}_{,m}: Cost for startup cost step n, mode m [$/h].

STH_{m}: Maximum number of hours that mode m can be off [h].

MU_{m}: Minimum up time for mode m [h].

MD_{m}: Minimum down time for mode m [h].

: Number of hours mode m has been off at t = 0 [h].

: Number of hours mode m has been on at t = 0 [h].

The unit commitment problem (UC) can be formulated as a minimization problem which main objective is to determine the generation dispatch to supply the demand and reserve requirements at minimum cost over a period of time. Mathematically can be represented as follow:

Subject to:

Combined cycle plants can be included into the general UC formulation by modifying the cost function and adding new constraints. The changes needed for the two components of the objective function represented by Equation (1), are described below.

Considering the incremental cost function represented by the piecewise function of

Subject to:

For all the segments inside the curve the constraints are:

For each mode, MW capacity restrictions apply:

Finally, the following restrictions related to the MW per segment, and the binary conditions of the index respectively are:

For the startup cost model, the transition cost not only from the off state but between modes needs to be considered.

First, the counter that takes into account the hours that the mode m has been off is represented and formulated as follow:

Then, based on the discretized approximation, a mathematical formulation for the start up cost is included per mode m and per simulation period t:

Subject to:

The constraints that relate the slack variable dv_{(m,t)} with the start up transition binary variable y_{(m,t)} and the startup segment activation w_{(n,m,t)} are:

These constraints are related to the logical variables as follow:

Then, the formulation explained in Subsection 3.1, can be reformulated to include combined cycle plants:

Subject to:

And the restrictions developed in this subsection and 3.3. In addition to these constraints which are related to the production and start up cost, further constraints are needed. These constraints are related to the relationship between configuration status, and on/off conditions.

For all simulation periods t, only one CC mode can be selected:

The restriction for the transition between mode “from” m_{f} and mode “to” m_{t} can be represented as follow:

_{m} for the 2CT-1ST case. This set is defined based on the state space transition diagram of

The modes also have restrictions due to the minimum time on and minimum time off that the generator needs to remain before the transition to another mode, these restrictions can be formulated as:

For the initial period, the number of periods the mode is on or off need to be considered:

Based on the general UC problem formulation, it was modified to include the combined cycle model into the MIP formulation, next section discusses the results considering a typical power system and the addition of combined cycle modeled as explained in this section.

Numerical results obtained by the proposed solution model are shown in this section.

The MIP problem is solved using GAMS, and the optimization engine selected is CPLEX 12.0. Tables 4 and 5 illustrate the CPLEX option setup.

The test system presented in [

teristics are modified in order to model them as a combined cycle model. This allows to study the differences between a typical CC modeled as an equivalent thermal plant and modeled taking into consideration the different modes of operation.

Then, a 20 generating system is built by duplicating the original system following the same pattern, the results for this system are shown in

In addition, the calculations and comparisons are repeated for different load profiles,

The development approach gives optimal, mutually exclusive commitment of the combined cycle plant configurations. Results illustrate the impact of explicitly model of combined cycle units on minimizing the cost of supplying the load. The additional computer time required to schedule a combined cycle unit with multiple configurations depends on the number of configurations, the number of transitions, the minumum up times, the transition times and the load profile.

Nowadays, the use of combined cycle plants becomes popular due to their advantages. Therefore, it is very important to have an accurate model to include combined cycle plants into a unit commitment problem. This paper presents a combined cycle model for optimal resource optimization problems based on mixed integer linear programming approach. Based on previous general models developed for thermal plants in [

incorporation of feasible and non convex constraints which are present in this type of studies and very difficult to solve.