1 Introduction

Recently, the number of combined-cycle gas turbines (CCGTs) in power systems has been substantially increasing because of their high efficiency, operational flexibility, lower natural gas prices, and fast response to mitigate uncertainty with increasing penetration levels of variable renewable generation [1]. A CCGT unit is composed of multiple combustion turbines (CTs) and steam turbines (STs) that can operate in different modes corresponding to different combinations of these turbines. Some steam turbines use the exhaust gas of CTs to generate electricity, which leads to higher efficiency compared to the traditional thermal units.

However, it is challenging to model the operational flexibility of CCGTs in the UC problem because the scheduling of CCGT units needs to decide the on/off status and power output of each CT and ST unit at every time interval. In practice, independent system operators (ISOs) use three typical methods to model CCGTs [2,3,4]: the aggregated modeling approach, in which the whole CCGT unit is modeled as a pseudo-thermal unit, ignoring all different operating configurations; the configuration-based approach, in which each commitment combination of CTs and STs is a particular configuration; and the component-based approach, in which each individual CT and ST unit is modeled separately. The dependency among CTs and STs is represented by a group of MW-steam constraints for CTs and steam-MW constraints for STs.

Although the aggregated model is simple and computationally efficient, the commitment results cannot be directly interpreted to the status of each CT and ST, and the solutions from this method might be physically infeasible [5]. The configuration-based method is deployed by several ISOs—such as California Independent System Operator, the Electric Reliability Council of Texas, and Midcontinent System Operator [6]—because the bidding curve framework is convenient, and the commitment results directly correspond to the status of CTs and STs; however, the physical limitations of configuration modes—such as minimum online/offline time and ramping rates—are approximated from the parameters of the CT and ST components. In market operations, it is difficult for CCGTs to generate these parameters accurately, which leads to inaccurate results. Although several improvements are available for the configuration-based model—such as the edge-based model [7] and tight model [8]—the computational burden for actual ISO systems is still high. In the component-based method (CBM), the CTs and STs are modeled individually, and all physical constraints of CTs and STs are respected; however, it is not likely to provide bidding curves of STs because they generate electricity from the exhaust gas [9] which depends on the status of the CTs.

This paper proposes a hybrid component and configuration model for CCGTs in the ISOs’ UC problem. The proposed model overcomes the above disadvantages of the current configuration-based and component-based models. In general, the advantage of the proposed hybrid method is modeling the day-ahead offer submission for CCGT units while respecting the physical constraints for each individual CT and ST component.

The rest of this paper is organized as follows. Section 2 proposes the mapping formulation from component status to configuration modes. Section 3 presents the UC model, including CCGT units and traditional thermal units. Section 4 presents a case study comparing the proposed method to the component-based model. Finally, Section 5 presents conclusions.

2 Mapping of component status to configuration mode

The components status of a CCGT unit can be mapped to a unique configuration mode (CM) in the following manner:

$$CM_{m} = \prod {V_{1} V_{2} \cdots V_{s} \cdots V_{S} }$$
(1)

where S is the number of components in this CCGT unit; Vs (s=1~S) represents the status of the s-th component in the m-th configuration mode (CMm). Here, assume that vs is the binary variable (i.e., an unknown binary variable for the UC problem) representing the commitment status of the s-th component. Then, if in the m-th mode, the s-th component (CT or ST) is online, Vs is vs; otherwise, Vs is 1-vs if the s-th component is offline in the m-th mode. Because both vs and 1-vs are binary variables, CMm in (1) can be bounded by these linear constraints:

$$\left\{ \begin{array}{l} CM_{m} \le V_{1} \hfill \\ CM_{m} \le V_{2} \hfill \\ \vdots \hfill \\ CM_{m} \le V_{S} \hfill \\ CM_{m} \ge \sum\limits_{s = 1}^{S} {V_{s} - (S - 1)} \hfill \\ \end{array} \right.$$
(2)

Here CMm can be defined as a continuous variable because Vs is binary and, because of the constraints in (2), CMm can be only 0 or 1. Therefore, adding CMm does not increase the number of binary variables in the UC problem, and the total number of binary variables will be less than those in the configuration-based model.

For instance, if there are 2 CT units and 1 ST unit, there will be 8 commitment combinations in the configuration model which leads to 8 configuration model binary variables at every time interval. But with the proposed method, the number of binary variables is 3 for three units. Table 1 shows the relationship between the configuration modes and component status for two CTs and one ST. Due to operational constraints (e.g. the ST cannot run unless at least one CT is running) the configuration modes do not explore the full set of commitment combinations.

Table 1 Mapping between modes and commitment for two CTs and one ST

The relationship between the configuration modes and component commitments is formulated as, for example:

$$CM_{1} = (1 - v_{1} )(1 - v_{2} )(1 - v_{3} )$$
(3)

which means only when v1, v2 and v3 are 0, CM1 is active.

CM2 to CM7 can be formulated in a similar manner. Because v1, v2, v3, 1 − v1, 1 − v2, and 1 − v3 are binary variables of the components’ status, CM1 to CM7 are bounded by the linear constraints in (2). For example, CM1 is bounded by the following constraints:

$$CM_{1} \le 1 - v_{1}$$
(4)
$$CM_{1} \le 1 - v_{2}$$
(5)
$$CM_{1} \le 1 - v_{3}$$
(6)
$$CM_{1} \ge 1 - v_{1} + 1 - v_{2} +1 - v_{3} - 2$$
(7)

Additional constraints on the configuration mode are formulated because the mode can be transitioned to only a limited number of possible modes, as shown in Fig. 1. For example, Mode 1 can only reach {Mode 1, Mode 2, Mode 3, Mode 4} in the next time interval. Therefore, the following constraint can be added:

$$CM_{i,t - 1,m} \le \sum\limits_{{n \in M_{m} }}^{{}} {CM_{i,t,n} }$$
(8)

where Mm is the possible next time interval configuration mode set of Mode m.

Fig. 1
figure 1

State transition graph for two CTs and one ST

If there are 2 CT units and 1 ST unit, the total number of configurations is 8, but the mode with only the ST unit online is impossible. To eliminate the impossible modes (here CM8) with only STs online, constraint (9) is added. The bilinear terms are linearized by (2) as follows:

$$CM_{8} = (1 - v_{1} )(1 - v_{2} )v_{3} = 0$$
(9)
$$CM_{8} \le 1 - v_{1}$$
(10)
$$CM_{8} \le 1 - v_{2}$$
(11)
$$CM_{8} \le v_{3}$$
(12)
$$CM_{8} \ge 1 - v_{1} + 1 - v_{2} + v_{3} - 2 = v_{3} - v_{1} - v_{2}$$
(13)

3 Unit commitment problem formulation

The general formulation of the UC problem including both traditional thermal and CCGT units is presented below.

3.1 Objective function

The objective of the UC problem includes the operational costs of traditional thermal and CCGT units represented by their generation cost and start-up and shutdown costs, as follows:

$$\hbox{min} \sum\limits_{t \in T}^{{}} {} \sum\limits_{i \in g}^{{}} {\left( {SU_{i} \cdot u_{i,t} + SD_{i} \cdot w_{i,t} + cp_{i,t} } \right)}$$
(14)

In (14), the start-up and shutdown costs of a CCGT unit can be calculated as the summation of the costs of its components, as for a set of traditional thermal units, or calculated according to its configuration [8]. The production cost of the CCGT units at each time interval is equal to the total production costs of all configuration modes at each time interval:

$$cp_{i,t} = \sum\limits_{m \in M}^{{}} {cp_{i,t,m} }$$
(15)

The cost of each configuration mode \(cp_{i,t,m}\) is formulated as:

$$cp_{i,t,m} \ge \mathop \alpha \nolimits_{i,m}^{n}\,\cdot\,CM_{i,t,m} + \mathop \beta \nolimits_{i,m}^{n}\,\cdot\,G_{i,t,m}$$
(16)

where \(\mathop \alpha \nolimits_{i,m}^{n}\) and \(\mathop \beta \nolimits_{i,m}^{n}\) are the coefficients of the bidding curve of the mth configuration mode.

The generation amount of a CCGT unit is the sum of all the generation of its modes, as shown in (17), considering that only one mode at a time can generate:

$$G_{i,t} = \sum\limits_{m \in M}^{{}} {G_{i,t,m} }$$
(17)

Each mode also has its own generation output limits, as:

$$\mathop G\nolimits_{i,m}^{\hbox{min} }\,\cdot\,CM_{i,t,m} \le G_{i,t,m} \le \mathop G\nolimits_{i,m}^{\hbox{max} }\,\cdot\,CM_{i,t,m}$$
(18)

3.2 Constraints for the single unit

The constraints for traditional thermal units are like those in [9] and are presented as follows for the sake of completeness:

$$u_{i,t} + w_{i,t} \le 1$$
(19)
$$v_{i,t} - v_{i,t - 1} \le u_{i,t} - w_{i,t}$$
(20)
$$\sum\limits_{{\tau = t - \mathop T\nolimits_{{i,{\text{min,up}}}} + 1}}^{t} {u_{i,t} } \le v_{i,t}$$
(21)
$$\sum\limits_{{\tau = t - \mathop T\nolimits_{{i,{\text{min,dn}}}} + 1}}^{t} {w_{i,t} } \le 1 - v_{i,t}$$
(22)
$$\mathop G\nolimits_{i,t}^{\hbox{min} } v_{i,t} \le G_{i,t} \le \mathop G\nolimits_{i,m}^{\hbox{max} } v_{i,t}$$
(23)
$$G_{i,t} - G_{i,t - 1} \le R_{i}^{U} v_{i,t - 1} + R_{i}^{SU} u_{i,t}$$
(24)
$$G_{i,t - 1} - G_{i,t} \le \mathop R\nolimits_{i}^{D} v_{i,t} + \mathop R\nolimits_{i}^{SD} w_{i,t}$$
(25)
$$G_{i,t} + SR_{i,t} \le \mathop G\nolimits_{i,t}^{\hbox{max} } v_{i,t}$$
(26)
$$SR_{i,t} \le \mathop {SR}\nolimits_{i,t}^{\hbox{max} }\,\cdot\,v_{i,t}$$
(27)
$$v_{i,t} ,u_{i,t} ,w_{i,t} \in \left\{ {0,1} \right\}$$
(28)

where \(v_{i,t}\), \(u_{i,t}\) and \(w_{i,t}\) are binary variables for the on/off, start-up and shutdown status; \(G_{i,t}\), \(\mathop G\nolimits_{i,t}^{\hbox{min} }\) and \(\mathop G\nolimits_{i,t}^{\hbox{max} }\) are the actual, minimum, and maximum power output of unit i at time t; \(\mathop R\nolimits_{i}^{U}\), \(\mathop R\nolimits_{i}^{D}\), \(\mathop R\nolimits_{i}^{SU}\) and \(\mathop R\nolimits_{i}^{SD}\) are the ramping-up, ramping-down, start-up ramping, and shutdown ramping capabilities; \(SR_{i,t}\) and \(\mathop {SR}\nolimits_{i,t}^{\hbox{max} }\) are spinning reserve capacity and the spinning reserve limit, respectively. Note that the configuration mode variables are applied only to CCGT units.

3.3 System energy balance, reserve constraints, and transmission constraints

Typically, system constraints include the system power and load balance, system reserve and transmission line limits:

$$\sum\limits_{i \in g}^{{}} {G_{i,t} } - \sum\limits_{b \in SB}^{{}} {D_{b,t} } = 0$$
(29)
$$\sum\limits_{i \in g}^{{}} {SR_{i,t} } \ge SR_{t}$$
(30)
$$- L_{l} \le \sum\limits_{i \in Lg}^{{}} {GSF_{l - i}\,\cdot\,G_{i,t} } - \sum\limits_{b \in Lb}^{{}} {GSF_{l - b}\,\cdot\,D_{b,t} } \le L_{l}$$
(31)

where \(GSF_{l - i}\) is the generation shift factor of bus i to line l; g is the generator set; Lg is the generator set for line l; and Lb is the demand set for line l.

4 Case studies

Because the CBM is currently the most accurate model of CCGTs in the UC problem, the CBM from [9] is employed as a benchmark to validate the effectiveness and efficiency of the proposed hybrid model. This section compares the proposed model’s performance on a modified IEEE 118-bus system. The test system has 54 traditional thermal units and 12 CCGT units (4001 to 4012), and the system data can be found in [7]–[9]. The models are solved by GUROBI 7.0.1 on a laptop with 2.40 GHz computer processing units and 12 GB RAM. The results of the two models are listed in Table 2.

Table 2 Computational results

Table 2 shows that the total cost of the hybrid model is very close to that of the CBM in both cases, “with” and “without” transmission constraints. The computational time of the hybrid model is less than 5% longer than the CBM for the case considering transmission constraints. This table shows that the total operation cost of the proposed hybrid model is within 0.6% of that of the CBM, which means that the proposed hybrid model accurately reflects the true cost of the system. Although the computational time of the hybrid model is slightly more than the CBM, the proposed model can be used to bid CCGT in ISO markets.

The CCGT schedules based on the two methods are very close, as shown in Table 3. The differences in the commitment status between the hybrid model and the CBM are because of the bidding curve approximation. Note, in Table 3, that only the results at time intervals 1 and 2 are listed, and the schedules are the same (Mode 7) for the rest of the time periods.

Table 3 Mode results from hybrid model and CBM

In the proposed method, the CCGT status changes from Mode 2 to Mode 5 between the first two intervals; whereas in the CBM, the status for several CCGT units changes from Mode 4 to Mode 7. The cost of Mode 4 is higher than that of Mode 2 under the same load level, but Mode 7 is cheaper than Mode 5. Therefore, the commitment cost results from the proposed method and CBM are close.

5 Conclusion

This paper proposed a hybrid component and configuration model for CCGT units in the UC process. The mapping between component status and configuration mode does not increase the number of binary variables in the UC formulation and only increases the total computation time slightly when compared to the component-based model, which is generally considered the most accurate traditional model for CCGTs in UC. Thus, the proposed hybrid model can more accurately represent the CCGT bidding and modeling issues encountered by ISOs in electricity markets, while still allowing an efficient UC modeling and computation.