Due to the advantages of fast response, high efficiency, and cleanliness, gas turbine units are becoming a key component in building new-type power system in China. In order to cope with the load and frequency fluctuations caused by the large-scale integration of new energy power sources such as photovoltaic power and wind power into the power grid, gas turbine units need to have a strong primary frequency modulation (PFM) capability, which are of great significance for the safe operation and frequency stability of the power grid.

In order to further enhance the PFM capability of gas turbine units, many scholars have conducted extensive research in terms of control strategy designs, control logic optimizations, and other aspects. In reference [1], a medium-pressure heating throttling scheme was proposed to optimize the PFM logic in order to increase the margin of load regulation for gas turbine units. In reference [2], a significant improvement in the response speed of the unit’s PFM was achieved by optimizing the speed control mode of the main control system, reducing the frequency modulation dead-zone and increasing the frequency modulation amplitude. In reference [3], the response speed of the unit to PFM commands was improved by adding locking logic and modifying frequency difference signals. For the problem that the response index of the PFM cannot meet the requirements of the power grid, an optimization logic scheme based on feedforward limiting and compensation for power closed-loop was proposed in reference [4] to improve the PFM capability.

The actual control logic of PFM in gas turbines is mostly based on the Proportional-Integral (PI) control strategy [5], due to the advantages of PI controllers, such as simple structure, excellent performance, simple implementation, and clear parameter meanings [6]. In addition, in order to further enhance the PFM capability of gas turbines, a new method based on Active Disturbance Rejection Control (ADRC) was proposed in reference [7], where ADRC parameters are optimized through a multi-objective genetic algorithm to achieve the improvement of gas turbine PFM capability. In addition, control strategies based on neural networks are also studied in the PFM system of gas turbines [8]. Due to the large amount of computation requirements for advanced control strategies, their engineering implementation still faces certain challenges. However, there is still no lack of potential for large-scale engineering applications given the arithmetic capabilities of current gas turbine control platforms [9]. For a long time in the future, the PI control strategy will still occupy the dominant position in the PFM system of gas turbines [10]. To enhance the control performance of PI in the PFM system of gas turbines, it is necessary to optimize the structure and parameter tuning of PI controllers. The DDE-PI controller has received increasing attention due to its adjustable parameters with feedforward coefficients, which can effectively balance the contradiction between tracking and anti-disturbance performance of closed-loop systems [11]. The DDE-PI controller further enhances the robustness of the closed-loop system while retaining the advantages of the classical PI strategy, and has been well applied in the super-heated steam temperature system [12] and the high-pressure heater water level system [13]. Running data show that DDE-PI has strong application potential.

In this study, a DDE-PI control strategy for the PFM system of a gas turbine is designed. Firstly, the composition of the MARK V heavy-duty gas turbine PFM system is introduced, and the control characteristics of the system are analyzed. Then, the DDE-PI control strategy is introduced, the calculation of the parameter stability domain is provided, and the practical parameter tuning procedure is summarized. Next, the DDE-PI control strategy proposed in this study is applied to the PFM control of MARK V heavy-duty gas turbine units. The simulation results show that the gas turbine unit using the method proposed in this study can ensure optimal PFM capability and a strong ability to cope with system uncertainty. Finally, the work of this study is summarized.

A frequency modulation system of an MS6001B heavy-duty gas turbine based on the MARK V control system was introduced in reference [14]. Taking the speed control system, fuel control system, and acceleration control system into consideration, the structural diagram of the PFM control system is shown in Fig. 1, where the acceleration control speed is generally kept constant in the control system. Reference [14] provides the physical meanings and corresponding values of some model parameters in Fig. 1, as shown in Table 1. The models of the other links are as follows:

(1)GT(s)=3.3s+1450s (2)Gz1(s)=12.5s+1 (3)Gz2(s)=215s+1+0.5

In addition, the controllers for speed and fuel in the system are G1(s) and G2(s), respectively. G1(s) and G2(s) can choose many control strategies, such as PI and ADRC.

The system model is built based on a liquid-fueled circulating single-shaft gas turbine, with a rated speed of 5100 r/min and rated power of 359 MW. The exhaust temperature, rated inlet temperature, and speed governing droop of the gas turbine unit are 550°C, 15°C, and 4%, respectively [5].

From Fig. 1, the output of the fuel control system of the gas turbine goes through a pure lag link and an inertia link. Combined with the integration effect, the speed of the gas turbine is obtained and serves as the output value of the speed control system, which is algebraically calculated as the set value of the fuel control system. This means that there is a coupling effect between the speed control system and the fuel control system due to their interaction. Note that Gf and n are fuel quantity and speed, respectively, and role as controlled variables. uGf and un are valve opening and unit speed of the PFM system, respectively, and role as manipulated variables. In addition, TR=550°C means the exhaust temperature. To enhance the control performance of the gas turbine PFM system, this study optimizes the parameters in Fig. 1, starting from the design of the control strategy.

Consider a general system described by the following equation: (4)Gp(s)=Δα0+α1s+⋯+αp−q−1sp−q−1+sp−qβ0+β1s+⋯+βp−1sp−1+spe−τswhere p, q and τ are the denominator order, the relative order and the high-frequency gain of the system, respectively. αj(j=1,2,⋯,p−q−1) and βk(k=1,2,⋯,p−1) are the coefficients of the numerator and denominator of the system transfer function, respectively. Considering system uncertainty, αj, βk and Δ are generally uncertain or unknown.

The following state space form is obtained by the state transformation of Eq. (4): (5)ϕ˙j=ϕj+1ϕ˙j=−∑j=0q−1λjϕj+1−∑j=0p−q−1ςjφj+1+Δu,j=1,⋯,q−1φ˙j=φj+1φ˙p−q=−∑j=0p−q−1αjφj+1+ϕ1,j=1,⋯,p−q−1y=ϕ1where ϕj(j=1,2,⋯,q) and φj(j=1,2,⋯,p−q) are the state variables of the system. λi(j=1,2,⋯,q−1) and ςi (j=1,2,⋯,p−q−1) are unknown parameters.

Define the observation extended state as: (6)f(ϕ,φ,u)=−∑j=0q−1λjϕj+1−∑j=0p−q−1ςjφj+1+(Δ−l)uwhere l is a parameter whose positive or negative sign is consistent with the symbol Δ. ϕ˙q in Eq. (5) can be described by the following equation: (7)ϕ˙q=f(ϕ,φ,u)+luwhen the relative order of the controlled object q equals 2, Eq. (5) is transformed to: (8){ϕ˙1=ϕ2ϕ˙2=f(ϕ,φ,u)+luy=ϕ1

The DDE strategy for the closed-loop system is designed as: (9)y¨+h1y˙+h0y=r

Combining Eq. (8), it can be obtained that the designed control law is: (10)u=−h0(ϕ1−r)−h1ϕ2−f^lwhere f^ is the observer’s observation of the extended state f(ϕ,φ,u), and can be estimated using the disturbance observer down below [15]: (11){f^=θ+kϕ2θ˙=−kθ−k2ϕ2−kluwhere k and θ are the gain and intermediate variables of the disturbance observer, and Eq. (10) is equivalent to: (12)u=−h0(ϕ1−r)−h1ϕ2−f^l=−θ+kϕ2+h0(ϕ1−r)+h1ϕ2l.

Considering the disturbance observer in Eq. (11), it can be obtained that: (13)θ˙=−kθ−k2ϕ2−klu=k[h0(ϕ1−r)+h1ϕ2]

Eq. (14) can be obtained by integrating both sides of Eq. (13). (14)θ=−k[h0∫(ϕ1−r)dt+h1ϕ1]

Considering Eqs. (13), (14) is equivalent to: (15)u=−k[h0∫(ϕ1−r)dt+h1ϕ1]+kϕ2l−h0(ϕ1−r)+h1ϕ2l

Considering that the set value r undergoes a step change in the actual system, r˙ is infinite and generally set to zero. Then we can get the tracking error as e=r−ϕ1 and e˙=r˙−ϕ˙1=−ϕ2, Eq. (15) is equivalent to: (16)u=kh1+h0le⏟P+kh0l∫edt⏟I+k+h1le˙⏟D−kh1lr

Similarly, when the controlled object turns out to be a system with a relative order of 1, its control law can be obtained as: (17)u=k+h0le⏟P+kh0l∫edt⏟I−klr

Thus, it is known that PI/PID control based on DDE is a two-degree freedom structure containing a feedforward and feedback controller, which is shown in Fig. 2 [16], bf is the feedforward controller and bf=kl (PI) or bf=kh1l (PID). Gc(s) is the feedback controller and Gc(s)=kp+kis (PI) or Gc(s)=kp+kis+kds (PID).

Since differentiation introduces amplification of the manipulated variable by measurement noise, DDE-PI controllers are adopted as G1(s) and G2(s) for the PFM system of MARK V heavy-duty gas turbine in this study.

According to the Mason transformation formula, the closed-loop system transfer function from r to y can be obtained as: (18)G(s)=y(s)r(s)=(Gc(s)−b)Gp(s)1+Gc(s)Gp(s)

As is shown in Eq. (17), the feedforward controller does not appear in the characteristic equation of the closed-loop system, that is to say bf does not affect the stability of the system, the stability only depends on the feedback controller Gc(s) and the controlled object Gp(s).

The frequency domain response of the controlled object described in Eq. (4) is decipted as follows [17]: (19)Gp(jω)=r(ω)eiϑ(ω)=a(ω)+jb(ω)where ω, a(ω) and b(ω) represent the angular frequency, real and imaginary parts of the controlled object, respectively.

The stability domain of the DDE-PI controller can be obtained using the D-partitioning method. The stability domain boundary of PI controller includes nonsingular boundary ∂Dω when ω∈(0,−∞)∪(0,+∞), and singular boundary ∂D0 and ∂D∞ when ω=0, ω=±∞ [18], respectively. By solving the stability domain boundary equations separately, the stability region of PI controller parameters can be obtained: (20){Ki=0ω+Kpa(ω)ω+Kib(ω)=0Kpb(ω)ω+Kia(ω)=0

Combining kp=k+h0l and ki=kh0l, and taking l≠0 into consideration, the stability region of the DDE-PI controller can be obtained: (21){kh0=0lω+(k+h0)a(ω)ω+kh0b(ω)=0(k+h0)b(ω)ω+kh0a(ω)=0

The parameters of the DDE-PI controller should be reasonably selected from the stability region described in Eq. (21). Through extensive simulations, the parameter tuning procedure is summarized as follows:

Determine the coefficients of the desired dynamic PI equation y˙+h0y=r according to control performance requirements. For example, given the desired adjustment time tsd, h0∈[4∼12]/tsd is obtained;

The parameter tuning procedure of the DDE-PI controller is shown in Fig. 3. It can be observed that the proposed tuning procedure has strict logic and simple steps even though it needs trials and errors.

To illustrate the simplicity and effectiveness of the proposed tuning method, take Gp(s)=12.5s+1e−0.3s in Eq. (2) as an example. Fig. 4 presents the stability region of the DDE-PI controller. The stability region provides the range of parameter selection, and then the tuning procedure above can be applied to obtain satisfactory parameters. The selected parameters of the DDE-PI controller are tsd=3, k=0.5, l=5 and h0=15. As a comparative controller, PI is tuned as KP=2 and KI=1. Fig. 5 shows the control performance for Gp(s)=12.5s+1e−0.3s, it can be learned that DDE-PI controller can obtain faster tracking performance and better disturbance rejection ability. To compare the control performance under different conditions, Monte Carlo experimental method is applied to test controller robustness [19]. By perturbing parameters of Gp(s) within ±20% range of their nominal values, one can obtain control performance with uncertain systems as shown in Figs. 6 and 7, it can be observed that the DDE-PI still obtains better control performance for uncertain systems. This discussion explains how to obtain the parameters of DDE-PI and illustrates the advantages of DDE-PI, and this provides a good foundation for the following applications for the gas turbine PFM system.

The DDE-PI controller proposed in this section will be applied to the gas turbine PFM system shown in Fig. 1, both G1(s) and G2(s) are DDE-PI controllers. The multi-objective optimized ADRC and PI control strategies designed in reference [7] are chosen as comparison control strategies. The three control strategies are denoted as DDE-PI, multi-objective optimized ADRC, and multi-objective optimized PI, respectively. The parameters of all three control strategies are listed in Table 2.

In order to improve the PFM capability of gas turbines, a DDE-PI controller is proposed in this study to enhance the control performance of the PFM system. Firstly, a typical PFM model of an MS6001B heavy-duty gas turbine is introduced, and its control characteristics are analyzed. Next, the DDE-PI control strategy is introduced, and an easy-to-implement and highly engineered parameter tuning procedure is summarized. Finally, the DDE-PI control method is applied to the PFM system of an MS6001B heavy-duty gas turbine, the simulation results show that the DDE-PI control strategy can achieve smaller overshoot with a fast response, demonstrating strong engineering application value. Based on the theoretical and simulation analysis, the subsequent work will further validate the effectiveness of the proposed method by completing the on-site realization of control logic, logical protection, parameter tuning, and putting into operation on-site of DDE-PI for actual units.