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Constrained nonlinear model predictive control for centrifugal compressor system surge including piping acoustic using closed coupled valve

ABSTRACT This paper deals with nonlinear model predictive control scheme for surge prevention in centrifugal compressors systems. To bring up the effects of the station’s piping system on the compressor surge, nonlinear dynamic of compression system is considered that involve acoustic of compressor system. Close-coupled-valve (CCV), as a common control input in compression systems, is considered as an actuator and the mathematical model describing the flow dynamic of compression system is derived in the presence of CCV. For controlling the compressor system surge instability, nonlinear model predictive control (NMPC) is applied, whose allow to consider constraints on CCV actuator and states of system, significantly enlarge the operating region of the compressor and enhance the authority of the control system. Numerical simulations show that the proposed control system is able to meet the desired specifications in avoiding and active control of surge, in the presence of different types of disturbances occurring along the pipeline.


Introduction
Compressors are essential machines in modern industrial processes for the pressing and transportation of gases and fluids. They are vital to the operation of key energy sectors, such as the oil and gas, nuclear, and hydroelectric. In addition, compressors are central components in the heating, ventilation, and air conditioning systems for homes and commercial buildings. With the current focus on improving the efficiency of energy usage in buildings and manufacturing systems, the compressor is an important link in the energy generation/consumption chain in our modern society that needs to be carefully studied. Surge is one of the main dynamic instability limiting the operation of centrifugal compressors (De Jager, 1995) and it is an axisymmtrical oscillation of the flow through the compressor and is characterized by a limit cycle in the compressor characteristic. This event forces the flow back toward the compressor inlet and initiates the surge limit cycle that affects the entire system. Surge can cause extensive structural damage to the machine because of the violent vibration and high thermal loads that generally accompany the instability (Pampreen, 1993).
Furthermore, any flow unsteadiness or periodic excitation in a centrifugal compressor station piping system can significantly decrease the compressor's surge margin.
Both acoustic resonance and system impedance are functions of the entire piping system connected to the compressor, including pipe friction, interface connections, valve/ elbow locations, pipe diameter, valve coefficients, etc. and can move the centrifugal compressor operating point into a surge or stonewall condition.
It is critically important to consider the impact of the station's piping system on the compressor dynamic behaviour. According to the findings of Brun and Kurz (2014) and Sparks (1983), the entire piping system connected to the compressor will have following effects on the compressor system: • operating point variation • amplification and speed up of surge occurrence • fluctuations transfer in the system • limit the stability range of compressor Thus, a careful acoustic and impedance effects evaluation over compression system should be performed to avoid impacting the operating range of the machine and to properly design the surge control system. Having an appropriate model from a compressor system is first step in studying the mentioned effects over surge instability and its controlling. There are several mathematical models developed over the years to describe the dynamics of the flow in compression systems, and extensive reviews of these models can be found in references (Gravdahl & Egeland, 1999b;Longley, 1994). Among these models, the most frequently referenced is the lumped parameter model introduced by Greitzer (1976aGreitzer ( , 1976b for axial compressors, and later demonstrated to be applicable to centrifugal compressors by Hansen, Jørgensen, and Larsen (1981).
Furthermore, as pointed out in Ref (Van Helvoirt & De Jager, 2007), the Greitzer model alone is not adequate to predict the dynamics associated with fluid flow in distributed systems, such as acoustic waves and flow pulsations in pipelines. The dynamics of compression systems for different configurations of the inlet and the exhaust piping was studied in Refs (Hagino, Kashiwabara, & Uda, 2005;Jungowski, Weiss, & Price, 1996;Sparks, 1983). Van Helvoirt and De Jager (2007) proposed to implement a transmission line model, first introduced by Krus, Weddfelt, and Palmberg (1994) for modelling hydraulic and pneumatic line systems, to describe the effects of the pipeline dynamics between the compressor and the plenum volume in the pressure oscillations during deep surge. The mathematical model for the compression system that captures the effect of the piping acoustics during both the stable and unstable operating conditions, along with the dynamics during the transition between these two states is expanded in (Yoon, Lin, Goyne, & Allaire, 2011).
Motivated by the potential benefits of controlling the surge, various measures have been introduced during the last few decades (Staroselsky & Ladin, 1979). A surge avoidance system is a widely used passive method, which possesses good reliability. In the system, a surge avoidance line is defined, which is located on the right of the surge line on the compressor map. Compressors are not allowed to operate in the region between the surge line and the surge control line. One reason of interest to study the surge phenomenon is that high efficiency operating points are usually around the surge line. The reliability of surge avoidance systems is obtained through the sacrifice of efficiency and operating range (Willems & De Jager, 1999;Gravdahl & Egeland, 1999b). Especially, the distance between the surge line and the surge control line is always conservative due to various and severe working conditions. A totally different solution for the surge problem is called active surge control, which was first presented by Epstein et al. in 1989. Active surge control systems do not attempt to avoid surge by limited the operating range of the compressor but by feedback control to stabilize surge itself. Several types of actuators could be used in active control systems. An air injector (Spakovszky et al., 1999), drive torque (Gravdahl, Egeland, & Vatland, 2002), closed-coupled control valve (Bartolini, Muntoni, Pisano, & Usai, 2008), movable plenum wall (including piston) (Uddin & Gravdahl, 2011), and bleed valve (Willems & De Jager, 1998) are reported frequently in the literature. Both the linear and nonlinear method can be used to design active surge controllers (Abed, Houpt, & Hosny, 1993;Moghaddam, Farahani, & Ananifard, 2011;Nayfeh & Abed, 2002;Weigl et al., 1998).
In nearly all cases, a compression system dynamic model is needed to develop an active surge controller. Although some achievements have been acquired by employing advanced algorithms like LQR, the Lyapunov method, and sliding mode variable structure control, however, none of these controllers have not considered the effects of pipe that is a negative point in model accuracy for stability of active controller. Based on a recently presented enhanced compression system model with variable impeller tip clearance and pipeline acoustics, a surge controller is designed (Yoon, Lin, & Allaire, 2014), but there are two objections in this case including the linear controller is not capable in capturing strongly nonlinear dynamics of compression system and also common actuators are the close-coupled valve (Gravdahl & Egeland, 1999a) and the throttle valve (Krstić, Fontaine, Kokotović, & Paduano, 1998). Considering optimization of control signal, as well as restrictions on actuators and states are another important topics that should be considered in the subject of controller design.
Overcoming the problems, we present new numerical results in the compressor surge NMPC using the CCV as an actuator in the compression system. The most important novelty of this paper is inference a model of compressor system which taking into account the effects of the pipe, uses the ccv actuator as a control input, and the NMPC is used to surge control. A surge controller is designed for a centrifugal compressor system based on the enhanced compression system model recently presented in (Yoon et al., 2011), and modified to support the CCV as actuator in this study. Both the avoiding and active control of surge considered in the simulation and the ability of the proposed controller is shown in stabilization of compressors operation. Therefore, this article is classified as follows. In the second part, the compressor system model taking into consideration the effects of pipeline and CCV actuator will be detailed. Reservations exactly how to a nonlinear predictive control and the simulation results for various cases will be discussed in third part.

System model
The dynamic model of the compression system considered here was presented in details in (Yoon et al., 2011). The compression system in the study of surge consists of three main components.
These are the compressor/plenum adding and storing the energy in the system, the throttle valve controlling the average flow rate, and the piping transporting the compressed gas/fluid. The nondimensional pressure rise ψ and mass flow rate are defined as the following functions of the mass flow rate m and the absolute pressure P: The constants in (1) are the impeller velocity U, the cross sectional area of the compressor duct A c , the inlet absolute pressure P o1 , and the density at the inlet ρ o1 .
The Greitzer model can capture the surge limit cycle for a compression system, and it serves as a good basis for development of surge control laws. As described in (Yoon et al., 2011), the acoustic resonance from the compressor pipeline is added to the model. The resulting block diagram for the compression system with pipeline dynamics is shown in Figure 1.

Compressor and plenum
The basis of the equations describing the flow in the compressor and the plenum volume comes from the Greitzer model in (Greitzer, 1976a(Greitzer, , 1976b. Assumptions for this model are low compressor inlet match number, low pressure rise compared with ambient pressure, isentropic compression process in the plenum with uniform pressure distribution, and negligible fluid velocities in the plenum. The use of a close coupled valve for surge control was accurately modelled and presented in (Gravdahl & Egeland, 1999b;Uddin & Gravdahl, 2015). The approach is to introduce a valve close to the plenum volume in the compressor. CCV means that there is no mass storage of gas between the compressor outlet and the valve, as can be seen from Figure 2.
Based on mentioned statement, the pressure increase in the compressor and the pressure drop across the valve can be combined into an equivalent compressor. This is needed in order enable the valve to control the characteristic of the equivalent compressor. The outlet mass flow from the compressor through a close coupled valve is given as follows: Where c r is valve constant, u r is the CCV opening percentage and p , c are plenum pressure rise and compressor pressure rise, respectively. The range value of u r is from 0% to 100%. Finally, according to (Gravdahl & Egeland, 1999b;Uddin & Gravdahl, 2015), equations for describing the pressure and mass flow in the constant speed centrifugal compressor can now be formulated as follows: where B is the Greitzer stability parameter and ω H is the Helmholtz frequency (Greitzer, 1976a). The state variables of the model are the compressor mass flow rate c and the plenum pressure rise p , which are nondimensionalized as shown in (1). The value of the plenum mass flow rate p is dependent on the dynamics of the piping. Finally, the steady-state compressor pressure rise c,ss is obtained from the compressor characteristic curve as a function of the compressor mass flow rate: The characteristic curve c,ss is divided into the stable and unstable part by the surge point at s . The coefficients A 1 , B 1 , and D 1 of the characteristic curve correspond to the stable flow region of the compressor, whereas the coefficients A 2 , B 2 , and D 2 meet unstable flow region, as (Yoon et al., 2014).

Piping
To describe the dynamics of the piping system, a modal approximation of the transmission line dynamics (Yang & Tobler, 1991) has been included in the compression system equations (Yoon et al., 2011). The single-mode state space representation of the piping equations can be expressed in the following form: for some matrix coefficients A ij ∈ R and B ij ∈ R, where P P and Q P are the upstream (plenum) pressure and volumetric flow rate, respectively. In the same way, P th and Q th are the downstream (throttle duct) pressure and flow rate. The above piping model represents the system in terms of the absolute pressure and volumetric flow rate. With the assumption that the change of density ρ of the gas in the pipeline because of the pressure and temperature fluctuation caused by the piping acoustics is small, the states of the piping model can be nondimensionalized as described in (1). After the coordinate transformation, the resulting piping equation with nondimensional states is expressed as follows:

Throttle valve
The flow rate through the throttle valve is a function of the pressure drop across the valve. Here, it is assumed the dynamics at the throttle duct section are much faster than the rest of the system, and only the steady-state behaviour is captured. The relationship between the pressure and the mass flow rate in the throttle valve section is given by where c th is the valve constant and u th is the throttle valve opening percentage.

Equations set
According to Figure 1, the nonlinear equations governing in compressor system can be obtained from combination of compressor, plenum, pipe and throttle equations. State variables are the compressor mass flow, the plenum pressure rise, the throttle section pressure rise, and the  (3), (6), and (7). By putting them together, the state space equations of the assembled system are as follows: The parameters of the theoretical model are summarized in Table 1. Also, the coefficients of the characteristic curve correspond to the stable and unstable operating regions of the compressor are given in this table. The corresponding matrix coefficients of the piping equation in (5) are found to be A 12 = 3.7 × 10 6 , A 21 = −1.92 × 10 −3 , A 22 = −8, B 12 = −3.7 × 10 6 , B 21 = 1.92 × 10 −3 , and B 22 = 7.98.

NMPC
In the compressor system control problem, we encounter with nonlinear system subject to physical and operational constraints on the input and state. Well known systematic nonlinear control methods such as feedback linearization (Isidori, 1989;Marino & Tomei, 1995;Nijmeijer & Van Der Schaft, 1990) and constructive Lyapunov-based methods (Krstic, Kanellakopoulos, & Kokotovic, 1995;Sepulchre, Jankovic, & Kokotovic, 1997) lead to very elegant solutions, but they depend on complicated design procedures that does not scale well to large systems and they are not developed in order to handle constraints in a systematic manner. From the concept of optimal control, the MPC controller does not take the saturations in the compression system into account and therefore cannot optimally predict. It is required that nonlinearities and constraints are explicitly considered in the controller for satisfying environmental and safety considerations, rewarding physical and operational constraints, and the operating of compressor system on the tighter performance specifications. Nonlinear predictive control, the extension of well-established linear predictive control to the nonlinear world, appears to be a well suited approach for this kind of problem.
NMPC allows the use of a nonlinear model for prediction and explicit consideration of state and input constraints and specified performance criteria is minimized on-line. Therefore, we use the NMPC for surge control in compression system.

NMPC formulation
Consider the 4nd-order state space equations of compressor system in (8), as mentioned above, with c as compressor mass flow rate, the plenum pressure rise p , the throttle section pressure rise th , the plenum mass flow rate p as state variables and r is mass flow through a close coupled valve in series with the compressor. The coefficients of the characteristic curve given in Table 1.
The control objective is to avoid surge, i.e. stabilize the system. This may be formulated as with α, β, κ, ρ ≥ 0 and the set point x * corresponds to an unstable equilibrium point, subject to the inequality constraints for t ∈ [0, T].
and the ordinary differential equation (ODE) given by d dt given initial condition x(t) ∈ X ⊂ R n . Valve capacity requires the constraint 0 ≤ u(t) ≤ 0.4 to hold, and the pressure constraint x 2 ≥ +0.6 − v avoids operation too far left of the operating point. The variable v ≥ 0 is introduced in order to avoid infeasibility and R = 10 is a large weight.
Remark 1: According to numerical analysis, the cost function is non-convex but considering linearity of constraints on u and v, the quadratic programming solution is feasible.

Simulation results
In this simulation, α = 1, β = 0, κ = 0.08 are considered and the horizon is chosen as T = 12. It should be remarked that the constraints on u and v are linear, such that any quadratic programming solution is feasible for the nonlinear programming. Two operating scenarios are used to demonstrate the ability of the controller. In the first scenario, simulation of compression system equipped with CCV by using parameters in Table 1 is given as follows. A compressor initially operate in steady state where the throttle valve openings equals to 20%. At time t = 8 s, the throttle is closed to 10%, such that the compressor is interred into surge if the controller had not activated, as shown in the Figure 3.
In this case, compressor will experience the deep surge limit cycle. When the CCV and controller are activated, the CCV is opening at the time of closing throttle, therefore plenum fluid is recycled to the inlet that decrease the plenum pressure and accelerate the compressor mass flow. Finally, the operating point acquire the stable area as shown in Figure 4.
The control signal is also shown in Figure 5. In second scenario, compressor initially is interred to the deep surge instability, and this situation continues until t = 8 s. In this time, the controller is allowed to operate and the CCV starts to move, as shown in Figure 6. Therefore, plenum fluid is returned to the inlet that cause reduction of plenum pressure and growth the compressor mass flow. Finally, compressor stable operating point is achieved instead of initial unstable states. The transient response of the compressor system is also shown in Figure 7.    The first scenario shows controller capability to comprehend the effects of pipe, throttle and the whole of compressor system, and the latter is after the surge occurrence to demonstrate the ability of controller.
Using of a non-linear predictive control and taking into account nonlinear dynamics of pipe, we'll be able to stabilize the compressor system under different conditions to prevent from the surge occurrence and also enlarging the range of compressor by the CCV as actuator.
The confusions and different working conditions can be cover for improving efficiency in the safe operation of the compressor with using of this controller.

Conclusion
The design of a CCV-based surge controller for centrifugal compressors were discussed. Many solutions for surge control have been proposed in the literature, but their effectiveness have been rare because of not considering the effects of pipe. In this paper, based on the model of the compression system with piping acoustics, we derived the model that support close coupled valve as the most common actuator, and presented the application of nonlinear model predictive control to perform stabilization of surge limit cycle. Proposed nonlinear model predictive control cover all of limitations over the states and CCV actuator. The controller was tested with extended proposed model in different types of simulation scenarios at different conditions and its capability in rejection the flow disturbances related to surge instability and stabilizing the compressor system was shown. Robust model predictive control against system uncertainties and other disturbances can be investigated in the future studies.