Draft: Nonlinear Controller Design with Bandwidth Consideration for a Novel Compressed Air Energy Storage System

To achieve both accumulator pressure regulation and generator power tracking for a Compressed Air Energy Storage (CAES) system, a nonlinear controller designed base on an energy based Lyapunov function. The control inputs for the storage system are the pump/motor displacements inside the hydraulic transformer and the liquid piston air compressor/expander. While the pump/motor inside the liquid piston has a low bandwidth, the other pump/motor inside the hydraulic transformer has a relatively higher bandwidth. On the other hand, the pneumatic energy storage path of open accumulator has high energy density, whereas the hydraulic path is more power dense. The nonlinear controller is then modified based on these properties. In the proposed approach, the control effort is distributed between the two pump/motors based on their bandwidths: Hydraulic transformer reacts to high frequency events, while the liquid piston air compressor/expander perform a steady storage/regeneration task. As the result, liquid piston air compressor/expander will loosely maintain the accumulator pressure ratio, while the pump/motor in hydraulic transformer precisely tracks the desired generator power. This control scheme also allows the accumulator to function as a damper for the storage system by absorbing power disturbances from the hydraulic path generated due to wind gusts. INTRODUCTION Extracting energy from wind is perhaps one of the most attractive industries in renewable energy generation. However, the main disadvantage and constraint in providing a good performance and capacity for the wind turbines is the intermittency and mismatch between available wind power and electrical power demand. Therefore, large scale energy storage systems can be significantly useful to improve the capacity factor of wind farms by providing the steady and predictable power for grid as well as capturing maximum available wind power in normal situations. Storing energy in high pressure compressed air is attractive since increasing the pressure ratio of the compressed air can result an appreciable energy density. For example, at a pressure ratio of 350 (35 MPa), 170MJ of energy can be stored in 1m3 of volume. Other major benefits of CAES systems are their low cost and long operation life. A novel CAES system has been proposed and modeled in [1, 2] (Fig. 1). The excess energy from the wind turbine is stored in the storage vessel prior to electricity generation, while the generator power is maintained at the desired value (demand power from electrical grid). This allows to downsize the electrical components and reduce the involved power electronics. In particular, downsizing the generator will consequently improve the capacity factor of the system defined based on the generator size. Two main challenges in the proposed CAES system are i) the low efficiency and power density of the air compressor/expander, ii) efficiency and power reduction of the compressor/expander when the pressure inside the storage vessel reduces as compressed air depletes. The first concern can be solved by deploying liquid piston air compressor/expander [3] with a chamber filled with porous materials, beside using optimal compression/expansion trajectory [4] and water spray cooling/heating method [5]. The open accumulator concept is a solution for the second issue [6]. Energy can be stored or extracted by pumping or releasing i) pressurized liquid similar to a conventional hydraulic accumulator or ii) compressed air similar to a conventional air receiver. In both cases, energy is stored in the compressed air. By coordinating the hydraulic and the pneumatic paths, the pressure can be maintained regardless of energy content. However, this coordination for the purpose of pressure regulation can affect the generator power due to tandem shaft connection between the pump/motors connected to the pneumatic and hydraulic paths of the open accumulator as well as induction generator. Therefore, a good control algorithm is essential for simultaneous achievement of the pressure regulation and generator demand power tracking. The efficient performance of the CAES system is significantly dependent on this controller design. Note that both objectives should be satisfied in presence of supply or demand power variations. In this paper, an energy-based controller is first developed to exponentially stabilize the system states and meet the abovementioned control objectives. However, this controller does not take advantage of the frequency characteristics of the system as well as meeting the bandwidth requirement for the liquid piston air compressor/expander unit. Therefore, a filter is introduced in the controller implementation to channelize the high-frequency and low-frequency commands to the hydraulic pump/motor and the liquid piston compressor/expander, respectively. In this scheme, the pump/motor responds to the fast changes in the input power due to wind gusts and the liquid piston compressor acts against long-term variations in either wind speed or electrical grid demand power. Simulation results have been presented for the combined wind turbine and CAES system while the turbine torque is controlled by the standard torque controller. Modeling A short summary of the system model is followed that describes different subsystems and their function in the combined wind turbine and storage system: A variable displacement pump (B) is directly (no gearbox) attached to the wind turbine (A) in nacelle which converts wind power to hydraulic power. Such a direct coupling requires a comparatively large displacement pump to transmit a large power (i.e. order of MW) since the wind turbine angular speed is low (≤ 20rpm). At the ground level, there is a tandem connection of a variable displacement hydraulic pump/motor (C), a near-isothermal liquid piston air compressor/expander (F) and a fixed speed induction generator (G), all driven by the pump (B). The liquid piston air compressor/expander, the main unit for storing/regenerating energy, consists of a compression/expansion chamber filled with some porous material (F1) and a liquid piston pump/motor (F2). The porous material is used in addition to optimal compression/expansion trajectory and water spray to enhance the heat transfer inside the chamber to improve its thermal efficiency which has a significant effect on the overall efficiency of the storage system. The open accumulator (E) which is in fact the storage vessel contains both air and liquid. Hydraulic path can be utilized to accommodate high power transient events such as wind gust or sudden power demand (from grid), whereas the pneumatic path can be reserved for steady storage/regeneration function. Figure 1. CAES SYSTEM ARCHITECTURE In summary, the overall dynamic of the combined wind turbine and storage system can be found as: Jrω̇r =− Dp 2π P0(r−1)−Γp(ωr)+ 1 2 ρ0πRrCP(β,λ) V 3 w ωr (1) Jgω̇g =− Dpm 2π P0(r−1)− Dl p 2π P0ln(r)η sgn(Dl p) trm −Γpm(ωg) −Γl p(ωg)−Tg(ωg) (2) V ṙ = Dl p 2π ωg + Dp 2π ωrr+ Dpm 2π ωgr−Ll p(Pw) − rLp(r)− rLpm(r) (3) V̇ =− ( Dp 2π ωr + Dpm 2π ωg ) +Lp(r)+Lpm(r) (4) where r, g, p, pm and l p are subscripts standing for turbine rotor, generator shaft, pump in nacelle, pump/motor in hydraulic transformer and the pump/motor connected to the liquid piston air compressor/expander chamber, respectively. In these equations, D is used to show the displacement for hydraulic actuators, Γ is the mechanical loss and L is the volumetric loss. Moreover, r is the pressure ratio of the air inside the accumulator and V is the volume of the compressed air. Additionally, Rr and Vw are the radius of the wind turbine rotor and the wind speed while Cp is the turbine power factor as a function of blade’s pitch angle (β) and tip speed ratio (λ). It should be noted that while the displacement of the actual liquid piston pump/motor in a cycle varies rapidly to achieve the desired compression/expansion profile [4], a cycle mean value is used here shown by Dl p. In the other words, cycle-by-cycle behavior is approximated by a time average model. ηtrm is then the thermodynamic efficiency of the compression/expansion chamber over one cycle. Finally, P0 and ρ0 are the ambient pressure and density of air and J is the angular moment of inertia. Controller Design The overall control objectives of the combined wind turbine and the storage system can be summarized as: i) Capturing maximum available power from wind by controlling the wind turbine angular speed; ii) Maintaining the accumulator pressure ratio over the storage or regeneration mode; and iii) Providing the required power demanded by the electricity grid. Since the wind turbine shaft and the generator shaft are not coupled in the proposed architecture, it would be possible to achieve all these goals at the same time. Wind turbine control is performed by utilizing standard torque control approach through the hydraulic pump located in the nacelle. The accumulator pressure and generator power will be controlled by the variable displacement pump/motors inside the hydraulic transformer as well as the liquid piston air compressor/expander unit. Storage System and Generator In conventional CAES systems, the pressure in the storage vessel reduces as compressed air in the storage vessel depletes, making it difficult for the air compressor/expander to maintain either its efficiency or power at all energy levels. However, in the open accumulator design, it is possible to maintain the pressure no matter how much compressed air is inside the vessel. So, it is important to design a controller such that it maintain the desired power on the generator shaft as well as the desired pressure ratio in the accumulator. Design of an appropriate nonlinear controller begins by choosing a suitable Lyapunov function. Here, the idea is to use an energy based Lyapunov function relying on the energy of generator shaft as well as compressed air inside the accumulator. Note that because the storage vessel is assumed to have enough space for the compressed air, the air volume dynamics given by Eqn. 4 will not be controlled at this level (i.e. a high level supervisory controller will control air volume). The Lyapunov function considering the generator shaft speed and accumulator pressure tracking errors is defined as: E = 1 2 Jgω̃g + ∫ V1