Transcript
Revenue Maximization of Electricity Generation for a Wind Turbine Integrated with a Compressed Air Energy Storage System Mohsen Saadat, Farzad A. Shirazi, Perry Y. Li Abstract— A high-level supervisory controller is developed for a Compressed Air Energy Storage (CAES) system integrated with a wind turbine. Complementary to the low-level controllers in our previous works to track generator power and pressure, this controller coordinates different subsystems to optimize the system performance. The control strategy is obtained by solving an optimal storage/regeneration problem in order to maximize the achievable total revenue from selling electricity to the electric grid. Dynamic Programming (DP) approach is used to solve the corresponding optimal control problem that accounts for all the major losses in the CAES system as well as its nonlinear dynamics. Results show that an increase of 51% in total revenue is achievable by using the CAES system for a conventional wind turbine. Furthermore, a case study has been conducted to investigate the effect of storage system sizing on the maximum revenue.
I. INTRODUCTION Large-scale cost effective energy storage technologies are receiving significant attention from researchers in recent years [1], [2]. This is due to the growing penetration of renewable sources of energy. Several studies have been done on the importance of energy storage systems for the electric grid. Assessment of potential benefits and economic market potential for energy storage used for electric-utility-related applications in [3] reveals positive impact of energy storage on the U.S. economy. In [4], a methodology for profit analysis and overall economic viability of the Battery Energy Storage System (BESS) for various applications is presented. The efficacies of batteries, flywheels and pump-hydro storage systems for mitigating the effect of wind intermittency are investigated in [2] using convex optimization techniques. The optimal power flow (OPF) problem in the presence of largescale energy storage systems is discussed in [5] showing significant reduction in generation costs. Among all different types of energy storage approaches, compressed air energy storage (CAES) systems offer many competitive features. Large power and energy capacity, high cycle life and fast response time make CAES systems particularly suited for energy storage purposes in the electric grid [6]. A novel CAES system for wind turbines has been introduced in [7]. In this new architecture (Fig. 1), the gearbox and generator of a conventional wind turbine are replaced by a variable displacement hydraulic pump in the nacelle, converting the wind power into hydraulic power. This hydraulic power drives a ground level storage/regeneration system consisting of a tandem connection between a variable The authors are with Mechanical Engineering Department, University of Minnesota, Minneapolis, MN 55455. Email:
[email protected]
displacement pump/motor, a near isothermal liquid piston air compressor/expander and a fixed speed induction generator. The storage/regeneration system uses the open accumulator architecture [8] that can exchange power hydraulically or pneumatically with a high pressure storage vessel. The storage vessel contains both liquid and compressed air at the same pressure. 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 [9]. Linear and nonlinear control laws have been designed for optimizing the turbine power, as well as tracking the demanded generator power and system pressure [10]. The proposed CAES system can store excess energy from off-shore wind turbines to alleviate power supply and demand imbalances during the day. This will also make availability of wind energy more reliable, predictable and less disruptive to the electric grid. Moreover, because the wind turbine and the generator shaft are decoupled, the electric generator can also work as an electric motor. Therefore, not only can excess wind power be stored in the storage vessel but it is also possible to extract and store energy from the grid and to regenerate it in the future based on the market needs. Although low level controllers have been designed in our previous works to achieve an individual task for each component in the combined system, a high level supervisory controller is needed to coordinate the whole system in order to maximize the overall performance. The high level control strategy can be defined as an optimal control problem over a given time horizon for the combined wind turbine and CAES system. One important objective is the total achievable profit from the combined system. Deterministic or stochastic approaches can be taken to formulate the corresponding optimization problem. In this work, we address the deterministic version of this problem by assuming that the wind speed and electricity price are known time-series over a given time interval. In addition, how the storage device sizing can influence the economic operation of a wind turbine connected to an electric grid with varying electricity price is investigated. The optimal storage/regeneration sequence is trivial in the case when the pump/motors are perfectly efficient, and there are no size and capacity constraints. The strategy reduces to storing energy when the electricity price is low and regenerating it when the electricity price is high. The optimal sequence in the presence of size and capacity constraints on the system components is much more challenging. Moreover, the option of storaging/regenerating energy using the hydraulic and/or
Fig. 1. Schematic of power flow inside the combined wind turbine and CAES system
pneumatic paths also increases the problem complexity as the choice involves tradeoffs between energy capacity, power capability and efficiency. The rest of the paper is structured as follows: The dynamics of the CAES system is discussed in section II followed by the mathematical formulation of the profit maximization problem. The solution approach using dynamic programming (DP) and a special case of the problem are presented in section III. In the special case, the pressure is assumed to be constant and the system components are considered to be ideal without any loss. This ideal case is formulated as a convex optimization problem with linear matrix inequality (LMI) constraints. It is done in order to validate the DP method and solution. This will be followed by case studies showing optimal energy storage/regeneration sequence for given wind and electricity price profiles when considering the inefficiencies of the system components. A comprehensive study is presented and discussed in section IV to show the effect of energy storage system sizing on the maximum profit. II. MODELING The full dynamics of the combined wind turbine and CAES system can be represented by four dynamic states: 1) turbine rotor speed ωr , 2) generator shaft speed ωg , 3) pressure ratio inside the storage vessel r and 4) air volume inside the storage vessel V . Here, we assume that the low level controllers are accomplishing their predefined tasks in maintaining the generator shaft speed (to produce electrical power at desired frequency) as well as tracking the optimal tip speed ratio of the turbine rotor (to capture maximum wind power when the wind speed is in region 2 [11]). Therefore, the remaining dynamics of the combined system can be represented by the air pressure ratio and the air volume in the storage vessel. Using the ideal gas law for air and assuming isothermal compression/expansion in the storage vessel, the air pressure ratio and volume dynamics are determined as follows ([9]): −1 u1 (t) r(t)u2 (t) r˙ = + (1) P0 V (t) ln(r(t)) r(t) − 1 u2 (t) V˙ = (2) P0 (r(t) − 1)
where u1 and u2 are the pneumatic and hydraulic powers coming out of the storage vessel and P0 is the ambient pressure (Fig. 1). Note that air pressure ratio and volume can be controlled independently due to two degrees of freedom in storage/regeneration power paths: i) pneumatic and ii) hydraulic. In order to maintain the generator frequency, the net power to the generator shaft must be zero at all times. Therefore, the generator power (Pg ) delivered to the electric grid can be calculated as: h i Pg (t) = η4 (t) η3 (t)u1 (t) + η2 (t) η1 (t)Pr (t) + u2 (t) (3) where Pr is the wind power captured by the wind turbine rotor. Moreover, η1 is the efficiency of the pump in the nacelle, η2 is the pump/motor efficiency, η3 is the efficiency of the liquid piston air compressor/expander unit (consisting of a hydraulic pump/motor and a liquid piston compression/expansion chamber) and η4 is the efficiency of the generator. Because Eqn. (3) is valid regardless of the power flow directions, efficiencies of the pump/motor, air compressor/expander and electrical generator satisfy: ( < 1 if η1 Pr + u2 > 0 η2 ≥ 1 if η1 Pr + u2 ≤ 0 ( η3 ( <1 η4 ≥1
<1 ≥1
if u1 > 0 if u1 ≤ 0
if (η1 Pr + u2 + η3 u1 ) > 0 if (η1 Pr + u2 + η3 u1 ) ≤ 0
These conditions simply express the fact that the presence of losses reduces the output power to input power ratio and increases the input power to output power ratio. In general, efficiency of a variable displacement pump/motor is a function of pressure, displacement and shaft speed. However, for the pump/motors in the proposed CAES system, the dependence on speed can be neglected since the generator shaft speed is under precise control to maintain the frequency of the generated electricity. A sample efficiency map for a variable displacement pump/motor is shown in Fig. 2. Assuming that the electricity price ($/MWhr) is a function of time given by S(t), the total revenue achieved by selling electricity to the grid over the time interval of [0, tf ] can be calculated as: Z tf J= Pg (t)S(t)dt (4) 0
The corresponding optimal control problem is now defined as the maximization of the revenue function given by Eqn. (4) using the control inputs u1 and u2 while the system dynamics are given by Eqns. (1) and (2). Using overline and underline to denote the allowable maximum and minimum of state variables, the physical constraints for this optimal
control problem can be summarized as follows: r ≤ r(t) ≤ r
(5)
V ≤ V (t) ≤ V
(6)
|η1 (t)Pr (t) + u2 (t)| ≤ P pm (r)
(7)
|u1 (t)| ≤ P ce (r)
(8)
|Pg (t)| ≤ P g
(9)
where Eqn. (5) and (6) are due to allowable pressure range of the CAES system and the maximum capacity of the storage vessel. Eqns. (7), (8) and (9) show the maximum power capability of the pump/motor, air compressor/expander and electrical generator/motor, where P pm , P ce and P g are the rated powers for each component, respectively. One additional constraint is required to guarantee that the initial and final energy level in the storage vessel are the same (i.e. total energy sold to the electric grid is obtained from the given time interval not from any prior storage). This constraint can be written as: Z tf (u1 (t) + u2 (t))dt = 0 (10) 0
t(i+1) . Now, by discretizing the state space (r-V ), the optimal sequence of r and V that maximizes the total revenue and satisfies all the constraints can be found by the DP search method. In order to check the performance and accuracy of the developed DP code, a benchmark case study has been defined and solved. In this benchmark case, we assume that one degree of control freedom is utilized to maintain the system pressure ratio at the desired value rd all the time. In this situation, u1 and u2 coordinate with each other to maintain the pressure ratio. This coordination can be obtained as: u2 (t) = αu1 (t) where α is: α=
1 − rd rd ln(rd )
(14)
The second assumption for the benchmark case is that all the efficiencies are equal to 100% (ideal system with no loss). In summary, the optimal control problem for this benchmark case can be mathematically formulated as: Z tf u∗1 = arg max u1 (t)S(t)dt (15) u1 (.)
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Fig. 2. Energetic efficiency of a pump/motor as a function of displacement and pressure ratio
III. SOLUTION APPROACH Due to the nonlinear system dynamics and complex efficiency mappings for different components in the integrated wind turbine and CAES system, analytical approaches to solve the optimal control problem are difficult to apply. Here instead, deterministic Dynamic Programming (DP) approach is used to find the optimal storage/regeneration strategy and power path (pneumatic and/or hydraulic). By discretizing the system dynamics in time and rearranging Eqns. (1) and (2), u1 and u2 can be written as: i P0 ln r(i) h u1 (i) = − r(i)∆V (i) + V (i)∆r(i) (11) ∆t(i) ∆V (i) u2 (i) = P0 r(i) − 1) (12) ∆t(i) where i is the discrete time index and ∆r and ∆V are the pressure ratio and volume change from time t(i) to
where e is the energy stored in the pressure vessel and can be calculated as [8]: e(t) = P0 V (t) rd ln(rd ) − rd + 1 (16) The significance of the benchmark case is that the optimal control problem can be formulated as a convex optimization problem, with a linear cost function and a number of linear matrix inequalities (LMI) describing all the constraints. In discrete time domain, using the forward difference method to evaluate time derivatives, we have: 1 U= DXDT − X (17) (α + 1)∆t where U = Diag u1 (t1 ), · · · , u1 (tn ) X = E − e(t0 )In×n E = Diag e(t1 ), · · · , e(tn ) D = Ln †
(18)
Note that t0 to tn are the discrete time nodes while we assume the state of charge for the pressure vessel at t0 † Lower shift matrix with ones only on the subdiagonal and zeroes elsewhere.
(20) where I is the identity matrix and W is the rotor power matrix defined as: W = Diag Pr (t1 ), · · · , Pr (tn ) (21) Note that in Eqn. (20) is a small positive value that determines the strictness of the equality constraint in Eqn. (10). To compare the results of DP and convex optimization methods, the optimal control problem is solved for the wind power and electricity price profiles shown in Fig. 3 (top). The wind power profile is calculated based on a series of 10-min average wind speed recorded at the elevation of 60m [12]. Assuming that the wind turbine has a rotor with radius of 65m and rated power of 2.5MW, the captured wind power can be calculated as: 1 3 Pr (t) = ρair πRr2 Cp∗ Vwind (22) 2 where ρair is the air density at ambient condition and Cp∗ ' 0.48 is the power coefficient at the optimal tip speed ratio (' 8.1) [11]. Electricity price profile is an hourly marginal price per electrical bus provided by the Electric Reliability Council of Texas (ERCOT) [13]. A total volume of 500m3 is considered for the pressure vessel. At pressure ratio of 200, this volume is equivalent to about 12MWhr of energy. In addition, the generator and the air compressor/expander rated powers are assumed to be 2.5MW and 2MW, respectively. Results of this benchmark case study are shown in Fig. 3 (middle and bottom). As can be observed, the DP and convex optimization solutions match well with each other. Note that for both approaches, a total number of 300 points is used to discretize time domain (∆t = 24min). In the case of no energy storage, the total revenue that can be achieved by selling the captured wind power to the electric grid is $7800 over the given 5-day time horizon. However, according to the optimal solution (DP or convex optimization), by providing a 12MWhr energy storage capacity for the wind turbine, the total revenue can raise up to $12000 over the same time range which is about 54% increase in the total revenue. Note that the electrical generator can function
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Moreover, all the constraints can be summarized as: − e(t0 )I ≤ X ≤ e − e(t0 ) I − min P g I + W, (α + 1)P ce I ∆t ≤ DXDT − X ≤ ≤ min P g I − W, (α + 1)P ce I ∆t T − (α + 1)∆t ≤ DT F XDT F − F T XF ≤ (α + 1)∆t
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Storage Vessel Energy (MWhr)
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Fig. 3. Comparison between DP solution and convex optimization approaches. Captured wind power and electricity price profiles (top); Generator power (middle); Energy level inside the storage vessel (bottom)
IV. CASE STUDIES As mentioned before, due to nonlinear system dynamics and complex efficiency mappings, DP is the only approach that will be used in this section to solve the optimization problem. Since the focus of this work is on the CAES system, constant efficiencies are assumed for the pump connected to the wind turbine in the nacelle (η1 = 90%) as well as the electric generator (η4 = 95%). The remaining efficiencies are corresponding to the pump/motor (η2 ) and the air compressor/expander unit (η3 ), both located inside the CAES system. Since the low level controller maintains the generator shaft speed, these efficiencies in the CAES system (η2 and η3 ) are only functions of the system pressure ratio (r) and their displacement (x). Note that for a given pressure and generator speed, there is a one-to-one relationship between displacement and power. Hence, the pump/motor efficiency can be plotted in the pressure-power domain (instead of pressure-displacement domain). For the given efficiency map shown in Fig. 2 and a maximum displacement of 2.5lit/rev for the pump/motor, the resulted efficiency in pressure-power domain is shown in Fig. 4 (left). Note that the generator
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frequency is assumed to be 60Hz. Here, we define the nominal power of the pump/motor to be the maximum hydraulic power flowing out of the machine (pumping mode) at the full displacement (x = 1) and the pressure of 200bar. In this way, the corresponding nominal power of the pump/motor will be about 3MW.
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Fig. 4. Pump/motor (left) and air compressor/expander (right) efficiency map as a function of system pressure ratio and power. Note that in order for Eqn. (3) to be valid regardless of power flow direction, the pump/motor efficiency in the pumping mode and the air compressor/expander efficiency in the compression mode are considered to be 1/η where 0 ≤ η ≤ 1 is the physical efficiency.
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The same approach has been taken to find and map the efficiency of the liquid piston air compressor/expander as shown in Fig. 4 (right). Note that the air compressor/expander unit includes a variable displacement pump/motor with a maximum displacement of 1.6lit/rev and a liquid piston compression/expansion chamber with a total heat transfer capability of 85kW/K [14]. These result in a nominal power of 2MW for the air compressor/expander used for the next case study. The optimal power flow problem for the given rotor power and electricity price shown in Fig. 3 (top) is solved using DP approach. Results are shown in Fig. 5 (top and middle). In order to better understand the significance of utilizing the energy storage system, a baseline case is also simulated and shown in Fig. 5 (bottom). In this case, the capacity of the storage vessel is assumed to be zero (considering the hydraulic power transmission between the wind turbine and electric generator, but no energy storage/regeneration). Let’s define the total efficiency as the ratio between the total energy delivered to the electric grid and the total energy entered into the integrated system over a given time interval. The total efficiency of the integrated system for the baseline case is about 79.5% and the total revenue is found to be $6650. This value is even less than the corresponding revenue for a conventional wind turbine (assuming 90% efficiency for the gearbox and 95% efficiency for the generator) which is about $7000 for the same wind speed and electricity price profiles. By introducing the CAES system with the storage vessel capacity of 300m3 and pressure range of 100 to 300bar, the total revenue can be increased to $9870 which is about 41% of improvement compared to the conventional wind turbine. The total efficiency of the CAES system is found to be 72.9%
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Fig. 5. Optimal storage/regeneration sequence. Air volume and air pressure ratio in the storage vessel (top); Captured wind power and generator electric power (middle); Baseline case where the capacity of the storage vessel is zero (bottom).
for this case. Note that the revenue can be boosted up to $10600 if the generator is also capable of functioning as an electric motor. Table I summarizes the total efficiency and revenue results for these different scenarios. TABLE I T OTAL EFFICIENCY AND REVENUE RESULTS Case Conventional Wind Turbine Integrated System with Zero Storage Capacity Integrated System with Storage Capacity of 300m3 Integrated System with Storage Capacity of 300m3
Generator Type G/M
Total Efficiency (%)
Total Revenue ($)
G
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7000
G
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6650
G
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9870
G/M
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10600
After solving the optimal power flow problem, we can investigate the effect of subsystem sizing on the overall performance of the integrated system. The CAES system consists of three main components: i) hydraulic transformer (pump/motor), ii) liquid piston air compressor/expander, and iii) storage vessel. Although there is no physical constraint on the size of storage vessel and air compressor/expander, the hydraulic transformer size is restricted by the wind turbine
rated power. Moreover, the main cost of the CAES system is due to the air compressor/expander and the storage vessel. Therefore, here we will only study the sizing effect of these last two subsystems and fix all the other parameters as well as the wind power and electricity price profiles. The result of this study is shown in Fig. 6. As shown, increasing the size of storage vessel and/or air compressor/expander nominal power result in a higher total revenue for the integrated system. However, after some point, any further increase in the size of subsystems will not cause a significant change in the total revenue. Therefore, by considering the manufacturing and maintenance cost of the CAES system, there exists an optimal size for the energy storage system that maximizes the achievable profit from the system. For example, by choosing a 1.5MW air compressor/expander and a 500m3 storage vessel, the total revenue is obtained as $10380 which is about 48% improvement compared to the conventional wind turbine. This is while doubling the sizes will increase the revenue to $11200 which is only 8% additional improvement. 1000
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Fig. 6. Effect of storage vessel capacity and air compressor/expander size on the achievable total revenue over a 5-day time interval.
V. CONCLUSIONS The optimal storage/regeneration sequence as well as the corresponding efficient power flow path were found for an integrated wind turbine and CAES system for given deterministic wind power and electricity price profiles. The problem was structured and solved as a convex optimization with LMI constraints in the case of a system with perfect efficiency and no pressure variation. However, to study the actual system, dynamic programming approach was used to solve the corresponding optimal control problem considering the losses in the CAES system as well as nonlinear volume and pressure dynamics inside the storage vessel. For the given set of parameters and component sizes, it was shown that a 51% improvement in total revenue can be achieved by equipping a conventional wind turbine with a CAES system. In this case, the total efficiency for the overall system
was found to be 74.5% over a 5-day time period when the electric generator is also capable of functioning as an electric motor. In addition, a size study was performed to investigate the effect of the storage vessel capacity and the air compressor/expander size on the total revenue. If combined with manufacturing and maintenance costs, these results can be used to determine the optimal economical size of a CAES system for a given wind turbine. In future studies, we will consider the random nature of the wind power and electricity price to solve the stochastic version of this optimal control problem. ACKNOWLEDGMENT This work is supported by the National Science Foundation under grant number EFRI-1038294.
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