Optimal Life Extending Control of a Boiler System

Transcription

1 Optimal Life Extending Control of a Boiler System Donglin Li Horacio J. Marquez Tongwen Chen Dept. of Electrical and Computer Engineering R. Kent Gooden Syncrude Canada Ltd. University of Alberta P.O. Bag 4009 MD 0019 Edmonton, Alberta Canada T6G 2V4 Fort McMurray, Alberta Canada T9H 3L1 {donglin, marquez, tchen}@ece.ualberta.ca gooden.kent@syncrude.com Abstract The objective of life extending control (LEC), also known as damage mitigating control, is to design a controller to achieve a good tradeoff between structural durability and dynamic performance in a system. This task involves both damage dynamics modeling and LEC design. In this paper, we propose a new hierarchical LEC structure and apply it to a typical boiler system. There are two damage models in this structure: Model I is for on-line LEC; Model II is for on-line life prediction of the critical components of the system. Finally, an optimal LEC scheme is proposed and simulation results show that the designed LEC substantially reduces the accumulated damage with a minimum loss of dynamic performance. Keywords: Fatigue Analysis, Damage Modeling, Life Extending Control, Boiler Systems, Co-Generation Systems. 1 Introduction During the first half of the 19-th century, the development of the steam engine led to increasing sources of repeated stress on metal parts and structural elements. Shortly thereafter, unex- This work was supported by the COURSE Program with the Alberta Energy Research Institute. Corresponding author. 1

2 plainable fractures particularly in locomotive axles became a great concern to engineers. About the middle of the century, experiments of Woehler and others showed the importance of the number of repetitions of stress (rather than duration of time) in causing failure of metals under relatively low stress [1]. Since then over 20,000 papers on fatigue analysis have been published throughout the world [2]; and a large amount of experiments have been done to provide useful data under various conditions. This data constitutes the basis of fatigue analysis and design. The main objective of early studies was to understand material properties in order to design reliable systems. Since the 1970 s, fatigue research has also been used as the basis for the development of life-prediction systems. Many methods for estimating fatigue life were proposed on which life-monitoring systems could be developed [3, 4, 5]. Parallel to this, most control systems are designed focusing on stability and performance only, and assuming that the materials involved have invariant properties. In other words, the control design process ignores the effects of aging, fatigue, and damage in the materials. Systems designed in this way may lose performance or even stability in the long term, yielding high risks in system operations. In 1991, Noll and co-workers [6] pointed out the need to address the trade-off between system performance and durability (of critical components); this formed the basis of the socalled damage mitigating control, or life extending control (LEC). LEC is an area of research involving the integration of two distinct disciplines: system science and mechanics of materials. Although a significant amount of research has been conducted in each of the individual areas of control and diagnostics, and analysis and prediction of materials damage, integration of these two has not received much attention [7]. An important step in LEC design is to construct damage models for the critical plant components. Because fatigue damage is cycle-dependent, fatigue damage models are usually based on stress-strain hysteresis loops. In contrast, most control theories, e.g., the linear quadratic regulation (LQR), are formulated in the time domain. To fit the framework of LQR, the damage dynamics should be expressed as a vector differential equation with respect to time. In 1994, Ray and co-workers [8] proposed a continuum fatigue/damage modeling method for use in LEC. The advantages of this approach are that it allows the damage model to be incorporated within the optimization problem and that the damage accumulated between any two instants of time can be derived even if the stress-strain hysteresis loop is not closed. However, because the so called rainflow cycle counting is used in this method, it is not quite suitable for real-time control (We will discuss this in 2.1). We find that an improved rainflow method proposed by Downing [9] is more suitable for real-time LEC. 2

3 Another important issue in the LEC research is to design a life extending controller. The challenge here is mainly in how to handle the two conflicting objectives of reducing damage and improving dynamic performance of the system. Ray [10] suggested an open-loop control policy and simulation results showed that it is possible to achieve a good compromise between performance and accumulated damage. Further research on LEC has been reported with different application areas. Dai and Ray [11] applied LEC in a reusable rocket engine controller design. Zhang and co-workers [12, 13] used a hybrid design method, Tangirala et al. [14] applied robust design theory to design LEC and verified on a laboratory test apparatus. Kallappa and co-workers [15, 16] designed LEC on a fossil fuel power plant. In [15], robust control theory was applied; in [16], a fuzzy control theory was used to implement a wide-range control. In all these results, a twotier architecture was used and was in the framework of feedforward and feedback control; the feedforward control was used to optimize the output tracking and obtain a good tradeoff between structural durability and dynamic performance, while feedback control was to suppress the effect of various disturbances. Hence LEC was added in the form of feedforward control. In this paper, we study LEC with application to a boiler system. This boiler system has seldom been shut down since it started, so it works in a steady state under various disturbances with an existing PI controller, which is for good performance. In view of this boiler system, a new system structure is proposed, in which the original controller of the system and LEC work together, and no additional change of the original controller is required. LEC is implemented in the form of feedback control. Moreover, to implement real time LEC, a fatigue modeling method which is suitable for on-line control is also discussed. In section 2, fatigue damage modeling methods are discussed. In Section 3, a new LEC closed-loop scheme is proposed and an optimal life extending controller is designed for the boiler system and simulation results are presented. 2 Modeling Damage Dynamics Most fatigue damage models from material science, are cycle-dependent; but cycle-dependent models are not suitable to be used within the optimal control law design. Ray in 1994 [8] proposed a continuous-time fatigue model as follows. Consider a critical component subject to cyclic loading; let ɛ r be the total strain corresponding to the reference stress σ r at the starting point R of a given reversal as determined from the rainflow cycle counting method, and ɛ and σ be the strain and stress of the point on 3

4 the same reversal with R respectively. We define ɛ = ɛ ɛ r, σ = σ σ r. The relationship between the stress σ and the strain ɛ is given [17] ( ) 1 ɛ 2 = σ σ + n, (1) 2E 2K where E is the modulus of elasticity, K the cyclic strength coefficient and n the cyclic strain-hardening coefficient, all determined experimentally based on material properties. According to the rainflow method, σ r is either a local minimum or maximum; so dσr =0. dt Further, since it is assumed that no damage occurs during unloading [8], the damage rate can be made equal to zero when σ<σ r. When σ σ r,wehave dδ e =2 d dt dσ dδ p dt =2 d dσ (( 1 ɛ f 2 ( σ σ r 2K σ σ r (σ f σ m ) b 1 dσ, (2) dt ) n 1 ) (1 σ m σ f ) cb ) 1c ) dσ, (3) dt where δ e and δ p are elastic and plastic damage respectively, σ f is fatigue strength, ε f fatigue ductility, b fatigue strength, c fatigue ductility exponent and σ m is the mean stress. The coefficients σ f, ε f, b and c can be determined experimentally based on material properties. For any time instant t in one cycle, the mean stress σ m in Eq. (2) and Eq. (3) can be calculated as: σ m(t) = 1 t t t 0 σ(τ )dτ, t t f, where t 0 and t f are the start and end times of one cycle respectively. Therefore if t = t f,we can find the mean stress of one cycle. dσ can be found from direct measurements of the strain dt rate by specific gauges, or by the finite element analysis (FEA). When the direct measurement and FEA are not available, Lu and Wilson suggested an alternate formula [18]: σ(τ) = Eα 1 ν [T m(τ) T(τ)], (4) with ν the Poisson s ratio, α the coefficient of linear expansion, T the temperature of the component at the critical point. T m is the mean temperature up to current time and it can be calculated as T m (t) = 1 t t 4 0 T (τ )dτ. (5)

5 Thus the total damage rate is the weighted sum of the elastic damage rate and the plastic damage rate: dδ dt = w dδ e +(1 dt w)dδ p dt. The weighting function w is based on the elastic and plastic strain amplitudes where ɛ re, the elastic part of ɛ r, is defined as w = ɛ e ɛ re ɛ ɛ r, ɛ re = σ r E. Therefore, the accumulated damage D can be found by in which n is the total number of cycles. time. If t = t f, we can find the damage of the kth cycle δ(k). n D = δ(k), (6) k=1 During each cycle, the damage δ is a function of Eq. (2) and Eq. (3) constitute model I. To predict the remaining life of the component monitored, model II should be constructed. Dowling [5] derived a formula by an analogous method to the modified Goodman diagram: ɛ a = σ f E ( 1 σ 0 σ f ) (2N ) b + ɛ f ( 1 σ 0 σ f ) (2N) c, (7) where ɛ a is the strain amplitude, σ 0 is the mean stress up to current time. Therefore the remaining life N (cycle number) can be predicted by Eq. (7) after ɛ a is calculated. We notice that this continuous-time model involves the use of the rainflow cycle counting method to specify the starting point of one cycle to be the reference point R. The rainflow method of cycle counting derived its name from an analogy used by Matsuishi and Endo in their early work on this subject. Several algorithms are available to perform the counting, [19]. However, to eliminate counting half cycles, the starting point should be at the strain value of greatest magnitude. For example, Fig. 1 shows a strain history, using a traditional rainflow method we cannot start counting from point A or B, we have to start with the point C which has the greatest amplitude. Thus the entire load history is required to be known before the counting process starts. For this reason, this continuous-time model method is not suitable for on-line control [20]. An improved rainflow cycle counting method, called the one pass method, was proposed by Downing [9], which allows taking any point as a temporal start point during the counting 5

6 Strain H B F D G A C E Time Figure 1: Variable amplitude strain history process; if a certain condition is satisfied, the starting point should be moved. For the same strain history shown in Fig. 1, using the one pass method we can start counting from point A. It is important to point out that this improved method can give the same result as the conventional rainflow method. Thus, incorporating this method in Ray s continuous model allows on-line implementation of LEC. Now we clarify our damage modeling process using the strain history shown in Fig. 1: 1. Transferring the strain history into cycles : DC, GF, BA and HE by the improved rainflow cycle counting method. 2. Modeling damage dynamics using Eq. (2) and Eq. (3), from the starting point to ending point of one cycle. - Since we use the improved rainflow cycle counting method, steps (1) and (2) can be executed simultaneously. Remark 1: For a boiler system, the high temperature and its variation are important causes of turbine damage, here we will only consider the thermal damage. 3 LEC Design for a Boiler System LEC has found several applications including rocket engines and fossil power plants in a framework of feedforward and feedback control [10, 15]. In this paper, we study LEC with application to a boiler system which is part of an industrial co-generation plant, owned and operated by Syncrude Canada s Ltd. Syncrude s Mildred Lake plant consists of a tarsand 6

7 mine, a bitumen extraction process, and an upgrader to produce a high quality synthetic crude oil. Its utility department is critical to plant operation because it supplies steam, water, electricity, air, and nitrogen to the various processes on the site. The main components of the steam system are three fuel gas fired utility boilers, two CO boilers and two once-through steam generators (OTSG). The objective of our research is to design an LEC to reduce the damage of the turbine in the utility boiler system. U3 Attemperator Y3 Steam Temperature SprayFlow Secondary Super Heater Furnance Primary Super Heater Y2 Drum Pressure Economizer U1 Feedwater Flow Steam Drum Y1 Drum level Heat Risers Mud Drum Downcomers U2--FuelFlow Y4--Excess Oxgen Induced Draft Fun U4 Air Flow Forced Draft Fun Figure 2: The operating principle diagram of the utility boiler The operating process of the utility boilers in Syncrude is shown in Fig. 2 [21]. The utility boilers in the plant are water tube drum boilers. This type of boiler usually comprises two separate subsystems: the steam water system (also called water side), and the fuel-air-fuel gas systems (also called the fire side). In the steam-water side, water is preheated by economizer and then is fed into the steam drum. Liquid water exits the steam drum and flows through the downcomer into the mud drum. The mud drum distributes the water to the risers, where the water is heated to saturation conditions. The saturated steam-water mixture then re-enters the steam drum in which the steam is separated from the water and exits the steam drum into the primary and secondary superheaters. In the two superheaters, the steam is further heated, and then is fed into one header followed by power generators. In between the two superheaters is an 7

8 attemperator which regulates the temperature of the steam exiting the secondary superheater by mixing water at a lower temperature with the steam from the primary superheater. In the fire side, fuel and air are thoroughly mixed and ignited in a furnace. The resulting combustion converts the chemical energy of the fuel to thermal or heat energy. The gases resulting from the combustion, known as the flue gases, pass through the superheaters, the risers, and the downcomer, and leave the boiler. In the present case, Syncrude Canada Ltd has available a simulation package, known as SYNSIM model [22]. SYNSIM is a computer simulation model that has been developed as a joint effort by members of the Departments of Chemical Engineering and Electrical Engineering at the University of Alberta, and of the technical staff of Syncrude Canada Ltd. The overall model includes submodels of the Utility and CO boilers with their controls, and associated letdown system controls, the turbine-generator sets with their controls, and the plant electrical load with dynamic load-shedding. The SYNSIM model was developed with the purpose of simulating certain upset conditions that have been sporadically detected, as well as a general tool for stability analysis. It has been extensively tested, and its predictions are consistent with measurements from the real plant. We should point out that while SYNSIM constitutes an invaluable analysis tool, it cannot be used directly for controller design due to the excessively large complexity of the models used in this package. Instead we design our LEC with a linear model which is linearized from the SYNSIM model at the normal operating condition. The final controller, however, will be simulated in SYNSIM. 3.1 LEC Closed-Loop Structure In many industrial facilities such as the boiler system at hand, control systems are already in operation for good dynamic performance. An important and practical question is: How to enhance the structural durability with existing control systems still in place? Fig. 3 is our proposed structure for LEC, in which the existing controller is kept in place for the plant, but a damage model and LEC are introduced on top at a higher level, which interact with both the existing controller and the plant. The damage models are used to estimate the accumulated damage and the damage rate within the system, based on measurable variables; the LEC is used to maintain the accumulated damage and damage rate within prescribed limits, while trading off little dynamic performance. This way we gain improved component durability and longer service life without affecting much of the routine operation. Incorporating the above idea, we propose a hierarchical LEC structure shown in Fig. 4. Here the inner feedback loop represents the existing control system, designed for system 8

9 Damage Model & LEC High level Existing Controller Low level Plant Figure 3: A schematic digram of LEC philosophy dynamic performance. Damage model I and the LEC block form the outer feedback loop. The signals involved in Fig. 4 are as follows: X is the state vector of the closed inner loop; ˆX is the estimate of X; Y is the plant output vector; U r is the reference input vector; U lec is the output variable from LEC; U is the control input vector; υ is the damage variable - it can be chosen as the stress σ or the damage rate δ (in our LEC design, it is the damage rate δ ); and N (remaining cycle number) is the output of damage model II for life prediction. Note that the controller is kept in place, and no changes are necessary. The LEC is introduced in order to reduce damage level of critical components in the system. Damage model I is used for on-line LEC feedback; while damage model II for life prediction. Both models should be suitable for on-line implementation. Because not all state variables are measurable, the state estimator is needed. This control structure has the following features: Hierarchical: The LEC is at a higher level with the existing controller in place; it provides U r + - U lec Controller K U Y N Plant Damage Model I Damage Model II State Feedback Controller Xˆ State Estimator υ L E C Figure 4: A hierarchical LEC structure 9

10 an additional control signal to enhance durability and safety within the system. (This feature would be welcome by practitioners and operators given the simplicity associated with implementing the LEC system.) Good dynamic performance: The LEC is designed to have a minimum effect on the dynamic performance already achieved by the existing control system. Optimal tradeoff: We target optimal tradeoff between component durability and dynamic performance. 3.2 LEC design In this section, an optimal life extending controller is designed for the boiler system. As discussed above, the system we study in this paper works in steady state without frequent start-up and shut-down. Therefore a linearization based method can be used. From Fig. 2 we see that in this system there are four inputs: feedwater flow (kg/s) u 1, fuel flow (kg/s) u 2, attemperator spray flow (kg/s) u 3 and air flow (kg/s) u 4 and three controlled outputs: drum level (m) y 1, drum pressure (KPa) y 2 and steam temperature ( 0 C) y 3. To simplify the control design problem, we assume the ratio of fuel-to-air flow rate is fixed; therefore the system becomes three inputs and three outputs. We linearized the system at the nominal operating conditions which are specified as follows: 1. Three utility boilers, two CO boilers and two OTSG s are on-line. 2. The total steam load is kg/s. 3. The total steam turbine electrical output is MW. At this operating condition, for each of the three utility boilers, we have u y u 0 = u 20 = , y 0 = y 20 = u 30 y 30. The process of the linearization can be found in [23]. The setup of the closed-loop system is shown in Fig. 4, where the plant dynamics, which are linearized from SYNSIM, can be described as y 1 y 2 y 3 = g 11 g 12 0 g 21 g 22 g 23 g 31 g 32 g u 1 u 2 u 3, (8)

11 where g 11 = s s s s s g 12 = s s g 21 = s g 22 = s g 23 = s s s s s g 31 = s s s g 32 = s s s g 33 = s s The controller K has already been designed to make the boiler generate enough steam, maintain the drum pressure and the steam temperature to their setpoints by K(s) = s The material type of the turbine blade is known to be ASTM A-516 Gr.70. For damage modeling, the experiment constants in Eq. (2) and Eq. (3) can be found in appropriate tables: ν =0.27, E = , α = , K =1193MPa, n =0.202, σ f =859, ɛ f =0.219, b = 0.108, c = 0.5. σ r is the reference stress at the starting point of each cycle, it can be calculated by Eq. (4). Thus model I can be constructed according to Eq. (2) and Eq. (3), and model II according to Eq. (7). To apply the LQR theory, the transfer function model in (8) should be transformed into the state space form. Because the damage dynamics represented by Eq. (2) and Eq. (3) is nonlinear, linearization (operating point δ0 = 0) should be performed first at the normal load conditions. To design the LEC via optimization, we have the cost function J 1 = 0 ( X T Q X + q δ 2 + U T RU)dt, (9) 11

12 in which X = X X 0, X 0 is the state at the operating point, which is corresponding to outputs at the operating point. X is the state vector of the closed loop system, which consists of the plant, the controller K and the unit negative feedback. After the minimal realization, X turns out to be a vector of dimension 19, in which we set the drum level as x 1, drum pressure as x 2, steam temperature as x 3, and damage rate δ as x 20, other states are not measurable. Q and q are the weighting matrices with the appropriate dimensions respectively. It should be mentioned that ideally X should have the dimension of 17, due to errors in calculation, two suppose-to-be canceled poles were not perfectly canceled by corresponding zeros, although they are very close. Fortunately, these poles did not affect the system performance too much because of the existence of the close zeros. Because not all states are measurable, so a state observer has to be designed before the LQ design. The pole assignment method is used to design an observer. We choose the observer to have the same poles with the resultant closed-loop system with the state-feedback controller. The Algebraic Riccati equation (ARE) based method is applied to obtain the following static state feedback controller: K LEC = Simulation results The purpose of this section is to evaluate our design and to study the properties of the LEC law in i) reducing accumulated damage, and ii) system performance. Before proceeding, we notice that the controller K currently working at the plant is not optimal. Thus, to make a fair comparison of system performance with and without the LEC law, we design a secondly LQ control law with cost function J 2 = 0 ( X T Q X + U T RU )dt, (10) It is immediate that the cost function J 1 and J 2 given in (9) and (10) are identical, except that J 2 does not include the damage rate δ and thus does not include the LEC law. We designed an LEC based on a linearized model, however, the final result should be tested under more rigorous conditions. Namely, it should be tested with a more accurate description which should incorporate the nonlinearities encountered in the true plant. Thus the final controller will be simulated in SYNSIM. Simulation results are shown in Fig. 5 to Fig. 8. It is clear that the LEC achieves its objective of significantly reducing the accumulated damage without significant deteriorating system performance. 12

13 1.2 without LEC with LEC Drum level (m) Number of sample data (Ts=0.05sec) x 10 4 Figure 5: Step responses of the drum level without LEC with LEC Drum Pressure (KPa) Number of sample data (Ts=0.05sec) x 10 4 Figure 6: Step responses of the drum pressure 13

14 515 without LEC with LEC 510 Steam Temperature(centigrade degree) Number of sample data (Ts=0.05sec) x 10 4 Figure 7: Step responses of the steam temperature 4.5 x without LEC with LEC 3 Accumulated damage Number of sample data (Ts=0.05sec) Figure 8: Comparison of the accumulated damage 14

15 Remark 2: In LEC design we included the damage rate δ in J 1 which weakened the weight of the drum level, so the performance (in overshoots and settling times) of the drum level was expected to become worse than that without LEC, while because the improvement of the performance of the steam temperature was in favor of the decreasing of the damage rate, the performance of the steam temperature should get better. The simulation results verified those. Remark 3: From Fig. 8, we see that with LEC the accumulated damage is reduced by about 50% compared with that without LEC. 4 Conclusions The contribution of this paper is as follows: we proposed a hierarchical structure for life extending control; we incorporated an improved rainflow cycle counting method with Ray s modeling theory to build a damage model for on-line model-based life extending control for an industrial boiler system; we proposed an LQR based optimal LEC design, and applied it to the boiler system, yielding promising results in simulation. References [1] Grover, H. J., Gordon, S. A., and Jackson, L. R.: Fatigue of metals and structures (U.S. Government Printing Office, 1954) [2] Pook, L. P.: The role of crack growth in metal fatigue (J. W. Arrowsmith Ltd, Bristol, 1983) [3] Dowling, N. E.: A discussion of methods for estimating fatigue life, Proc. of SAE Fatigue Conference, 1982, pp [4] Socie, D. F.: Fatigue life prediction using local stress-strain concepts, Experimental Mechanics, SESA, 1997, pp [5] Dowling, N. E.: Fatigue life prediction for complex load versus time histories, Journal of Engineering Materials and Technology, 1983, 105, pp

16 [6] Noll, T. E., Austin, S. D., Graham, G., Harris, T., Kaynes, I., Lee, B., and Sparrow, J.: Impact of active controls technology on structural integrity, 32nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 1991, pp [7] Ray, A., and Lorenzo, C. F.: Damage-mitigating control: an interdisciplinary thrust between controls and material science, Proc. of ACC Conference, 1994, 3, pp [8] Ray, A., Wu, M. K., Carpino, M., and Lorenzo, C. F.: Damage-mitigating control of mechanical systems: part I conceptual development and model formulation, ASME J. Dynamic Syst., Measurement, Contr., 1994, 116(3), pp [9] Downing, S. D., and Socie, D. F.: Simple rainflow counting algorithms, Int. J. Fatigue, 1982, pp [10] Ray, A., Wu, M. K., Carpino, M., and Lorenzo, C. F.: Damage-mitigating control of mechanical systems: part II formulation of an optimal policy and simulation, ASME J. Dynamic Syst., Measurement, Contr., 1994, 116(3), pp [11] Dai, X., and Ray, A.: Damage-mitigating control of a reusable rocket engine: parts I and II, ASME J. Dynamic Syst., Measurement, Contr., 1996, 118(3), pp [12] Zhang, H., and Ray, A.: Robust damage-mitigating control of mechanical structures: experimental validation on a test apparatus, ASME J. Dynamic Syst., Measurement, and Contr., 1999, 121(3), pp [13] Zhang, H., Ray, A., and Phoha, S.: Hybrid life-extending control of mechanical systems: experimental validation of the concept, Automatica, 2000, 36, pp [14] Tangirala, S., Holmes, M., Ray, A., and Carpino, M.: Life-extending control of mechanical structures: experimental verification of the concept, Automatica, 1998, 34, pp [15] Kallappa, P., Holmes, M. S., and Ray, A.: Life-extending control of fossil fuel power plants, Automatica, 1997, 33, pp [16] Kallappa, P. and Ray, A.: Fuzzy wide-range control of fossil power plants for life extension and robust performance, Automatica, 2000, 36, pp [17] Ellyin, F.: Fatigue damage, crack growth and life prediction (Chapman & Hall, 1997) 16

17 [18] Lu, S., and Wilson, B.: On-line stress calculation and life monitoring systems for boiler components, Trans. Inst. MC, 1998, 20(1), pp [19] Anzai, H., and Endo, T.: On-site indication of fatigue damage under complex loading, Int. J. Fatigue, 1979, 1(1), pp [20] Tangirala, C.: Stochastic modeling of fatigue damage for on-line monitoring and lifeextending control (Ph.D. Thesis, Pennsylvania State University, 1996) [21] Tan, W., Marquez, H. J., Chen, T., and Gooden, R. K.: H control design for an industrial boiler, Proc. of ACC Conference, 2001, pp [22] Rink, R. D., White A., and Leung, R.: SYNSIM: a computer simulation model for the Mildred Lake steam/electrical system of Syncrude Canada Ltd. (Technical Report, University of Alberta, 1996) [23] Tan W., Marquez H. J., and Chen T.: Multivariable robust controller design for a boiler system, IEEE Trans. on Control Systems Technology, 2002, 5, pp