Ind. Eng. Chem. Res. 2008, 47, 9931–9940
9931
Controllability Analysis and Decentralized Control of a Wet Limestone Flue Gas Desulfurization Plant A. L. Villanueva Perales,* F. J. Gutie´rrez Ortiz, P. Ollero, and F. Mun˜oz Gil Departamento de Ingenierı´a Quı´mica y Ambiental, UniVersidad de SeVilla, Camino de los Descubrimientos s/n, 41092, SeVille, Spain
Presently, decentralized feedback control is the only control strategy used in wet limestone flue gas desulfurization (WLFGD) plants. Proper tuning of this control strategy is becoming an important issue in WLFGD plants because more stringent SO2 regulations have come into force recently.3 Controllability analysis is a highly valuable tool for proper design of control systems, but it has not been applied to WLFGD plants so far. In this paper a decentralized control strategy is designed and applied to a WLFGD pilot plant taking into account the conclusions of a controllability analysis. The results reveal that good SO2 control in WLFGD plants can be achieved mainly because the main disturbance of the process is well-aligned with the plant and interactions between control loops are beneficial to SO2 control. 1. Introduction Wet limestone flue gas desulfurization is the most widespread technology to control SO2 emission in coal-fired power stations. In the wet limestone desulfurization process the flue gas is scrubbed with slurry containing limestone particles (CaCO3). The SO2 absorbed from the flue gas reacts with limestone particle, and gypsum (CaSO4 · 2H2O) is produced according to the following overall reaction: SO2 + CaCO3(s) + 0.5O2 + 2H2O f CaSO4 · 2H2O(s) + CO2 (1) At present, operating conditions of WLFGD plants have to be changed for achieving higher SO2 removal and meeting more stringent SO2 regulations that have come into force recently.3 In fact, SO2 removal target is usually set somewhat higher than necessary to provide a safety margin against process disturbances. However, the larger the safety margin the higher the operating cost. How much the safety margin can be reduced depends on proper control strategy selection and design. Controllability analysis is a valuable tool for that purpose because it reveals the control limitations of the plant to be controlled. However, controllability analysis of WLFGD plants has not been performed in the literature so far. The main components of a WLFGD plant are the absorber, where flue gas is brought into contact with the scrubbing slurry, and the oxidation tank, where air is sparged to promote the oxidation of the SO2 absorbed to gypsum (Figure 1). The most important controlled variable in a WLFGD plant is the SO2 concentration in the desulfurized gas because regulation SO2 emission limits must be fulfilled. The oxidation tank pH must also be controlled because it has a great influence on limestone utilization and, therefore, on gypsum quality. The higher the oxidation tank pH, the lower the limestone utilization and the higher the residual limestone content on gypsum.4 If commercial-quality gypsum is to be produced, the content of residual limestone must be kept lower than 3 wt %. Besides, the pH must be kept within a safety range because scaling, plugging, or corrosiveness problems in the plant could occur.5 The main manipulated variables are the fresh limestone slurry flow rate to the oxidation tank and the slurry * To whom correspondence should be addressed. E-mail: villanueva@ esi.us.es. Phone: +0034 954487223.
recycle flow rate in the absorber. The main disturbance in a WLFGD plant is the inlet SO2 load to the absorber. In EPRI’s FGD manual5,11 some control strategies for WLFGD systems are briefly presented. Among them, the most suitable strategy for rejecting changes in the inlet SO2 load is shown in Figure 1.2,5 The fresh limestone flow rate is mainly determined by a static feedforward controller from the inlet SO2 load to the absorber and the SO2 removal efficiency desired. The SO2 load is estimated from the product of the inlet SO2 concentration and the power plant boiler load. The fresh limestone supply is corrected by a PI controller to keep the pH in a design range. The slurry recycle flow rate is only adjusted when the inlet flue gas flow rate greatly varies to keep the liquid-gas ratio in the absorber in a design range.5 This is done manually varying the number of recirculation pumps in service corresponding to the different spray levels in the absorber. With this control strategy the correct amount of limestone is added for the SO2 removal desired, keeping the residual limestone content low and the pH in a favorable range. The main disadvantage of this control strategy is that the outlet SO2 concentration is not directly controlled and other disturbances unlike the inlet SO2 load (e.g, limestone reactivity, chemical composition of the slurry,7 etc.) can change the relation between the fresh limestone supply, SO2 removal efficiency, and pH. Another important disadvantage is that only three to five discrete values of slurry recycle flow rate can be set in a spray scrubber. The later results in transitory SO2 removal efficiencies greater than required and unnecessary pumping power consumption when large changes in the flue gas flow rate occur.6 Oosterhoff2 simulated a modification of the previous control strategy in which a PI controller, manipulating the fresh limestone flow rate, directly controlled the SO2 concentration in the desulfurized gas. He compared the performance of that strategy with the case in which only feedforward control was used and better control performance was observed. However, the proposed strategy also has disadvantages because the pH is not controlled and aforementioned operational problems could occur. Anyway, the simulations were carried out using a very simple model of a wet limestone flue gas desulfurization unit which was not even validated. Thus, his conclusions on dynamic and control of WLFGD units must be considered cautiously. Mitsubishi Heavy Industries1 presented a decentralized control strategy (Figure 2) in which the gypsum purity and SO2
10.1021/ie800801a CCC: $40.75 2008 American Chemical Society Published on Web 11/12/2008
9932 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008
Figure 1. Typical control strategy of a WLFGD industrial plant.
Figure 2. Decentralized control strategy using a limestone slurry analyzer and controllable pitch-vane pumps.
concentration in the desulfurized gas were controlled simultaneously. The gypsum purity is maintained at a fixed value by directly controlling the limestone concentration in the oxidation tank which is measured with a proprietary continuous limestone slurry analyzer. This is done by a feedback controller which manipulates the raw limestone supply to the oxidation tank. The outlet SO2 concentration is controlled by manipulating the recycle slurry flow rate with controllable vane-pitch pumps. A Smith predictor is used for that purpose. Besides, a feedforward controller, based on the inlet SO2 load, is also used in both loops. The advantages of this control strategy are that the excess of limestone in the slurry is directly controlled and continuous adjustment of the recycle flow rate is possible due to the controllable vane-pitch pumps, which allows a precise control of the outlet SO2 concentration thus avoiding unnecessary power pumping consumption. The control strategy performance was evaluated in a FGD unit of the 600 MWth Tohoku power plant. Good control of outlet SO2 concentration and limestone
concentration was observed for large changes in the boiler load. Nevertheless, issues such as the analysis of control properties of the plant and design of the control strategy were not shown. The aim of this work is to show on the basis of a WLFGD pilot plant how a controllability analysis of a full-scale WLFGD plant would be performed and how the results of the analysis would be used for proper control system selection and design. For those purposes, the paper is conceptually divided into three parts. First, the WLFGD pilot plant is briefly described, including the control system. Second, a controllability analysis of the pilot plant is carried out from a multivariable linear model of the pilot plant previously obtained by the authors.4 Some general conclusions about controllability of WLFGD plants are withdrawn. Finally, a decentralized feedback control strategy is designed and applied to the pilot plant. Control performance is evaluated for inlet SO2 load rejection and outlet SO2 concentration and oxidation tank pH reference tracking.
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9933
Figure 3. Pilot plant flowsheet. AT, SO2 sensor; M, variable speed motor; FT, gas flowmeter; pHT, pH sensor.
2. Wet Flue Gas Desulfurization Pilot Plant The pilot plant (Figure 3) was thoroughly described in a previous paper4 so only a brief description is provided here. Flue gas is generated in a pyrotubular boiler equipped with a gas-oil modular burner. Carbon disulphide (CS2) is added to gas-oil to obtain a desired SO2 concentration in the flue gas. A fraction of the flue gas generated is cooled by water quenching and sent to the absorber. The rest fraction of the flue gas bypasses the absorber. In the absorber, flue gas is brought into countercurrent contact with limestone slurry, and SO2 absorption takes place. The slurry leaving the absorber is sent the oxidation tank where fresh limestone slurry is continuously supplied and a slurry bleed avoids accumulation of gypsum, which is sent to a storage tank. Fresh limestone and slurry bleed are pumped using peristaltic pumps. Most of the slurry from the oxidation tank is recycled to the top of the absorber using a centrifugal pump. The pilot plant can be monitored and manually controlled by means of a SCADA computer. For executing the control algorithms developed in this work a new computer (control station) was added to the pilot plant communication network. The control algorithms were implemented in Simulink which was installed in the control station. Communication between Simulink and the programmable logic controller (PLC) of the plant is by means of a Profibus-DP fieldbus using OPC protocol. For this purpose, an OPC client for Matlab and an OPC server were installed and configured in the control station. Real-time execution of Simulink is possible using a real time blockset8 which synchronizes Simulink with the computer processor clock. 3. Linear Input-Output Controllability Analysis The main control problem in a WLFGD plant is the rejection of its most important disturbance, the inlet SO2 load to the absorber. In coal-fired power stations this disturbance is a consequence of large changes in boiler load to meet power grid demand and also changes in sulfur content of coal, boiler air intake, and so forth. On the other hand, the setpoints of SO2
Table 1. Nominal Operating Point of the Desulfurization Pilot Plant treated flue gas flow rate, G (Nm3/h) inlet flue gas SO2 concentration (ppmv) L/G ratio in the absorber (L/(Nm3)) oxidation tank pH SO2 removal efficiency (%) outlet flue gas SO2 concentration (ppmv) reagent ratio (Ca/S)
250 1500 12 3.9 95 70 1.03
and pH are not significantly changed because outlet flue gas SO2 concentration is restricted by SO2 emission limit and oxidation tank pH must be kept in a narrow operating range to yield commercial-quality gypsum, respectively. Therefore, from a process control point of view, disturbance rejection is much more important than reference tracking in WLFGD plants. Anyway, in this paper both disturbance rejection and reference tracking are studied to find out what control performance can be expected using decentralized control. Before tackling the design of the control strategy, it is important to carry out an input-output linear controllability analysis of the WLFGD plant to get an idea of how easy the plant is to control.9 For that purpose, a linear model of the WLFGD plant is necessary. An identification methodology for WLFGD plants previously proposed by the authors was applied to the pilot plant, and a linear multivariable model was identified and validated.4 The model takes into account the main controlled, manipulated, and disturbance variables in a WLFGD plant. Plant identification was carried out around a nominal operating point (Table 1) which was previously determined so that the usual WLFGD operating requirements were met; that is, (European) SO2 emission limit for large power plant stations was met3 (outlet SO2 concentration e70 ppmv), and commercial-quality gypsum was obtained. The linear model of the pilot plant is shown in eq 2: where SO2 is the outlet flue gas SO2 concentration (ppmv), pH is the oxidation tank pH, Q is the rotating speed of the fresh limestone supply peristaltic pump (rpm), L is the rotating speed of the slurry recirculation pump motor (Hz), and Load represents the SO2 load to the absorber (Nm3 · ppmv/h), which is calculated as the product of the inlet flue gas flow rate
9934 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 Table 2. Multivariable Poles and Zeros of the Pilot Plant (s-Domain) poles
zeros -4
-4.45 × 10 -6.90 × 10-4 -7.78 × 10-4 -10.35 × 10-4 -83.75 × 10-4
and inlet flue gas SO2 concentration. The circumflex accent means that the variables are dimensional. The time scale of the model is seconds. 3.1. Scaling. To carry out the controllability analysis the linear model of the plant must be scaled by dividing each variable by its maximum expected or allowable change9 so that the magnitude of each scaled variable is equal to or less than 1: d ) dˆ/dˆmax
u ) uˆ/uˆmax
y ) yˆ/eˆmax
r ) rˆ/eˆmax (3)
where dmax is the largest expected disturbance change, umax is the maximum allowable input change, and emax the maximum allowable control error. The scaled reference r is not less than one in magnitude, but it is usually expressed using a scaled reference r˜ less than one in magnitude: r˜ ) rˆ/rˆmax
rˆmax rˆ r) ) r˜ ) Rr˜ eˆmax eˆmax
(4)
Applying the scaling procedure to the pilot plant model, the scaled transfer functions G and gd are obtained as follows: ˆ Du , gd ) De-1gˆdDd G ) De-1G
(5)
where De )
[
] [
]
ˆ max 0 eˆmax pH ^ Q 0 , D ) , Dd ) Loadmax u eˆmax SO2 0 0 Lˆmax (6)
Thus, the scaled control error can be calculated from eq 7: e ) y - r ) Gu + gdd - Rr˜ R ) De-1Dr, Dr )
[
rˆmax pH 0 rˆmax SO2 0
(7)
]
10-4 × (-8.34 + 2.74j) 10-4 × (-8.34-2.74j) -3.01 × 10-4
(iii) êmax pH is chosen as 0.12 because changes in the oxidation tank pH of that order do not affect the process. (iv) The maximum expected change in SO2 reference (rˆmax SO2) is chosen as 60 ppmv just in a hypothetical case that SO2 removal efficiency had to be increased beyond 99% (10 ppmv outlet SO2 concentration). The maximum expected change in pH reference (rˆmax pH) is selected as 0.7 which is a typical pH variation range in wet flue gas desulfurization units.10 (v) The maximum allowable change in the recycle pump motor frequency (Lˆmax) is upper limited to +6 Hz due to flooding problems in the packed tower. The maximum allowable change ˆ max) is selected in the rotating speed of the peristaltic pump (Q as 10 rpm which would suffice to reject maximum changes in disturbance and references. Using these scaling factors, the scaled transfer functions are
[
-6.6 6.96 966s + 1 1448s + 1 G) 4 -4.73e-120s -0.218(4.69 × 10 s + 1) 5 2 1284s + 1 2.68 × 10 s + 2364s + 1 -0.66 988s + 1 gd ) 1.78 236s + 1
[ ]
(9)
From the scaled linear model the following properties of the pilot plant are studied: multivariable poles and zeros, singular values, disturbance condition number, control limitations due to input saturation,and relative gain array. 3.2. Multivariable Poles and Zeros. Poles and/or zeros located in the right half of the s-plane limit the achievable bandwidth.9 In Table 2 poles and zeros of the plant are shown (note: scaling does not change the poles and zeros of a system). It can be observed that there are neither poles nor zeros in the right half-plane so the bandwidth is not limited by neither poles nor zeros. 3.3. Maximum and Minimum Singular Values. The singular values give information about the gains of a multivariable plant.9 The maximum and minimum singular values are the largest and smallest gains for any input direction, respectively, that is,
(8)
The scaling factors depend on control requirements but also on constraints of plant equipment and operating conditions. In the case of the WLFGD pilot plant, the following scaling factors were used: (i) The maximum expected change in the treated flue gas flow rate is 20 Nm3/h. Inlet flue gas SO2 concentration is supposed to be constant at its nominal value (1500 ppmv). Therefore, the maximum expected change in the inlet SO2 load is Loadmax) 30000 Nm3 · ppmv/h. (ii) êmax SO2 is selected as 50 ppmv which corresponds to changes of (3.3% in the SO2 removal efficiency. This maximum allowable error is sensible due to the significant and fast effect of the inlet SO2 load on the outlet SO2 concentration in WLFGD plants.4
]
σ(G) e
||GV||2 e σ(G) ||V||2
(10)
where σ j (G) and σ _(G) are the maximum and minimum singular values, respectively, and V is any input direction different from zero. The minimum singular value is a useful measure for evaluating the feasibility of achieving acceptable control. The larger σ _(G), the better the control performance that can be achieved.9 In Figure 4 the maximum and minimum singular values of the pilot plant are shown as a function of frequency. The minimum singular value is greater than one for a large range of frequencies, so input saturation for reference tracking is not expected.9 In Table 3 singular value decomposition of the pilot plant model at low frequency (steady state) is shown. At steady state the lowgain output direction is mainly in the outlet SO2 direction, but this
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9935
Figure 4. Maximum and minimum singular values of the pilot plant.
Figure 5. Condition number and disturbance condition number of the plant as a function of frequency.
Table 3. Singular Value Decomposition of the Pilot Plant Model at Steady State singular values
input direction [Q, L]T
output direction [pH, SO2]T
σ j (G) ) 10.21 σ _(G) ) 3.21
[-0.8, +0.6]T [-0.6, -0.8]T
[-0.93, +0.36]T [+0.36, +0.93]T
does not necessarily imply that SO2 is difficult to control because the minimum singular value is larger than 1. On the other hand, the high-gain output direction at steady state is mainly in the pH direction. Therefore, at steady state, pH control is expected to be easy. Anyway, control performance for disturbance rejection can depend strongly on the direction of the disturbance.12 This is studied by the disturbance condition number. 3.4. Disturbance Condition Number. The disturbance condition number of a plant G (γd(G)) measures the magnitude of the manipulated variables needed to reject a disturbance in the direction z (z ) gdd) relative to rejecting a disturbance with the same magnitude but in the direction of the high-gain output direction of the plant.12 γd(G) )
||G-1z||2 ||G-1gd||2 σ(G) ) σ(G) ||z||2 ||gd||2
Figure 6. Magnitude of inputs needed for perfect rejection of largest expected change in SO2 load as a function of frequency.
(11)
The closer the disturbance direction is to the high-gain output direction of the plant, the smaller the changes that are needed in the manipulated variables to reject the disturbance. Therefore, the disturbance condition number can be used to get an idea of how easy it is to reject a disturbance. It holds that9 1 e γd(G) e γ(G)
(12)
If the disturbance condition number as a function of frequency is nearly one, the disturbance direction practically matches the high-gain output direction of the plant while if the disturbance direction matches the low-gain output direction, γd(G) equals the plant condition number (γ(G) ) σ j (G)/σ _(G)). How well the disturbance is aligned with the high-gain output direction is especially important when dealing with control of illconditioned plants.13,14 In Figure 5 the condition number of the pilot plant is shown as a function of frequency. It can be observed that γ(G) is not large relative to 1 so the plant is not ill-conditioned. Therefore, disturbance direction is not critical for achieving a good control performance. In Figure 5 disturbance condition number of the pilot plant is also shown as a function of frequency. On one hand, the disturbance condition number is small at high frequency so changes in SO2 load can be easily rejected initially. This is expected also to be true in any WLFGD plant because the large and fast effect of slurry recycle on outlet SO2 concentration can counteract effectively the large and fast effect of disturbance on the outlet SO2 concentration. On the other hand, at low frequencies
Figure 7. RGA elements of the pilot plant.
(steady state) changes in the SO2 load seem to be more difficult to reject. The steady state disturbance direction is [pH, SO2] ) [-0.76, 1.78] which is mainly in the SO2 direction while the high-gain plant output direction at steady state is mainly in the pH direction (Table 3). This could result in input saturation which is checked in the next point. 3.5. Input Saturation for Disturbance Rejection. The input needed for perfect disturbance rejection can be obtained from eq 7 with e ) 0 and r˜ ) 0: u ) G-1gdd
(13)
The most unfavorable case is when ||d||∞ ) 1; that is, the largest expected disturbance change is considered. In that case, input saturation does not occur if
9936 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008
||u||∞ ) ||G-1gd||∞ e 1
(14)
The magnitude of scaled inputs needed for perfect rejection of the largest expected change in SO2 load as a function of frequency is shown in Figure 6. The maximum changes of the scaled inputs are lower than 1, so input saturation in the pilot plant is not expected even for perfect disturbance rejection. 3.6. Relative Gain Array. The relative gain array (Λ(G)) is a well-known tool for pairing selection in decentralized control, but it is also a very useful tool for assessing achievable control quality and controller structure selection.9,14 The relative gain array (RGA) is defined for a 2 × 2 plant as
[
] [
λ11 λ12 λ11 1 - λ11 ) λ21 λ22 1 - λ11 λ11 1 λ11 ) g12g21 1g11g22
Λ(G) )
]
Figure 8. Effect of largest expected SO2 load change on outlet SO2 concentration and tank pH (magnitude of gd2 and gd1, respectively).
(15)
Some important conclusions from RGA elements have been presented in the literature: (i) Plants with large RGA elements (>5-10) are difficult to control, and inverse-based controllers should not be used.14 (ii) For decentralized control pairing selection the following rules are suggested:9 (1) input-output pairings on negative steady state RGA elements should be avoided. (2) Prefer pairings for which the RGA matrix is close to identity in the crossover region. In a previous paper by the authors,4 it was demonstrated that RGA elements of WLFGD plants are not large due to the dynamic properties of this kind of plant. This is also true for the RGA of the pilot plant which is shown in Figure 7. Therefore, inverse-based controllers are expected to be used in WLFGD plants without problems related to input uncertainty.14 With regard to pairing selection in decentralized control, the steady state RGA of the pilot plant is shown in eq 16: Λ(G(0)) )
[
0.047 0.953 0.953 0.047
]
(16)
It can be seen that steady state RGA elements are positive. Moreover, this is also expected to be true for any WLFGD plant. For example, let us consider the λ11 element which is the openloop gain between Q and pH divided by the gain between the same two variables when the SO2 loop is under “perfect control”. The steady state open-loop gain between Q and pH is positive because adding more fresh limestone to the oxidation results in an increase of the pH tank. Furthermore, adding more fresh limestone to the oxidation tank results in more SO2 absorbed due to the higher oxidation tank pH. Therefore, if SO2 were controlled, at steady state it would be necessary to decrease the recycle flowrate to keep constant the outlet SO2 concentration. This would result in a further increase of the oxidation tank pH. Thus, both open- and closed-loop gain between Q and pH are positive, and close-loop gain is larger than open-loop so 0 < λ11(0) < 1, resulting in all RGA elements to be positive. Because all steady state RGA elements are positive, neither pairing can be disregarded initially. Then, pairing selection in WLFGD plants should be done according to the second pairing rule which is based on the closed-loop crossover frequency. The crossover frequency should be selected to ensure acceptable control (|e| < 1) for the largest expected disturbance change. In Figure 8 the scaled control errors of pH and SO2 due to the largest expected SO2 load change, that is, the gains of gd1 and gd2, respectively, are shown as a function of frequency. Without any control system, the maximum expected change in the SO2
load would result in acceptable pH deviation, but unacceptable SO2 deviation would be obtained for frequencies below 6.5 × 10-3 rad/s. Therefore, the crossover frequency should be larger than 6.5 × 10-3 rad/s, and then, according to the second pairing rule and Figure 7, SO2-L and pH-Q pairings should be selected. The second pairing rule just suggests pairing variables that are “close to each other”; that is, the dynamic between paired variables are fast, so that the responses of each loop are decoupled in the crossover region. The former is certainly true for the pairing SO2-L due to the fast effect of the recycle flow rate on the outlet SO2 concentration. Therefore, these pairings seem to be sensible for WLFGD plants in general. 4. Decentralized Control Strategy of the Desulfurization Pilot Plant The design of a decentralized control system is usually carried out in two steps: (1) pairing selection and (2) design of a controller for each loop. Pairing selection was carried out previously. In this section the design of the control loops is considered. First, a controllability analysis focused on decentralized control9,15 is carried out to assist in the tuning of the controllers. Then, the controllers are tuned using the IMC-PID tuning rules as a guideline. Lastly, the performance of the control system is tested for reference tracking and disturbance rejection in the pilot plant. 4.1. Controllability Analysis for Decentralized Control. A lower bound on controller gain for acceptable reference tracking and disturbance rejection in decentralized control has been presented by Skogestad et al.9,15,16 They proved that the control error of each loop (ei) can be estimated by eq 17 for frequencies below the closed loop bandwidth (ωB):16 ei ≈ -(1 + giiki)-1γijrj + (1 + giiki)-1δikdk
ω e ωB (17)
where ki is the controller of loop i, γij is the ij element of the performance relative gain array (PRGA), δik is the ik element of the closed-loop disturbance gain array (CLDG), rj is the reference of loop j, and dk is the kth disturbance. The PRGA and CLDG arrays are calculated as follows: ˜ ) diag{gii} (18) ˜ G-1, CLDG ) ΓGd, G PRGA } Γ ) G In eqs 17 and 18 the elements of G are ordered so the pairings are located in the diagonal. From eq 17 it can be proved that the following bound must be fulfilled for each loop i to obtain acceptable control9,16 considering the maximum expected changes in disturbances (|dk| ) 1) and references (|rj| ) Rj): |1 + giiki| > max{|δik|, |γij| · |Rj|} ∀ i k,j
(19)
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9937
medium frequencies (γ12 > 1) but favorable to the SO2 loop because γ22 is smaller than one. From the PRGA and CLDG elements it can be concluded that for the pairings selected interactions are beneficial to SO2 control for both disturbance rejection and reference tracking but they are not beneficial to pH control. This is convenient because the outlet SO2 concentration is the most important controlled variable. 4.2. Design of Controllers. Disturbance rejection is the most important objective of the control system in a wet flue gas desulfurization unit. Therefore, the controller of each loop was tuned to fulfill the bound imposed by eq 19 with regard to SO2 load rejection: Figure 9. Comparison of CLDG and gd elements.
|1 + giiki| > δi
i ) 1, 2
(20) 17,18
Figure 10. PRGA elements as a function of frequency.
where Rj is the jth element of the diagonal of the matrix R. With regard to the pilot plant, loop 1 corresponds to the pairing pH-Q while loop 2 corresponds to the pairing SO2-L. Besides, there is only one disturbance in the pilot plant so k ) 1; that is, Gd is a vector. The elements of the CLDG and PRGA are very useful for understanding the effect of interactions in decentralized control. In Figure 9 the CLDG and gd elements of the desulfurization pilot plant are shown as a function of frequency. δi is the “apparent” disturbance gain as seen from loop i when the system is controlled using decentralized control9 while gdi is the openloop disturbance gain. It can be observed that interactions reduced the effect of the SO2 load on the outlet SO2 but increase its effect on the oxidation tank pH. A physical explanation can be given for this behavior. For example, let us study the response of the control system for an increase in the SO2 load. Initially, the outlet SO2 concentration would increase significantly while the oxidation tank pH would decrease slightly. Then, the SO2 controller would increase the recycle flow rate to decrease the outlet SO2 concentration. This would result in an additional decrease of the oxidation tank pH. Therefore, we can observe that the effect of the SO2 load on the oxidation tank pH increases due to interactions. On the other hand, the pH controller would increase the fresh limestone supply to the oxidation tank to keep the pH in its setpoint. The later would result in an additional decrease of the outlet SO2 so that the interactions reduce the effect of the disturbance on the outlet SO2 concentration. In Figure 10 the PRGA elements are shown. γij is a measure of performance degradation in loop i due to a change in the reference of loop j.16 For pH reference changes, interactions are favorable to both pH and SO2 loop because γ11and γ21 are smaller than 1, respectively. However, for SO2 reference changes, interactions are unfavorable to pH loop at low and
The IMC-PID tuning rules developed by Rivera were chosen because they are reliable and easy to use, although they are not specifically oriented to disturbance rejection (Table 4). However, due to the pairings selected, the interactions decrease the effect of the disturbance on SO2, and the IMC-PID tuning rules provided a good first estimate of controller settings. These settings were slightly modified before adjusting the lambda parameters. Thus, the integral time of the pH controller, k1(s), was reduced from 966 s, suggested by the tuning rules, to 600 s to avoid a very sluggish disturbance rejection. On the other hand, the derivative term of SO2 controller, k2(s), was removed because the performance of the controller did not change due to the high derivative action of g22. Next, the lambda parameters were first adjusted until the disturbance bounds (eq 20) were fulfilled and then fine-tuned through simulations until acceptable control was obtained for the maximum expected change in the SO2 load. The lambda parameters finally used are λ(k1) ) 1043.56 s and λ(k2) ) 272 s. The final design of the controllers is shown in eq 21 in scaled form: 1 600s 1 1 k2(s) ) -39.78 1 + (21) 2389s 55253s + 1 In Figures 11 and 12 fulfillment of controller gain bounds (eq 19) for pH and SO2 control loops is shown, respectively. Disturbance bounds (eq 20) are met for both controllers except within a small frequency range at high frequencies. This is not a problem because disturbances in that range of frequency are not expected. This was checked by simulating the response of the control system for the maximum expected step change in the SO2 load. It was confirmed that acceptable control was obtained for disturbance rejection. The IMC-PID tuning rules suggest using a filter to cancel the derivative term of the g22 transfer functions, and therefore, the resulting controller k2(s) does not correspond to any common type of industrial PID controller. However, this controller can be easily implemented in commercial DCS (digital control systems) used in many process plants and power stations. In the pilot plant both controllers were implemented in discrete form. The pH and SO2 controllers were discretized using the Euler (backward difference) and Tustin’s approximation, respectively. A sampling time of 30 s was used. The discretized controllers are shown in eq 22.
(
k1(s) ) 0.133 1 +
(
)
)(
)
1.05 - z-1 1 - z-1 1 + 0.01248z-1 - 0.988z-2 k2(z) ) -0.01088 1 - 1.999z-1 + 0.995z-2
k1(z) ) 0.133
(22)
9938 Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 Table 4. PID-IMC Tuning Rules Used as Guideline for pH and SO2 Loops, Respectivelya controller
[
k(s) ) kc 1 +
plant model
][
1 1 + τDs τIs τFs + 1
]
k τs + 1 k(βs + 1) , τ2s2 + 2τδs + 1
a
kck
τI
τ/λ
τ
2τδ/λ
2τδ
τD
τF
τ/2δ
β
β>0
λ is an adjustable tuning parameter.
Figure 11. Comparison of gain bounds and controller gain for pH loop. Figure 13. Control system response for 80% of maximum expected step change in the disturbance (SO2in kept constant at 1500 ppmv). Setpoints: pH ) 3.8, SO2 ) 52 ppmv.
obtained from simulations. To improve disturbance rejection, feedforward controllers were added to the pH loop (kˆd1) and SO2 loop (kˆd2; eq 24). A sampling time of 30 s was used for each controller. gˆd1 0.0759 - 0.067z-1 ) 3.33 × 10-4 gˆ11 0.8133 - 0.720z-1 gˆd2 0.7098 - 0.9327z-1 + 0.2463z-2 kˆd2 ) ) 2 × 10-4 gˆ22 2.808 - 4.49z-1 + 1.685z-2
kˆd1 ) -
Figure 12. Comparison of gain bounds and controller gain for SO2 loop.
(24)
The validity of the approximations was checked by simulations which confirmed that the control system performance using the continuous or discrete form of the controllers was the same. The discrete controllers were implemented in the plant without scaling as shown in eq 23.
The performance of the decentralized control system with feedforward control for the same change in the SO2 load is shown in Figure 14. Now, acceptable control of SO2 is obtained. The largest SO2 deviation is decreased 45% (33 ppmv) with regard to control without feedforward. Besides, the time that the outlet SO2 concentration is above the setpoint is decreased from 1000 to 400 s, although the SO2 settling time does not vary. On the other hand, pH control performance is barely improved. In conclusion, SO2 control greatly improves when feedforward control is added to the decentralized control system, and therefore, it is recommended to use it in desulfurization plants. SO2 Reference Tracking. Control performance for SO2 reference tracking at different values of tank pH has also been assessed. Step changes of 80 ppmv in SO2 reference at tank pH of 4 and 4.5 are shown in Figures 15 and 16, respectively. The magnitude of the changes is 20% larger than the maximum expected one, but it has been selected to study SO2 reference tracking widely around the nominal outlet SO2 concentration (70 ppmv). It can be observed that control performance at different values of pH is very similar. In all cases acceptable pH control is achieved. Acceptable pH control for SO2 reference
-1
1.05 - z kˆ1(z) ) 11.08 1 - z-1 1 + 0.01248z-1 - 0.988z-2 kˆ2(z) ) -0.0013 (23) 1 - 1.999z-1 + 0.995z-2 4.3. Pilot Plant Experiments. The performance of the decentralized control was studied in the desulfurization pilot plant for disturbance rejection and reference tracking. The results are shown next. SO2 Load Rejection. The response of the control system for 80% of the maximum expected step change in the SO2 load is shown in Figure 13. Acceptable control of pH is obtained, but the largest outlet SO2 concentration deviation from the SO2 setpoint is 60 ppmv, 20% larger than 50 ppmv, the maximum allowable deviation. This result slightly differs from that
Ind. Eng. Chem. Res., Vol. 47, No. 24, 2008 9939
Figure 14. Response of the control system with feedforward for 80% of maximum expected step change in the disturbance (SO2in kept constant at 1500 ppmv). Setpoints: pH ) 3.8, SO2 ) 52 ppmv.
Figure 15. SO2 reference tracking for 80 ppmv step changes at tank pH of 4.
Figure 16. SO2 reference tracking for 80 ppmv step changes at tank pH of 4.5.
changes was expected from Figure 11 but only around nominal pH tank (≈3.9), where the pilot plant was identified. Concerning SO2 control, a rising time of 500 s and 18% overshoot are observed at both values of tank pH. From the results it can be concluded that the dynamic of the plant does not change significantly in the pH range 3.9-4.5 and good control performance can be obtained for SO2 reference tracking. It must be pointed out that good SO2 reference tracking was possible due to the filter considered in the PID-IMC tuning rules for the
Figure 17. pH reference step change from 3.8 to 4.0. SO2 setpoint: 32 ppmv.
Figure 18. pH reference step change from 4.0 to 4.5. SO2 setpoint: 32 ppmv.
L-SO2 controller. This filter tries to cancel the high derivative action of the zero of the L-SO2 transfer function, which is very near to the jω axis. If the zero is not canceled, very aggressive responses are obtained for SO2 reference tracking but not for pH reference tracking and disturbance rejection. Therefore, a standard PID without filter can be used in the SO2 loop if the control system only has to deal with disturbance rejection and/ or pH references tracking. pH Reference Tracking. Results for pH reference step changes of different magnitude (+0.2, +0.5) are shown in Figures 17 and 18, respectively. In both experiments, the pH closed-loop time constant is 800 s, about 80% of the pH openloop time constant, but a large pH overshoot is obtained (30%). Therefore pH control is not very good. Anyway, this is not really important because very good SO2 control is achieved. The maximum SO2 deviation from the setpoint is 20 ppmv which is very low compared with 50 ppmv, the maximum allowable error. From Figures 17 and 18 it is noticed that the fresh limestone supply (Q) has not changed at steady state regardless of the pH reference step changes. Only the recycle flow rate has changed at steady state. This may be confusing to the reader because of the pairings selected in the decentralized control system. However, it is easy to understand if the steady state matrix of the plant (eq 25) is analyzed.
[
]
6.96 -6.6 (25) -4.73 -0.218 Because the recycle flow rate barely affects the outlet SO2 concentration at steady state (g22(0) , 1), steady state control G(0) )
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of SO2 practically only depends on manipulation of the fresh limestone supply. When the SO2 reference is kept constant (Figures 17 and 18), the fresh limestone supply barely changes at steady state and the pH reference change is practically carried out by the recycle flow rate. 5. Conclusions The dynamic properties and control limitations of a wet limestone flue gas desulfurization plant have been studied by means of an input-output linear controllability analysis. The analysis reveals that (1) WFGD plants are not difficult to control because RGA elements are not large.4 Moreover, WLFGD plants are well aligned with the main disturbance of the process, the inlet SO2 load to the absorber, so disturbance rejection does not pose a serious problem. (2) For decentralized control, pairing the recycle flow rate with the outlet SO2 concentration and fresh limestone supply with tank pH is beneficial to SO2 control and avoids instability at high frequencies due to interactions. Furthermore, steady state control is greatly influenced by interactions between control loops. Finally, a decentralized control strategy with the aforementioned pairings has been designed using a controllability analysis and evaluated in the desulfurization pilot plant for disturbance rejection and reference tracking. The results reveal that acceptable control of a WLFGD plant can be obtained in a wide operating condition range using a decentralized control strategy. These results will be used as a benchmark for designing advanced control strategies that could outperform the decentralized feedback control strategy. Acknowledgment This work is part of the research project “Advanced control of wet flue gas desulfurization units” (PPQ2001-1106) funded by the Science and Technology Ministry of Spain. Nomenclature CLDG ) closed loop disturbance gain dk ) kth disturbance D ) scaling matrix e ) vector of control errors gdik ) ikth element of Gd gij ) ijth element of G G ) plant transfer matrix Gd ) disturbance transfer matrix ki ) controller of loop i kdi ) feedforward controller of loop i L ) rotating speed of the slurry recirculation pump motor (Hz) Load ) SO2 load to absorber (Nm3 · ppmv/h) pH ) oxidation tank pH Q ) rotating speed of fresh limestone supply peristaltic pump (rpm) r ) vector of reference outputs (setpoints) Rj ) jth element of R matrix R ) scaled reference matrix SO2 ) outlet flue gas SO2 concentration (ppmv) u ) vector of manipulated inputs y ) vector of outputs Greek letters δik ) ikth element of CLDG γ(G) ) condition number γd(G) ) disturbance condition number γij ) ijth element of Γ λij ) ijth element of RGA
σ j (G) ) maximum singular value σ _(G) ) minimum singular value ω ) frequency ωB ) closed loop bandwidth Λ(G) ) relative gain array (RGA) Γ(G) ) performance relative gain array (PRGA) Vector norms |a|2 ) (∑i|ai|2)0.5, 2-norm (Euclidian norm) |a|∞ ) maxi |ai|, infinity-norm Symbols xˆ ) non-scaled variable (accent on “x”) } ) equal by definition
Literature Cited (1) Okino, S.; Sakamoto, R.; Kono, S.; Oshima, M.; Inoue, S. Development of Automatic Calcium Carbonate Slurry Analyzer for Optimum Control System of Desulfurization Plant. Mitsubishi HeaVy Industries Technical ReView 1997, 34, 15–19. (2) Oosterhoff, A. T. Modelling and simulation of a wet flue gas desulphurization installation. Journal A 1990, 31 (2), 3–8. (3) DirectiVe 2001/80/EC of the European Parliament and of the Council; Office for Official Publication of the European Communities: Luxembourg, 2001. (4) Villanueva Perales, A. L.; Ollero, P.; Gutie´rrez Ortiz, F. J.; Vidal, F. Dynamic analysis and identification of a wet limestone flue gas desulfurization pilot plant. Ind. Eng. Chem. Res. 2008, doi:10.1021/ ie071582x. (5) Radian International LLC. Electric Utility Engineer’s FGD Manual: Volume 1, FGD Process Design; U.S. Department of Energy, Office of Fossil Energy, 1996. (6) Villanueva, A. L. Control avanzado de unidades de desulfuracio´n hu´meda de gases de combustio´n. Ph.D. Thesis. Department of Chemical and Environmental Engineering, University of Seville, Spain, 2007. (7) Ukawa, N.; Okino, S.; Iwaki, T.; Oshima, M.; Watanabe, Y. The effects of fluoride complexes in wet limestone flue gas desulfurization. J. Chem. Eng. Jpn. 1992, 25, 146–152. (8) Daga, L. Real-Time Blockset for Simulink, 2004. http://digilander. libero.it/LeoDaga/Simulink/RTBlockset.htm, accessed Sept, 2004. (9) Skogestad, S.; Postlethwaite, I. MultiVariable Feedback Control; Wiley: New York, 2005. (10) Warych, J.; Szymanowski, M. Optimum values of Process Parameters of the Wet Limestone Flue Gas Desulfurization System. Chem. Eng. Technol. 2002, 25, 427–432. (11) Radian International LLC. Electric Utility Engineer’s FGD Manual: Volume 2; U.S. Department of Energy, Office of Fossil Energy, 1996. (12) Skogestad, S.; Morari, M. Effect of Disturbance Directions on Closed-loop Performance. Ind. Eng. Chem. Res. 1987, 26, 2029–2035. (13) Skogestad, S.; Morari, M. Robust control of ill-conditioned plants: High-purity distillation. IEEE Trans. Autom. Control 1988, 33 (12), 1092– 1105. (14) Skogestad, S.; Morari, M. Implications of Large RGA Elements on Control Performance. Ind. Eng. Chem. Res. 1987, 26, 2323–2330. (15) Hovd, M.; Skogestad, S. Procedure for Regulatory Control Structure Selection with Application to the FCC Process. AIChE J. 1993, 39, 1938– 1953. (16) Hovd, M.; Skogestad, S. Simple Frequency-dependent Tools for Control System Analysis, Structure Selection and Design. Automatica 1992, 28, 989–996. (17) Rivera, D. E.; Morari, M.; Skogestad, S. Internal Model Control. 4. PID Controller Design. Ind. Eng. Chem. Process Des. DeV. 1986, 25, 252–265. (18) Rivera, D. E.; Jun, S. K. An integrated identification and control design methodology for multivariable process system application. IEEE Control System Magazine 2000, 25–37.
ReceiVed for reView May 19, 2008 ReVised manuscript receiVed September 12, 2008 Accepted September 29, 2008 IE800801A
