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Transient power control in dynamic positioning - governor feedforward and dynamic thrust allocation
9th IFAC Conference on Manoeuvring and Control of Marine Craft, 2012 The International Federation of Automatic Control September 19-21, 2012. Arenzano, Italy
Transient power control in dynamic positioning - governor feedforward and dynamic thrust allocation Aleksander Veksler ∗ Tor Arne Johansen ∗ Roger Skjetne ∗∗ ∗
Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU), Trondheim, Norway. (e-mail: [email protected], [email protected]) ∗∗ Department of Marine Technology, Norwegian University of Science and Technology (NTNU), Trondheim, Norway. (e-mail: [email protected]) Abstract: A thrust allocation algorithm with capability to assist the power management system on a vessel with diesel-electric propulsion has recently been proposed. In the work presented here, the results show that the use of this thrust allocation algorithm produces a significant improvement in the stability of the electric bus frequency in the presence of large power load variations, as long as those load variations are small compared to the kinetic energy of the vessel at low speed. Further reduction in the frequency variations are achieved through use of a feedforward signal to the diesel engine governors, which enables them to react to surges in power consumption before they affect the frequency on the electric bus. 1. INTRODUCTION Maintaining a position is an important operational mode for ships that perform functions such as supplying offshore installations, drilling and oil production, laying pipes, supporting divers or ROV operations, or performing surfacebased measurements or exploration. Dynamic positioning systems use the thrusters of the vessel to maintain a steady position relative to an external position reference such as GPS or the position of another vessel. Coordination of the thrusters to keep the operational envelope close to the physical limitations while minimizing the power consumption has been explored in the past two decades, see for example Sørdalen (1997), Johansen et al. (2003), Jenssen and Realfsen (2006), Ruth et al. (2007) and Lindegaard and Fossen (2003). A vessel with a well-designed dynamic positioning capability should have thrusters capable of producing large amounts of moment and force quickly in any direction. This is best achieved by using thrusters that can dynamically change the direction in which they apply force, and by locating the thrusters as far apart as practical. This makes it impractical to power those thrusters with a mechanical connection to the power plant. In the current industry practice for ships that are expected to spend a significant part of their service time in dynamic positioning, electrical power distribution is implemented almost universally. The thruster and propulsion system constitutes the largest consumer in terms of maximal power requirements for most ships, depending on what other large power consumers are installed on a particular vessel. A diesel-driven power plant operates optimally when the load is close to being constant. The diesel engine may be physically unable to respond to a sudden load increase due to the relatively slow turbocharger dynamics. Depending on the tuning, the governor may also be slow to ©2012 IFAC
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respond to changes in load, resulting in frequency variations. Although small frequency variations are tolerated, the classification societies typically set a limit of +/-10% on the maximal frequency deviation from the nominal value. If this limit is exceeded, generators and some of the consumers will disconnect from the power grid. Variations in loading also lead to increased NOX emissions and soot. A thrust allocation algorithm that attempts to even out load between the switchboards is considered in Jenssen and Realfsen (2006). Thrust allocation algorithms available in the literature do not consider variations in the total power consumption by the thrusters, although several do consider variations in force, and by implication, in power of the individual thrusters. Several methods to reduce the consequences of power variation are being explored in the industry. A recent development is using a large battery bank on the vessel to provide additional power during load spikes. Also, an emerging practice is to supply the propulsion units, generators and essential auxiliaries with emergency power during a blackout. In this case, they will be able to resume operation quickly. A thrust allocation algorithm that attempts to assist the power management system in reducing load variations on the power plant by counteracting the load variations from other consumers on the ship has been introduced in Veksler et al. (2012). This algorithm accepts a request to increase or decrease power consumption by the thrusters. It will then attempt to fulfill this request as long as it does not cause the vessel to deviate too far off position, or incur large costs for other aspects of the operation such as fuel consumption. This algorithm exploits the fact that the hull of the vessel itself effectively stores large amounts of energy: for the vessel simulated for this paper, the kinetic energy of the vessel moving at 0.5 m/s is equivalent to the total power plant output for 0.3 seconds. The main contribution of this paper is a case study that 10.3182/20120919-3-IT-2046.00027
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
Description Current time, i.e. time when the thrust allocation problem is solved. Deviation in respectively velocity and position of the vessel from what they would have been if thrust command was allocated exactly, as functions of time. verr (t), xerr (t) ∈ R3 contain longitudinal, lateral and heading components; ∆
ations in the power consumption. This method is welldocumented in the literature - although usually with quadratic power cost function; see Fossen (2002).
Letter T verr (t), xerr (t), verr, T , xerr, T
3/2
Pmin = min Pc K |f | f,s
2
+ kskQ1
(1)
subject to
∆
verr, T = verr (t = T ), xerr, T = xerr (t = T ) Maximal allowed values for verr (t) and xerr (t) Lower and upper limits for the integrals in (6), (7) which calculate deviations in velocity and position at time Te . Respectively actual and desired angular frequency of the voltage on the electrical network. Typically, ω0g = 2π · 60. Thruster configuration matrix. It is a function of the vector α consisting of orientations of the individual thrusters. In this paper, α is assumed to be constant. f ∈ RN , the force produced by individual thrusters. The elements of f are normalized into range [−1, 1]. K ∈ RN ×N such that Kf is the vector of forces in Newtons. Pc ∈ R1×N such that (4) holds. Ψ 0, quadratic cost matrix of variation in force produced by individual thrusters. Θ ∈ R+ is the cost of variation in total power consumption. The total power consumed by the thrusters per (4) The desired rate of change of power consumption by the thrusters. This signal can be used to reduce either frequency or load variations on the electrical network. Minimal power consumption by the thrusters needed to produce commanded thrust. Actual and desired generalized force produced by all thrusters. τ, τd ∈ R3 contain surge and sway forces, and yaw moment.
B(α)Kf = τd + s
verr, max , xerr, max Ts , Te
(2)
f ≤f ≤f (3) The notation used here and in the following is that ∆ p p p T 2 ∆ kxkA = xT Ax and |x|p = x1 x2 · · · xN for any x, A and p of suitable dimension. The cost matrix Q1 must be large enough to ensure that s is significantly larger than zero only when constraints (2)-(3) would otherwise be infeasible. The solution to this optimization problem provides a minimum Pmin to which the power consumption can be reduced while delivering the requested thrust τd , at least as long as the condition s ≈ 0 holds. Power consumption in the thrusters is estimated by the nonlinear relationship
ωg , ω0g
B(α)
f
K
3/2
Pth = Pc K |f | (4) which is similar to what was used in Johansen et al. (2004). In Veksler et al. (2012), the following thrust allocation optimization problem is used for thrust allocation when there is no power bias requirement:
Pc Ψ Θ Pth P˙f f
3/2
min Pc K |f |
f,τe ,s1 ,s2
2 kτe kQ2
Pmin
2 2
+ K f˙ + Θ P˙th − P˙f f + Ψ
+
2 ks1 kQ3
ˆ
Te
(5)
2
+ ks2 kQ4
subject to
τ , τd
−verr, max ≤ M
Table 1. Variables used in the thrust allocation algorithm.
−1
B(α)Kf (t) − τd dt + s1 ≤ verr, max Ts
ˆ
includes a power plant simulation with dynamic variations in frequency and voltage, as well as a feedforward term in the diesel engine governors.
(6) Te
−xerr, max ≤
verr (t)dt + s2 ≤ xerr, max
B(α)Kf = τd + τe
(8) 3/2
Pmax ≥ Pc K |f |
2. THRUST ALLOCATION ALGORITHM The variables used for the thrust allocation algorithm are described in Table 1. The trust allocation algorithm in Veksler et al. (2012) expands on the idea of allowing the propulsion system to deviate from the commanded thrust over a short time in order to improve the dynamics of the power distribution system. This idea has been explored in Radan et al. (2008), where the local thruster controllers are allowed to deviate from the orders they receive from the thrust allocation algorithm in order to counteract the frequency variations on the network. Coordinating the deviation from the dynamic positioning orders in the thrust allocation algorithm makes it possible to estimate and limit the resulting deviations in the velocity and the position of the ship. As the first step, the thrust allocation problem is solved for minimal power consumption without regard to vari159
(7)
Ts
f ≤f ≤f
(9) (10)
Thruster bias is not used in this work, but if it was then (9) would have to be constrained from below as well. This optimization problem includes cost for variations in power consumption and in force produced by individual thrusters. It uses a smaller cost Q2 on deviation from thrust allocation command τd , that is Q2 Q1 . This is to allow the produced generalized force τ to deviate from τd when beneficial. The velocity is constrained within a pre-defined range by imposing (6). The error in position is calculated by integrating the velocity error, and it is constrained by (7). Both (6) and (7) assume that the orientation of the vessel is approximately constant. A limit on maximal power consumption has to be imposed; it is introduced as Pmax in (9). This limit necessitates the
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
slack variables s1 and s2 in the constraints (6) and (7), with cost matrices Q3 and Q4 large enough to ensure that s1 and s2 will significantly deviate from zero only if the constraints (6) and (7) would otherwise be infeasible. The middle terms of equations (6) and (7) represent deviations in the ship’s velocity and position, at time Te , from what they would have been if the thrust allocation command were to be executed exactly. In an exact physical interpretation of Ts , it must be the time when the thrust allocation started running. In practice it is desired to let the thrust allocation operate on a faster time scale than the dynamic positioning controller, since the dynamic positioning controller will also attempt to correct a deviation in position. In the implementation, the choice was made to let integration start five seconds in the past relative to when the thrust allocation is solved. The end time of the integral is discussed in Section IV of Veksler et al. (2012). Without the P˙f f signal, the third term in (5) would be zero if the thrust allocation consumed exactly the same amount of power as in the previous iteration of the algorithm (assuming forward Euler discretization). The power feedforward term Pf f signals a “soft” requirement for thrust allocation to increase or decrease its power consumption compared to power consumption in the previous iteration. Two applications for this signal are discussed in Veksler et al. (2012); however, the only way this signal is used in this work is to compensate for other power consumers that rapidly vary their consumption in predictable patterns. The signal P˙f f is used to reduce variations in the total power consumption by setting P˙f f = −P˙others
(11)
where Pothers is the power consumption by other consumers on the vessel. Since the power plant is able to handle rapid load reductions much better than rapid load increases, in this paper the cost of load variation downwards is set to a fraction of load variation upwards, by changing the value of Θ in (5) depending on whether P˙th − P˙f f is positive or negative. 3. DIESEL ENGINE DYNAMICS The most important physical limitation for a diesel-electric power plant’s ability to accept large and rapid load increases within the combined rated power of the currently active generator sets is the dynamics of the turbocharger. The purpose of the turbocharger is to compress air prior to entering the combustion cylinders. With more air in the cylinder more fuel may also be injected thus increasing the power rating. The turbocharger is driven by the exhaust gases, so when the engine is running on low load, the turbocharger will also run at a fraction of its top velocity. If more fuel is injected into the cylinders without giving time for the turbocharger to accelerate, there will not be enough air in the cylinders to react with the fuel. 3.1 Diesel Engine Model The most accurate models for the dynamics of a turbocharged diesel engine would include a CFD simulation of the process fluids in the engine, as well as the dynamic behavior of the mechanical parts throughout the combustion cycle. A diesel engine deployed in a power plant is 160
controlled by its governor in a tight feedback loop, which counteracts much of the dynamic behavior of the engine. For the purpose of testing the performance of the power system, the model of the diesel engine needs to accurately represent the most important dynamical properties of the engine as well as the physical limitations which are impossible for the governor to correct. The authors could not find a model that would suit those requirements in the literature, so a model was developed for this paper. It is based on Xiros (2002); Skjetne (2011); Boldea (2005); Roy et al. (1993). The model presented in this work can be seen as a simplification of the model in Xiros (2002). It is cyclemean in that all the state variables that are considered are averaged out through the combustion cycle, and it is quasisteady in thermodynamic parameters, which in reality are distributed and vary throughout those volumes. The benefit of this model compared to other models available in the literature is that situations when the engine experiences large load variations are represented with a fair degree of fidelity, while in most other respects the model remains fairly simple. Assumptions and simplifications Compared to the model in Xiros (2002), the following assumptions and simplifications are made in the present work: • The angular velocity of the turbine is assumed to depend on the generated power only. In reality this relationship is quite dynamic, with other factors such as thermodynamic relationships incorporated in the exhaust manifold. Still, both generated power and the exhaust volume that drives the turbocharger depend upon how much fuel is burned per unit of time, and both relationships are linear to some degree. • To calculate the Air-to-Fuel ratio (AF) after each injection, it is assumed that the fuel injected into the cylinder in each cycle is proportional to the fuel index position. The amount of air entering the cylinder is linearly dependent on the velocity of the turbocharger compressor. If the compressor velocity is zero, then the amount of air entering will be ma,0 , and it will linearly increase to a maximum value as the velocity of the compressor approaches its maximal value. • There is a delay in the order of (60/N )·(2/zc ) seconds from fuel index change until the corresponding change of torque on the drive shaft. The main cause of the delay is that it takes time before the new measure of fuel is injected into the next cylinder in the firing sequence. The nominal RPM of the engines in the simulation was around N = 1800, so this delay had little practical consequence. • On older engines, setting a new value for the fuel index involved moving an actual fuel rack, which resulted in a certain amount of lag. On newer engines with direct fuel injection there is no physical fuel rack, so this delay is not included in the model. • Performance of a diesel engine during a large transient is limited by the performance of the turbocharger, which needs time to increase the pressure in the intake manifold. Until it does, the concentration of oxygen in the combustion chamber will limit the combustion. • The damping due to friction is mostly a function of the current engine RPM. Since the engine in a power plant normally operates in a narrow RPM
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
range, this friction is not important for the dynamical performance of the engine and was not modelled. Variables The variables used for the diesel engine model are described in Table 2. Description Break mean effective pressure in the cylinders (p.u.) Total mechanical torque from an engine (p.u.) Electrical torque (p.u.) Rated BMEP (Pa) Instantaneous crank shaft RPM Nominal engine RPM Number of cylinders Cylinder volume (m3 ) Combustion efficiency (non-dimensional, p.u.) Fuel rack position (nondimensional, p.u.) Turbocharger rotational velocity (p.u.) Air flow (mass) without the turbocharger as fraction of the maximal airflow Nominal air-to-fuel ratio on max turbocharger velocity and max BMEP Air-to-fuel ratio at which the combustion stops due to excessive in-cylinder cooling from the injected fuel. Air-to-fuel ratio at which full combustion is achieved. Typical values: 20-27 for HFO, 17-20 for Diesel Oil Current engine power output (Watt) Power consumed by the load (Watt) Rated engine power (Watt) Moment of inertia of the rotating mass in the genset (kg · m2 ) Inertia constant of the engine, represented as the time needed for the engine running at nominal power to produce the energy equivalent to the kinetic energy in the rotating mass at nominal speed.
Letter pe tm te pe,r N Nr zc Vh ηc Fr ωt ma,0 AFn AFlow AFhigh P Pl Pr I H
Table 2. Variables used for the diesel engine model. Formulas AF =
ηc =
ma,0 + (1 − ma,0 )ωt · AFn Fr
1
AF −AFlow AFhigh −AFlow
0
AF ≥ AFhigh AFlow < AF < AFhigh AF ≤ AFlow
(12)
(13)
tm = pe = ηc Fr
(14)
ω˙ t = −κ1 (ωt − pe )
(15)
P = pe,r pe zc Vh N/60 = Pr tm N/Nr
(16)
H= N˙ =
1 2I
2πNr 2 60 Pr
1 2 Nr (tm
(17)
a PID controller, which measures the deviation in the electric frequency from a drooped setpoint, and controls the fuel index to correct this deviation. In the work described here, a feedforward functionality was implemented and tests were conducted both with and without this functionality. Without the feedforward, the governor can only respond to changes in load after these changes affect the frequency. This leads to frequency variations that do not originate in the physical limitations of the system. The proposed feedforward signal measures the electric load, distributes it between the active generator sets, calculates the approximate fuel index position which would produce the electric power currently consumed, and add this value to the output of the PID controller. In this way, when the power consumption changes, the fuel index instantaneously changes to a value close to what is needed to match the produced mechanical power and the consumed electrical power. With these nearly balancing each other out, the torques on the rotating parts of the generating set will approximately match, resulting in a near-constant rotational velocity. The remaining deviation is corrected by the PID controller. As mentioned in Subsection 3.1, the density of the air injected into the cylinders limits how much fuel can be effectively injected into the cylinder. In the present work it is assumed that the diesel engine informs the governor about the maximum efficient fuel index, and the governor is never allowed to exceed this value. A more practical governor should estimate this value with for instance observers. The thrust allocation algorithm described in Veksler et al. (2012) reduces the load variations in the network essentially by delaying some of the power consumption. In situations with large and rapid load increases, this results in the governor first reacting less than it would have with a standard algorithm, for instance the one described by (1)-(3). Then it is unable to move the fuel index enough to deliver power for the delayed consumption due to the limitations mentioned above. This situation often resulted in an unnecessary large frequency drop. To avoid this, the thrust allocation algorithm was modified to inform the PMS both about how much power it is consuming and how much power it would have consumed with a standard thrust allocation algorithm. Since an amount similar to that difference is likely to be requested by the thrusters shortly, it is prudent for the governor to prepare for the coming load increase. In this paper, this was done by integrating the power difference to an energy quantity, and changing the setpoint frequency so that the resulting change in the kinetic energy of the rotating machinery would be equivalent to the energy difference produced by the thrust allocation algorithm. 4. SIMULATED VESSEL
− te )
(18) H These formulas are used to calculate the torque and RPM of the engine in the model. 3.2 Governor The governor used in the simulation is based on the governor described in Skjetne (2011). That governor is 161
The characteristics of the simulated vessel, including its propulsion system and the power plant, are described in this section. 4.1 Hull and thruster system The vessel simulated in this paper is SV Northern Clipper, featured in Fossen (2002). It is 76.2 meters long, mass of 4.591 · 106 kg. It has four thrusters, with two tunnel
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
thrusters near the bow and two azimuth thrusters at the stern. The maximal force for each thruster was set to 1/40 of the ship’s dry weight.
Bus frequency
61
60.5
4.2 Power plant and distribution
60 Bus frequency (Hz)
The power plant installed on the simulated vessel consists of three generator sets. Two of them are rated at 1125 kVA apparent power, and the third one at 538 kVA. All gensets are connected to a single distribution bus. The engine governors were set in droop mode with the setpoint frequency of 60 Hz and a 5% droop.
59.5
59
4.3 Dynamic positioning algorithm The dynamic positioning algorithm used in this work is a set of three PID controllers, one for each degree of freedom. Better algorithms are available in for example Fossen (2002) Chapter 7. For the purposes of this paper the simple PID controllers were sufficient.
58.5
58 340
360
380
400 Time (s)
420
440
460
Fig. 1. Bus frequency with and without PMS-assisting TA, feedforward disabled
The simulated vessel was subjected to a steady environmental disturbance equivalent to 1% of the ship’s weight, forcing the thrusters to maintain a persistent load. This is essential to physically allow the thrust allocation algorithm to compensate for the load spikes. In addition to the thrusters, the load consisted of a constant load of 300 kVA and periodic load spikes of 1.4 MVA, which after two seconds dropped to 0.2 MVA and after two additional seconds to zero. The power factor was set to 0.95 for the thrusters and 0.75 for the other consumers. Simulations were run both with and without the load feedforward to the governors. All measurements were taken after the transients subsided. Figure 1 shows the bus frequency without the use of feedforward. The PMS-assisting functionality in thrust allocation was turned off about 390 seconds into the simulation, after which the power consumption in the thrusters had no connection to the power consumption elsewhere. Figure 2 shows the fuel rack position for one of the diesel engines. Figure 3 shows the total load on the electric network and Figure 4 shows the deviation in the position of the vessel. Figure 5 shows the bus frequency for the simulation with the feedforward enabled. The PMS-assisting functionality in thrust allocation was turned off about 260 seconds into the simulation. Figure 6 shows the fuel rack position for one of the generators. The load on the network and the deviation in position were qualitatively equivalent to what was observed without the feedforward. 6. CONCLUSION The PMS-assisting thrust allocation algorithm was demonstrated to successfully reduce frequency variations on the electric network of a vessel with diesel-electric propulsion. It is most efficient when combined with a feedforward functionality in the governors. 7. FUTURE WORK Although the algorithm proposed in Veksler et al. (2012) improves on the stability of the electrical network, further improvement is possible within the operational constraints. Further research will be directed towards the 162
Fuel rack position
0.9
Max efficient position Position 0.85
0.8
0.75 Fraction of full open
5. RESULTS
0.7
0.65
0.6
0.55
0.5
0.45 340
360
380
400 Time (s)
420
440
460
Fig. 2. Fuel rack position with and without PMS-assisting TA, feedforward disabled development of an algorithm that considers a probable future trajectory of the variables that describe kinetic and electrical states of the vessel. While this approach will always have an inherent uncertainty due to both operator action and other unforeseeable circumstances, we hope that this will improve the system performance in most situations and still fulfill the operational requirements in all situations. 8. ACKNOWLEDGMENTS The authors of this paper are funded by the Research Council of Norway. The project is done in close cooperation with Kongsberg Maritime, represented by Eirik Mathiesen and Bjørnar Realfsen, and with significant contributions from Trond Toftevaag of SINTEF. REFERENCES Boldea, I. (2005). Synchronous Generators. CRC Press.
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
Total load on the bus
2400
Bus frequency
60.6 60.4
2200
60.2 2000
Bus frequency (Hz)
Load on the bus (W)
60 1800
1600
1400
59.8 59.6 59.4 59.2
1200
59 1000
800 340
58.8
360
380
400 Time (s)
420
440
58.6 200
460
Fig. 3. Total (real) load on the electric network with and without PMS-assisting TA, feedforward disabled
240
260 Time (s)
280
300
320
Fig. 5. Bus frequency with and without PMS-assisting TA, feedforward enabled Fuel rack position
Position of the vessel
0.2
220
0.9 Max efficient position Position
x y ψ
0.15
0.85
0.8
Fraction of full open
position, (m), (rad)
0.1
0.05
0
−0.05
0.7
0.65
0.6
0.55
−0.1
−0.15 340
0.75
360
380
400 Time (s)
420
440
0.5 200
460
220
240
260 Time (s)
280
300
320
Fig. 6. Fuel rack position, feedforward enabled
Fig. 4. Position of the vessel, feedforward disabled Fossen, T.I. (2002). Marine Control Systems. Tapir Trykkeri. Jenssen, N.A. and Realfsen, B. (2006). Power optimal thruster allocation. In Proc. Dynamic Positioning Conference, Houston. Johansen, T.A., Fuglseth, T.P., Tøndel, P., and Fossen, T.I. (2003). Optimal constrained control allocation in marine surface vessels with rudders. In IFAC Conf. Manoeuvring and Control of Marine Craft, Girona. Johansen, T.A., I., F.T., and Berge, S.P. (2004). Constrained nonlinear control allocation with singularity avoidance using sequential quadratic programming. IEEE Trans. Control Systems Technology, 12, 211–216. Lindegaard, K.P. and Fossen, T.I. (2003). Fuel efficient control allocation for surface vessels with active rudder usage: Experiments with a model ship. IEEE Trans. Control Systems Technology, 11, 850–862. Radan, D., Sørensen, A.J., ˚ Adnanes, A.K., and Johansen, T.A. (2008). Reducing power load fluctuations on ships using power redistribution control. SNAME Journal of 163
Marine Technology, 45, 162–174. Roy, S., Malik, O., and Hope, G. (1993). Adaptive control of speed and equivalence ratio dynamics of a diesel driven power-plant. Energy Conversion, IEEE Transactions on, 8(1), 13 –19. doi:10.1109/60.207400. Ruth, E., Sørensen, A.J., and Perez, T. (2007). Thrust allocation with linear constrained quadratic cost function. In Proc. IFAC Conf. Control Applications in Marine Systems, Bol, Croatia. Skjetne, R. (2011). Modeling a diesel-generator power plant. Lecture notes in course TMR4290, 2011, NTNU. Sørdalen, O.J. (1997). Optimal thrust allocation for marine vessels. Control Engineering Practice, 5, 1223– 1231. Veksler, A., Johansen, T.A., and Skjetne, R. (2012). Thrust allocation with power management functionality on dynamically positioned vessels. In American Control Conference 2012. Xiros, N. (2002). Robust Control of Diesel Ship Propulsion. Springer.
Transient power control in dynamic positioning - governor feedforward and dynamic thrust allocation Aleksander Veksler ∗ Tor Arne Johansen ∗ Roger Skjetne ∗∗ ∗
Department of Engineering Cybernetics, Norwegian University of Science and Technology (NTNU), Trondheim, Norway. (e-mail: [email protected], [email protected]) ∗∗ Department of Marine Technology, Norwegian University of Science and Technology (NTNU), Trondheim, Norway. (e-mail: [email protected]) Abstract: A thrust allocation algorithm with capability to assist the power management system on a vessel with diesel-electric propulsion has recently been proposed. In the work presented here, the results show that the use of this thrust allocation algorithm produces a significant improvement in the stability of the electric bus frequency in the presence of large power load variations, as long as those load variations are small compared to the kinetic energy of the vessel at low speed. Further reduction in the frequency variations are achieved through use of a feedforward signal to the diesel engine governors, which enables them to react to surges in power consumption before they affect the frequency on the electric bus. 1. INTRODUCTION Maintaining a position is an important operational mode for ships that perform functions such as supplying offshore installations, drilling and oil production, laying pipes, supporting divers or ROV operations, or performing surfacebased measurements or exploration. Dynamic positioning systems use the thrusters of the vessel to maintain a steady position relative to an external position reference such as GPS or the position of another vessel. Coordination of the thrusters to keep the operational envelope close to the physical limitations while minimizing the power consumption has been explored in the past two decades, see for example Sørdalen (1997), Johansen et al. (2003), Jenssen and Realfsen (2006), Ruth et al. (2007) and Lindegaard and Fossen (2003). A vessel with a well-designed dynamic positioning capability should have thrusters capable of producing large amounts of moment and force quickly in any direction. This is best achieved by using thrusters that can dynamically change the direction in which they apply force, and by locating the thrusters as far apart as practical. This makes it impractical to power those thrusters with a mechanical connection to the power plant. In the current industry practice for ships that are expected to spend a significant part of their service time in dynamic positioning, electrical power distribution is implemented almost universally. The thruster and propulsion system constitutes the largest consumer in terms of maximal power requirements for most ships, depending on what other large power consumers are installed on a particular vessel. A diesel-driven power plant operates optimally when the load is close to being constant. The diesel engine may be physically unable to respond to a sudden load increase due to the relatively slow turbocharger dynamics. Depending on the tuning, the governor may also be slow to ©2012 IFAC
158
respond to changes in load, resulting in frequency variations. Although small frequency variations are tolerated, the classification societies typically set a limit of +/-10% on the maximal frequency deviation from the nominal value. If this limit is exceeded, generators and some of the consumers will disconnect from the power grid. Variations in loading also lead to increased NOX emissions and soot. A thrust allocation algorithm that attempts to even out load between the switchboards is considered in Jenssen and Realfsen (2006). Thrust allocation algorithms available in the literature do not consider variations in the total power consumption by the thrusters, although several do consider variations in force, and by implication, in power of the individual thrusters. Several methods to reduce the consequences of power variation are being explored in the industry. A recent development is using a large battery bank on the vessel to provide additional power during load spikes. Also, an emerging practice is to supply the propulsion units, generators and essential auxiliaries with emergency power during a blackout. In this case, they will be able to resume operation quickly. A thrust allocation algorithm that attempts to assist the power management system in reducing load variations on the power plant by counteracting the load variations from other consumers on the ship has been introduced in Veksler et al. (2012). This algorithm accepts a request to increase or decrease power consumption by the thrusters. It will then attempt to fulfill this request as long as it does not cause the vessel to deviate too far off position, or incur large costs for other aspects of the operation such as fuel consumption. This algorithm exploits the fact that the hull of the vessel itself effectively stores large amounts of energy: for the vessel simulated for this paper, the kinetic energy of the vessel moving at 0.5 m/s is equivalent to the total power plant output for 0.3 seconds. The main contribution of this paper is a case study that 10.3182/20120919-3-IT-2046.00027
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
Description Current time, i.e. time when the thrust allocation problem is solved. Deviation in respectively velocity and position of the vessel from what they would have been if thrust command was allocated exactly, as functions of time. verr (t), xerr (t) ∈ R3 contain longitudinal, lateral and heading components; ∆
ations in the power consumption. This method is welldocumented in the literature - although usually with quadratic power cost function; see Fossen (2002).
Letter T verr (t), xerr (t), verr, T , xerr, T
3/2
Pmin = min Pc K |f | f,s
2
+ kskQ1
(1)
subject to
∆
verr, T = verr (t = T ), xerr, T = xerr (t = T ) Maximal allowed values for verr (t) and xerr (t) Lower and upper limits for the integrals in (6), (7) which calculate deviations in velocity and position at time Te . Respectively actual and desired angular frequency of the voltage on the electrical network. Typically, ω0g = 2π · 60. Thruster configuration matrix. It is a function of the vector α consisting of orientations of the individual thrusters. In this paper, α is assumed to be constant. f ∈ RN , the force produced by individual thrusters. The elements of f are normalized into range [−1, 1]. K ∈ RN ×N such that Kf is the vector of forces in Newtons. Pc ∈ R1×N such that (4) holds. Ψ 0, quadratic cost matrix of variation in force produced by individual thrusters. Θ ∈ R+ is the cost of variation in total power consumption. The total power consumed by the thrusters per (4) The desired rate of change of power consumption by the thrusters. This signal can be used to reduce either frequency or load variations on the electrical network. Minimal power consumption by the thrusters needed to produce commanded thrust. Actual and desired generalized force produced by all thrusters. τ, τd ∈ R3 contain surge and sway forces, and yaw moment.
B(α)Kf = τd + s
verr, max , xerr, max Ts , Te
(2)
f ≤f ≤f (3) The notation used here and in the following is that ∆ p p p T 2 ∆ kxkA = xT Ax and |x|p = x1 x2 · · · xN for any x, A and p of suitable dimension. The cost matrix Q1 must be large enough to ensure that s is significantly larger than zero only when constraints (2)-(3) would otherwise be infeasible. The solution to this optimization problem provides a minimum Pmin to which the power consumption can be reduced while delivering the requested thrust τd , at least as long as the condition s ≈ 0 holds. Power consumption in the thrusters is estimated by the nonlinear relationship
ωg , ω0g
B(α)
f
K
3/2
Pth = Pc K |f | (4) which is similar to what was used in Johansen et al. (2004). In Veksler et al. (2012), the following thrust allocation optimization problem is used for thrust allocation when there is no power bias requirement:
Pc Ψ Θ Pth P˙f f
3/2
min Pc K |f |
f,τe ,s1 ,s2
2 kτe kQ2
Pmin
2 2
+ K f˙ + Θ P˙th − P˙f f + Ψ
+
2 ks1 kQ3
ˆ
Te
(5)
2
+ ks2 kQ4
subject to
τ , τd
−verr, max ≤ M
Table 1. Variables used in the thrust allocation algorithm.
−1
B(α)Kf (t) − τd dt + s1 ≤ verr, max Ts
ˆ
includes a power plant simulation with dynamic variations in frequency and voltage, as well as a feedforward term in the diesel engine governors.
(6) Te
−xerr, max ≤
verr (t)dt + s2 ≤ xerr, max
B(α)Kf = τd + τe
(8) 3/2
Pmax ≥ Pc K |f |
2. THRUST ALLOCATION ALGORITHM The variables used for the thrust allocation algorithm are described in Table 1. The trust allocation algorithm in Veksler et al. (2012) expands on the idea of allowing the propulsion system to deviate from the commanded thrust over a short time in order to improve the dynamics of the power distribution system. This idea has been explored in Radan et al. (2008), where the local thruster controllers are allowed to deviate from the orders they receive from the thrust allocation algorithm in order to counteract the frequency variations on the network. Coordinating the deviation from the dynamic positioning orders in the thrust allocation algorithm makes it possible to estimate and limit the resulting deviations in the velocity and the position of the ship. As the first step, the thrust allocation problem is solved for minimal power consumption without regard to vari159
(7)
Ts
f ≤f ≤f
(9) (10)
Thruster bias is not used in this work, but if it was then (9) would have to be constrained from below as well. This optimization problem includes cost for variations in power consumption and in force produced by individual thrusters. It uses a smaller cost Q2 on deviation from thrust allocation command τd , that is Q2 Q1 . This is to allow the produced generalized force τ to deviate from τd when beneficial. The velocity is constrained within a pre-defined range by imposing (6). The error in position is calculated by integrating the velocity error, and it is constrained by (7). Both (6) and (7) assume that the orientation of the vessel is approximately constant. A limit on maximal power consumption has to be imposed; it is introduced as Pmax in (9). This limit necessitates the
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
slack variables s1 and s2 in the constraints (6) and (7), with cost matrices Q3 and Q4 large enough to ensure that s1 and s2 will significantly deviate from zero only if the constraints (6) and (7) would otherwise be infeasible. The middle terms of equations (6) and (7) represent deviations in the ship’s velocity and position, at time Te , from what they would have been if the thrust allocation command were to be executed exactly. In an exact physical interpretation of Ts , it must be the time when the thrust allocation started running. In practice it is desired to let the thrust allocation operate on a faster time scale than the dynamic positioning controller, since the dynamic positioning controller will also attempt to correct a deviation in position. In the implementation, the choice was made to let integration start five seconds in the past relative to when the thrust allocation is solved. The end time of the integral is discussed in Section IV of Veksler et al. (2012). Without the P˙f f signal, the third term in (5) would be zero if the thrust allocation consumed exactly the same amount of power as in the previous iteration of the algorithm (assuming forward Euler discretization). The power feedforward term Pf f signals a “soft” requirement for thrust allocation to increase or decrease its power consumption compared to power consumption in the previous iteration. Two applications for this signal are discussed in Veksler et al. (2012); however, the only way this signal is used in this work is to compensate for other power consumers that rapidly vary their consumption in predictable patterns. The signal P˙f f is used to reduce variations in the total power consumption by setting P˙f f = −P˙others
(11)
where Pothers is the power consumption by other consumers on the vessel. Since the power plant is able to handle rapid load reductions much better than rapid load increases, in this paper the cost of load variation downwards is set to a fraction of load variation upwards, by changing the value of Θ in (5) depending on whether P˙th − P˙f f is positive or negative. 3. DIESEL ENGINE DYNAMICS The most important physical limitation for a diesel-electric power plant’s ability to accept large and rapid load increases within the combined rated power of the currently active generator sets is the dynamics of the turbocharger. The purpose of the turbocharger is to compress air prior to entering the combustion cylinders. With more air in the cylinder more fuel may also be injected thus increasing the power rating. The turbocharger is driven by the exhaust gases, so when the engine is running on low load, the turbocharger will also run at a fraction of its top velocity. If more fuel is injected into the cylinders without giving time for the turbocharger to accelerate, there will not be enough air in the cylinders to react with the fuel. 3.1 Diesel Engine Model The most accurate models for the dynamics of a turbocharged diesel engine would include a CFD simulation of the process fluids in the engine, as well as the dynamic behavior of the mechanical parts throughout the combustion cycle. A diesel engine deployed in a power plant is 160
controlled by its governor in a tight feedback loop, which counteracts much of the dynamic behavior of the engine. For the purpose of testing the performance of the power system, the model of the diesel engine needs to accurately represent the most important dynamical properties of the engine as well as the physical limitations which are impossible for the governor to correct. The authors could not find a model that would suit those requirements in the literature, so a model was developed for this paper. It is based on Xiros (2002); Skjetne (2011); Boldea (2005); Roy et al. (1993). The model presented in this work can be seen as a simplification of the model in Xiros (2002). It is cyclemean in that all the state variables that are considered are averaged out through the combustion cycle, and it is quasisteady in thermodynamic parameters, which in reality are distributed and vary throughout those volumes. The benefit of this model compared to other models available in the literature is that situations when the engine experiences large load variations are represented with a fair degree of fidelity, while in most other respects the model remains fairly simple. Assumptions and simplifications Compared to the model in Xiros (2002), the following assumptions and simplifications are made in the present work: • The angular velocity of the turbine is assumed to depend on the generated power only. In reality this relationship is quite dynamic, with other factors such as thermodynamic relationships incorporated in the exhaust manifold. Still, both generated power and the exhaust volume that drives the turbocharger depend upon how much fuel is burned per unit of time, and both relationships are linear to some degree. • To calculate the Air-to-Fuel ratio (AF) after each injection, it is assumed that the fuel injected into the cylinder in each cycle is proportional to the fuel index position. The amount of air entering the cylinder is linearly dependent on the velocity of the turbocharger compressor. If the compressor velocity is zero, then the amount of air entering will be ma,0 , and it will linearly increase to a maximum value as the velocity of the compressor approaches its maximal value. • There is a delay in the order of (60/N )·(2/zc ) seconds from fuel index change until the corresponding change of torque on the drive shaft. The main cause of the delay is that it takes time before the new measure of fuel is injected into the next cylinder in the firing sequence. The nominal RPM of the engines in the simulation was around N = 1800, so this delay had little practical consequence. • On older engines, setting a new value for the fuel index involved moving an actual fuel rack, which resulted in a certain amount of lag. On newer engines with direct fuel injection there is no physical fuel rack, so this delay is not included in the model. • Performance of a diesel engine during a large transient is limited by the performance of the turbocharger, which needs time to increase the pressure in the intake manifold. Until it does, the concentration of oxygen in the combustion chamber will limit the combustion. • The damping due to friction is mostly a function of the current engine RPM. Since the engine in a power plant normally operates in a narrow RPM
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
range, this friction is not important for the dynamical performance of the engine and was not modelled. Variables The variables used for the diesel engine model are described in Table 2. Description Break mean effective pressure in the cylinders (p.u.) Total mechanical torque from an engine (p.u.) Electrical torque (p.u.) Rated BMEP (Pa) Instantaneous crank shaft RPM Nominal engine RPM Number of cylinders Cylinder volume (m3 ) Combustion efficiency (non-dimensional, p.u.) Fuel rack position (nondimensional, p.u.) Turbocharger rotational velocity (p.u.) Air flow (mass) without the turbocharger as fraction of the maximal airflow Nominal air-to-fuel ratio on max turbocharger velocity and max BMEP Air-to-fuel ratio at which the combustion stops due to excessive in-cylinder cooling from the injected fuel. Air-to-fuel ratio at which full combustion is achieved. Typical values: 20-27 for HFO, 17-20 for Diesel Oil Current engine power output (Watt) Power consumed by the load (Watt) Rated engine power (Watt) Moment of inertia of the rotating mass in the genset (kg · m2 ) Inertia constant of the engine, represented as the time needed for the engine running at nominal power to produce the energy equivalent to the kinetic energy in the rotating mass at nominal speed.
Letter pe tm te pe,r N Nr zc Vh ηc Fr ωt ma,0 AFn AFlow AFhigh P Pl Pr I H
Table 2. Variables used for the diesel engine model. Formulas AF =
ηc =
ma,0 + (1 − ma,0 )ωt · AFn Fr
1
AF −AFlow AFhigh −AFlow
0
AF ≥ AFhigh AFlow < AF < AFhigh AF ≤ AFlow
(12)
(13)
tm = pe = ηc Fr
(14)
ω˙ t = −κ1 (ωt − pe )
(15)
P = pe,r pe zc Vh N/60 = Pr tm N/Nr
(16)
H= N˙ =
1 2I
2πNr 2 60 Pr
1 2 Nr (tm
(17)
a PID controller, which measures the deviation in the electric frequency from a drooped setpoint, and controls the fuel index to correct this deviation. In the work described here, a feedforward functionality was implemented and tests were conducted both with and without this functionality. Without the feedforward, the governor can only respond to changes in load after these changes affect the frequency. This leads to frequency variations that do not originate in the physical limitations of the system. The proposed feedforward signal measures the electric load, distributes it between the active generator sets, calculates the approximate fuel index position which would produce the electric power currently consumed, and add this value to the output of the PID controller. In this way, when the power consumption changes, the fuel index instantaneously changes to a value close to what is needed to match the produced mechanical power and the consumed electrical power. With these nearly balancing each other out, the torques on the rotating parts of the generating set will approximately match, resulting in a near-constant rotational velocity. The remaining deviation is corrected by the PID controller. As mentioned in Subsection 3.1, the density of the air injected into the cylinders limits how much fuel can be effectively injected into the cylinder. In the present work it is assumed that the diesel engine informs the governor about the maximum efficient fuel index, and the governor is never allowed to exceed this value. A more practical governor should estimate this value with for instance observers. The thrust allocation algorithm described in Veksler et al. (2012) reduces the load variations in the network essentially by delaying some of the power consumption. In situations with large and rapid load increases, this results in the governor first reacting less than it would have with a standard algorithm, for instance the one described by (1)-(3). Then it is unable to move the fuel index enough to deliver power for the delayed consumption due to the limitations mentioned above. This situation often resulted in an unnecessary large frequency drop. To avoid this, the thrust allocation algorithm was modified to inform the PMS both about how much power it is consuming and how much power it would have consumed with a standard thrust allocation algorithm. Since an amount similar to that difference is likely to be requested by the thrusters shortly, it is prudent for the governor to prepare for the coming load increase. In this paper, this was done by integrating the power difference to an energy quantity, and changing the setpoint frequency so that the resulting change in the kinetic energy of the rotating machinery would be equivalent to the energy difference produced by the thrust allocation algorithm. 4. SIMULATED VESSEL
− te )
(18) H These formulas are used to calculate the torque and RPM of the engine in the model. 3.2 Governor The governor used in the simulation is based on the governor described in Skjetne (2011). That governor is 161
The characteristics of the simulated vessel, including its propulsion system and the power plant, are described in this section. 4.1 Hull and thruster system The vessel simulated in this paper is SV Northern Clipper, featured in Fossen (2002). It is 76.2 meters long, mass of 4.591 · 106 kg. It has four thrusters, with two tunnel
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
thrusters near the bow and two azimuth thrusters at the stern. The maximal force for each thruster was set to 1/40 of the ship’s dry weight.
Bus frequency
61
60.5
4.2 Power plant and distribution
60 Bus frequency (Hz)
The power plant installed on the simulated vessel consists of three generator sets. Two of them are rated at 1125 kVA apparent power, and the third one at 538 kVA. All gensets are connected to a single distribution bus. The engine governors were set in droop mode with the setpoint frequency of 60 Hz and a 5% droop.
59.5
59
4.3 Dynamic positioning algorithm The dynamic positioning algorithm used in this work is a set of three PID controllers, one for each degree of freedom. Better algorithms are available in for example Fossen (2002) Chapter 7. For the purposes of this paper the simple PID controllers were sufficient.
58.5
58 340
360
380
400 Time (s)
420
440
460
Fig. 1. Bus frequency with and without PMS-assisting TA, feedforward disabled
The simulated vessel was subjected to a steady environmental disturbance equivalent to 1% of the ship’s weight, forcing the thrusters to maintain a persistent load. This is essential to physically allow the thrust allocation algorithm to compensate for the load spikes. In addition to the thrusters, the load consisted of a constant load of 300 kVA and periodic load spikes of 1.4 MVA, which after two seconds dropped to 0.2 MVA and after two additional seconds to zero. The power factor was set to 0.95 for the thrusters and 0.75 for the other consumers. Simulations were run both with and without the load feedforward to the governors. All measurements were taken after the transients subsided. Figure 1 shows the bus frequency without the use of feedforward. The PMS-assisting functionality in thrust allocation was turned off about 390 seconds into the simulation, after which the power consumption in the thrusters had no connection to the power consumption elsewhere. Figure 2 shows the fuel rack position for one of the diesel engines. Figure 3 shows the total load on the electric network and Figure 4 shows the deviation in the position of the vessel. Figure 5 shows the bus frequency for the simulation with the feedforward enabled. The PMS-assisting functionality in thrust allocation was turned off about 260 seconds into the simulation. Figure 6 shows the fuel rack position for one of the generators. The load on the network and the deviation in position were qualitatively equivalent to what was observed without the feedforward. 6. CONCLUSION The PMS-assisting thrust allocation algorithm was demonstrated to successfully reduce frequency variations on the electric network of a vessel with diesel-electric propulsion. It is most efficient when combined with a feedforward functionality in the governors. 7. FUTURE WORK Although the algorithm proposed in Veksler et al. (2012) improves on the stability of the electrical network, further improvement is possible within the operational constraints. Further research will be directed towards the 162
Fuel rack position
0.9
Max efficient position Position 0.85
0.8
0.75 Fraction of full open
5. RESULTS
0.7
0.65
0.6
0.55
0.5
0.45 340
360
380
400 Time (s)
420
440
460
Fig. 2. Fuel rack position with and without PMS-assisting TA, feedforward disabled development of an algorithm that considers a probable future trajectory of the variables that describe kinetic and electrical states of the vessel. While this approach will always have an inherent uncertainty due to both operator action and other unforeseeable circumstances, we hope that this will improve the system performance in most situations and still fulfill the operational requirements in all situations. 8. ACKNOWLEDGMENTS The authors of this paper are funded by the Research Council of Norway. The project is done in close cooperation with Kongsberg Maritime, represented by Eirik Mathiesen and Bjørnar Realfsen, and with significant contributions from Trond Toftevaag of SINTEF. REFERENCES Boldea, I. (2005). Synchronous Generators. CRC Press.
IFAC MCMC 2012 September 19-21, 2012. Arenzano, Italy
Total load on the bus
2400
Bus frequency
60.6 60.4
2200
60.2 2000
Bus frequency (Hz)
Load on the bus (W)
60 1800
1600
1400
59.8 59.6 59.4 59.2
1200
59 1000
800 340
58.8
360
380
400 Time (s)
420
440
58.6 200
460
Fig. 3. Total (real) load on the electric network with and without PMS-assisting TA, feedforward disabled
240
260 Time (s)
280
300
320
Fig. 5. Bus frequency with and without PMS-assisting TA, feedforward enabled Fuel rack position
Position of the vessel
0.2
220
0.9 Max efficient position Position
x y ψ
0.15
0.85
0.8
Fraction of full open
position, (m), (rad)
0.1
0.05
0
−0.05
0.7
0.65
0.6
0.55
−0.1
−0.15 340
0.75
360
380
400 Time (s)
420
440
0.5 200
460
220
240
260 Time (s)
280
300
320
Fig. 6. Fuel rack position, feedforward enabled
Fig. 4. Position of the vessel, feedforward disabled Fossen, T.I. (2002). Marine Control Systems. Tapir Trykkeri. Jenssen, N.A. and Realfsen, B. (2006). Power optimal thruster allocation. In Proc. Dynamic Positioning Conference, Houston. Johansen, T.A., Fuglseth, T.P., Tøndel, P., and Fossen, T.I. (2003). Optimal constrained control allocation in marine surface vessels with rudders. In IFAC Conf. Manoeuvring and Control of Marine Craft, Girona. Johansen, T.A., I., F.T., and Berge, S.P. (2004). Constrained nonlinear control allocation with singularity avoidance using sequential quadratic programming. IEEE Trans. Control Systems Technology, 12, 211–216. Lindegaard, K.P. and Fossen, T.I. (2003). Fuel efficient control allocation for surface vessels with active rudder usage: Experiments with a model ship. IEEE Trans. Control Systems Technology, 11, 850–862. Radan, D., Sørensen, A.J., ˚ Adnanes, A.K., and Johansen, T.A. (2008). Reducing power load fluctuations on ships using power redistribution control. SNAME Journal of 163
Marine Technology, 45, 162–174. Roy, S., Malik, O., and Hope, G. (1993). Adaptive control of speed and equivalence ratio dynamics of a diesel driven power-plant. Energy Conversion, IEEE Transactions on, 8(1), 13 –19. doi:10.1109/60.207400. Ruth, E., Sørensen, A.J., and Perez, T. (2007). Thrust allocation with linear constrained quadratic cost function. In Proc. IFAC Conf. Control Applications in Marine Systems, Bol, Croatia. Skjetne, R. (2011). Modeling a diesel-generator power plant. Lecture notes in course TMR4290, 2011, NTNU. Sørdalen, O.J. (1997). Optimal thrust allocation for marine vessels. Control Engineering Practice, 5, 1223– 1231. Veksler, A., Johansen, T.A., and Skjetne, R. (2012). Thrust allocation with power management functionality on dynamically positioned vessels. In American Control Conference 2012. Xiros, N. (2002). Robust Control of Diesel Ship Propulsion. Springer.