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Shipboard Power Management Using Constrained Nonlinear Model Predictive Control

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Shiboard Power Management Using Constrained Nonlinear Model Predictive Control Phili Stone and Daniel F. Oila GE Power Conversion Pittsburgh, PA USA Hyeongjun Park and Jing Sun University of Michigan Ann
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Shiboard Power Management Using Constrained Nonlinear Model Predictive Control Phili Stone and Daniel F. Oila GE Power Conversion Pittsburgh, PA USA Hyeongjun Park and Jing Sun University of Michigan Ann Arbor, MI, USA Steve Pekarek and Ray DeCarlo Purdue University West Lafayette, IN, USA Eric Westervelt, James Brooks, and Gayathri Seenumani GE Global Research Center Schenectady, NY, USA Abstract Both new and existing naval vessels of all sizes face ever-increasing ower suly requirements to suort advanced mission loads including high ower sensors, weaons, and launchers. Adding additional conventional generators to suort these loads is infeasible given size and weight constraints and given the ulsed nature of those new loads. Instead, an otimization-based Power Management Controller (PMC) is used to dynamically control ower system sources and loads in real time in order to serve system needs with a minimal amount of ower suly equiment. In this aer, a Model Predictive Control (MPC) aroach is used to dynamically coordinate sources and loads based on future demand. A cost function is used to rioritize various shi goals and objectives, and constraints are added to reflect hardware limitations. A Constrained Nonlinear MPC algorithm is then used to minimize the cost over a finite future horizon and generate control commands in real-time. The PMC is demonstrated to successfully control and imrove system erformance on a hardware test bed for shi ower system research. I. INTRODUCTION The ower generation caacity of a shiboard ower system is inherently restricted by the weight and footrint limitations of a shi hull. Modern advanced defense systems such as the air and missile defense radar (AMDR) [], electromagnetic rail guns [2], electromagnetic launch systems [3], and solid-state and free electron lasers [4] reresent high ower electrical loads that are beginning to lace a high demand on the limited caacity [5]. As the resective technologies mature, these weaons are becoming increasingly critical to countering hostile surface, air, and missile targets and maintaining state-ofthe-art offensive caability should it be necessary. Many of these weaons are tyically ulsed with a demand on the order of seconds [4]. However, the sum of the total ower caacity of each of these weaons lus the shi service loads could quickly exceed the ower generation caacity of the shi and thus limit the number of weaons that can be installed unless the available generation caacity is intelligently allocated. To enable the effective emloyment of multile high ower weaons within the shi ower system, a ower management controller (PMC) is essential. A shiboard PMC dynamically coordinates the shi ower sources (main generators, auxiliary generators, UPS devices, energy storage modules, etc.) and loads to most effectively rovide ower to the weaons only when they need it. The PMC thus enables multile high ower weaons, a reduction of system mass/volume through reduction of ower generation requirements, and reduced troo levels due to automated decision-making. The roosed PMC aroach emloys model redictive control (MPC) [6] that resides at a suervisory level to rovide ower to loads as they need them while otimizing shi erformance. MPC considers a known model of the lant (the shi and its loads/weaons) and looks ahead over a rediction horizon to determine the otimal control references to rovide to the local controllers of the ower system elements. In this way the PMC can reare the ower system for a short-term high ower weaon by taking reemtive action such as raming u generators, raming non-vital loads, charging energy storage devices, and reconfiguring the system so as to otimize system erformance and deliver the requested ower at the required time of the high ower ulse (weaon firing). This concet of a PMC is not limited to future shi arrangements with multile high ower weaons. Among other caabilities, existing sohisticated shi ower managers can reconfigure breakers and add or shed sources and loads based uon re-defined load riorities and following a set of rules [7]. The roosed PMC can do these things as well as other tasks such as tracking otimal efficiency oints for multile gas turbine generators to decrease fuel consumtion and thus cost. The redictive PMC considers the defined goals and riorities of the shi to ram u/down generation sources and enforce/relax load demands as necessary on-the-fly. The model redictive aroach enables the PMC to look ahead as far as the established rediction horizon and begin comensating for ulse loads before they actually occur and then gracefully recover. Instead of installing a surlus of energy storage modules on a shi, the PMC can coordinate a reduced number of modules along with existing generation caacities to accommodate the shi load demands. The roosed PMC is demonstrated on the medium voltage DC test bed for shi ower system research at Purdue University and this test bed is described in Section II. The MPC aroach as it ertains to a shiboard PMC is discussed in further detail in Section III. The roosed PMC is tested by establishing a baseline behavior and then attemting to imrove that behavior by adjusting the relative riorities of the shi goals with two successive test cases. The test cases for demonstrating the effectiveness of the PMC on the test bed are outlined in Section IV. The testing results are resented and analyzed in Section V. Conclusions regarding the level of success in emloying the MPC aroach to shiboard ower management are discussed in Section VI. II. TEST BED FOR SHIP POWER SYSTEM RESEARCH In order to validate research relative to shi ower systems a medium voltage (MV) DC test bed has been constructed at Purdue University [8]. A one-line diagram of the ortion of this test bed used for testing the roosed PMC is shown in Figure. The system consists of two sources totaling 7 kw of caacity and three loads connected in arallel to a 75 V DC bus. Generation system (GS-) consists of a four quadrant dynamometer serving as the rime mover (PM-) that regulates a 59 kw wound rotor synchronous generator (SG-) to 8 rm. AC-DC conversion is erformed by a assive diode rectifier (R-) and the field voltage of the generator is controlled by a voltage regulator (VR-) to control the outut voltage. GS- is intended to reresent the rime shi ower source such as a main gas turbine generator (GTG) set. Generation system 2 (GS-2) consists of a four quadrant dynamometer that regulates the seed of an kw ermanent magnet synchronous generator (SG-2) to 36 rm. AC-DC conversion is erformed by an IGBT-based active rectifier (R- 2) and the outut voltage of the generator may be controlled through the active rectifier by a voltage regulator (VR-2). Alternatively, the PMC may rovide a ower command to the GS-2 which is then translated to a torque/current command at the local level. GS-2 is intended to reresent a smaller, faster shi ower generation source such as a diesel generator. The rimary load on the system is the shi roulsion system (SPS) which consists of a roulsion drive (PD) and a 37 kw induction machine (IM). A dynamometer tyically regulates the IM seed to the rated 8 rm, but for PMC Figure. Purdue MVDC test bed one-line diagram. Figure 2. Square wave ulse ower load (SWPPL) rofile. testing the seed will be regulated to 9 rm limiting the effective maximum roulsion ower to 8.5 kw. The PD consists of an IGBT-based inverter with seed and torque control modes; the latter is emloyed for PMC testing. The ulse ower load (PPL) emulates a ulse-ower weaon with an energy storage buffer between the load and DC bus. A buck converter charges a caacitor from the bus until a threshold voltage is reached at which time the caacitor disconnects from the bus and discharges into a resistive load. Thus the ulse rofile as seen by the bus is a ram that increases from to 8 kw in ~ seconds. To lace an even greater burden on the system, the square wave ulse ower load (SWPPL) may be utilized. The SWPPL consists of a high ower buck converter (PS-) that stes the voltage down to 5 V across a urely resistive load. The SWPPL can sink u to 8 kw for second on, second off intervals for u to 7 cycles before needing time to recover thermally. The rofile as seen by the bus can be seen in Figure 2. The PMC testing will only consider the more severe SWPPL as a disturbance to the system. The PMC is imlemented in MATLAB Simulink and is converted to real-time-comatible code via Simulink Coder to reside on a B&R 8 industrialized PC. This PC interfaces with the test bed comonents via a common network switch. The PMC functions in real time using the MATLAB xpc kernel and communicates using the user datagram rotocol (UDP). The GS-2 and SPS receive ower commands from the PMC via the switch and convert those ower commands first to torque commands and then to inverter current commands which they attemt to track using hysteresis control. Each comonent rovides the necessary measurements (e.g., rotor seed, electrical ower, mechanical ower, DC voltage) back to the PMC via the same switch. The GS- is treated as a slack generator by the PMC. Only the droo gain (D GS ) of the voltage controller is exosed as a control inut to the PMC. The GS- rovides the ower necessary to balance the ower on the bus such that the sum of the ower sourced by the GS- and the GS-2 is equal to the sum of the ower sunk by the SPS and the SWPPL. The PMC thus has three controls: the GS- droo gain command D GS, the GS-2 ower command P GS2 *, and the SPS ower command P SPS *. III. MODEL PREDICTIVE CONTROL APPROACH The PMC otimizes the shi erformance to achieve a set of re-secified objectives while resecting erformance and oerability constraints. To do so, a multi-objective cost function denoted by J is minimized subject to system dynamic and safety constraints over a given eriod of time. In this work, the PMC is formulated as a discrete-time otimal control roblem described by the following: P( x ) : min J( x( ), u( )) n x: [, N] R, m u: [, N] R N where J( x(), u()) =Φ ( x( N)) + L( x( k), u( k)) k = () n+ m n st.. x( k+ ) = f( x( k), u( k)), f : R R, n x() = x, x R, n+ m l Cx ( ( k), u( k)), C: R R, k =,, N where, xk ( ), uk ( ) are the ower system states and control inuts, resectively, and N is the length of the rediction horizon over which the cost function (J) is minimized and the oint wise state and inut constraints need to be satisfied. The equality constraint is the dynamic state evolution that describes the ower system f ( xk ( ), uk ( )), while the inequality constraint reresents the bounds on the states and the control inuts. A model redictive control aroach is adoted in order to solve the PMC roblem. The aroach emloys a receding horizon based MPC, where the above otimization roblem is solved reeatedly at every time ste k and the inut elements of the otimal control sequence corresonding to the first time index are alied as the control inuts to the lant. The MPC considers a model of the ower system and can leverage a- riori information about known load disturbances such as ulse loads in order to otimize shi erformance. Comuting the solutions to () in deterministic real-time is difficult, so two methods were considered to solve the PMC roblem in this rogram, namely, Interior Point Otimizer (IOt) (see [9] and references there-in) and Integrated Perturbation Analysis Sequential Quadratic Programming (IPA-SQP) []. The IOt is an interior-oint based general urose nonlinear rogramming solver which is alicable to large scale systems. More secifically, the algorithm is comutationally efficient for roblems where the gradient and hessians associated with the otimization roblem have sarse structure which is the case for the otimal control roblem associated with the PMC. In addition, the IOt enables using imlicit discretization for the equality dynamic constraint which alleviates any numerical issues associated with the discretization. The IPA-SQP is an algorithm develoed at the University of Michigan that secifically addresses the comutational efficiency of otimization arising in the MPC formulation. It exloits the comlementary features of erturbation analysis and sequential quadratic rogramming, and rovides the otimal solution in the redictor-corrector form. In both methods, the otimal control sequence comuted at any given instant can be used as a feasible initial guess for the otimization solved at the successive time instant, thereby imroving the comutational efficiency. A third algorithm emloying deterministic dynamic rogramming (DDP) was also develoed under the Comact Power Conversion Technologies rogram that funded this work. This aroach differs from the MPC aroach by recomuting most of the otimal solution offline. Due to this feature the DDP algorithm is relatively fast and deterministic, but less flexible than an MPC algorithm. Due to sace limitations, this work will only show results from the IOt algorithm to sufficiently rove the efficacy of model redictive control as a shi ower management tool. Further details on this control algorithm and test results are available []. IV. POWER MANAGEMENT CONTROLLER TEST PLAN Shi erformance is otimized by minimizing a cost function which defines the erformance goals of the shi and assigns relative riority to each goal. The cost function emloyed by the roosed PMC for the Purdue DC test bed is * 2 2 * 2 Lxk ( ( ), uk ( )) = ( Vbus V ) 2 3( 2 2 )... k bus + PTotal + P k GS P k G S + (2) * 2 * 2 2 4( PSPS P ) 5( ) 6( )... k SPS + ωsps ω k SPS + DGS D k GS + k ( P P ) + ( P P ) + ( P P ) GSk GSk 8 GS 2k GS 2k 9 SPSk SPSk Each arameter of this cost function reresents an interest of the ower system. The weighting factor,, for each arameter assigns a relative riority to the arameter. The descrition of each arameter is shown in Table. The main bus voltage is V- bus, the owers P Total, P GS, P GS2, and P SPS are the total, GS, GS2, and SPS owers resectively. An asterisk on each reresents a target or desired ower. The droo gain for GS is D GS, and the SPS shaft seed is ω. SPS Table. PMC test lan. Quantity Purdue Bed Equivalent A B C Bus voltage deviation DC bus voltage deviation Power tracking error Pulsed load ower deviation Fuel efficiency GS-2 tracking to efficiency oint Shi velocity deviation SPS induction machine ower α SPS induction machine seed GS- droo gain ram rate Ram rate of GS- droo gain GS- rime mover ram rate Ram rate of GS- electrical ower 7 7 α 7 α GS-2 rime mover ram rate Ram rate of GS-2 electrical ower SPS rime mover ram rate Ram rate of SPS electrical ower The error between the bus voltage measurement and the target value of 75 V is critical to the roer oeration of ower system elements. The total ower tracking error has some overla with the bus voltage deviation term and adds another layer of stability by ensuring that the ower at the bus is balanced between generation and load. It is assumed that an efficiency curve for the GS-2 is known and that P * GS2 is the most efficient oint on the curve. The equivalent of shi velocity deviation is accounted for with the target deviation of the SPS ower and rotor seed. The ram rate of the GS-outut ower is included to rovide a means of limiting the burden on this unit. The GS- reresents a main turbine generator in this test bed, and therefore it is of interest to limit the ower ram rate so as to limit the wear of the machine and increase its exected lifetime. Finally, ram rate terms for each of the PMC control inuts are included to revent large, sudden changes in demand. Based uon the arameters included in the cost function, metrics are develoed to quantify the erformance of the PMC. The metrics used for the PMC testing on the Purdue test bed were chosen to be the maximum and average deviations of the GS-2 outut ower, the SPS outut ower, and the bus voltage from their resective targets. Other metrics are the maximum ram rate of the GS- ower and the total time during which the ram rate exceeds a threshold value. The ram rate is a basic sloe calculation that considers the resent value and the value. seconds ahead. The threshold is an arbitrary value that would be chosen by the user based uon the secific generator tolerance levels. The test lan to demonstrate the effectiveness of the PMC consists of three test cases which are outlined in Table. Before testing the PMC, baseline measurements are observed during a full SWPPL ulsing cycle (7 ulses as shown in Figure 2) without a PMC and constant ower targets being fed to GS-2 and SPS in oen loo. After closing the control loo between the PMC and the test bed, the weighting factors are tuned so that the test bed behavior is similar to the behavior without a PMC, thus establishing a baseline PMC behavior as the first PMC test case, A. Two follow u test cases seek to imrove uon this behavior. The second test case ( B) consists of increasing the enalty on the GS- ower ram rate, which is the equivalent of extending the lifetime of the generator. The third test case ( C) maintains the new GS- ower ram rate enalty and increases the SPS tracking enalty to decrease the deviation from the target. This is the equivalent of asking the shi to maintain a velocity closer to a set oint. A trade-off exists between the selection of the PMC samling time, the rediction horizon over which the PMC can look ahead to assess the otimal control inuts, and the hardware caabilities. Considering these trade-offs and the fastest/slowest dynamics of the system, the test cases consider a model-redictive PMC with a samling time of 6 ms and a rediction horizon of 6 stes (equivalent to.36 s). V. RESULTS AND ANALYSIS The test lan was carried out on a low-order simulation model of the test bed described in Sec. II to assess the exected PMC behavior and reduce risk before testing in the lab. The results for simulation and lab testing are shown and conclusions are drawn based uon comarison of both waveforms and metrics. The chosen set oints for the test cases are for GS-2 to generate 3 kw and SPS to sink kw. 5 GS- Maximum Ram Rate Time that GS- Ram Rate Exceeds Threshold (6 kw/s) kw/s 5 seconds.5 6 SPS Maximum Power Deviation 3 SPS Average Power Deviation GS-2 Maximum Power Deviation 5 GS-2 Average Power Deviation Vbus Maximum Deviation.8 Vbus Average Deviation Figure 3. PMC simulation results: a) GS- electrical ower, b) SPS electrical ower, and c) GS-2 electrical ower. A. Simulation Results With the PMC in a closed control loo with the model of the test bed, the SWPPL rofile was introduced to the system as a known disturbance. The disturbance is known to the PMC in the sense that the rofile of the disturbance (magnitude and ulse width) is stored offline and the PMC is given a warning signal that the rofile is going to be introduced to the system within a reset amount of time (5 seconds in this set of test cases). The amount of warning time could be as low as the rediction horizon without imacting the PMC erformance. The simulated electrical ower waveforms for GS-, SPS, and GS-2 are shown in Figure 3 and the chosen metrics for quantifying the system erformance are shown in Figure 4. The Figure 4. PMC simulation metrics. system was disturbed with the 4 second ulse cycle shown in Figure 2, but because the results for each ulse were identical only the results due to the first ulse in the cycle are shown in Figure 3. The 8 kw ulse disturbs the system just after 6 seconds of simulation time and the first ulse lasts for one second. Observing the GS- electrical ower in Figure 3(a) it is demonstrated that under A the GS- is absorbing most of the ulse disturbance ower resulting in a very high ram rate (73. kw/s). This abusive ram rate would limit the exected lifetime of a shi GTG, so under B the enalty on the GS- ram rate is increased in the PMC cost function. The results under B (shown in red) demonstrate that increasing the GS- ram rate enalty results in a decreased GS- ram rate. The ram rate is reduced by 45 down to 4.4 kw/s as ref
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