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On Distributed Economic Model PredictiveControl of Nonlinear Process Systems
Timothy Anderson, Matthew Ellis andPanagiotis D. Christofides
Department of Chemical & Biomolecular EngineeringDepartment of Electrical EngineeringUniversity of California, Los Angeles
AIChE Annual MeetingAtlanta, GA
November 17, 2014
PROCESS ECONOMICS AND CONTROLBackground
• Hierarchical approach to process economic optimization and control
• Upper layer: economic optimization
⋄ Real-time optimization (RTO) (TE Marlin and AN Hrymak, AIChE Symposium, 1997)
⋄ Optimizes process economics via a steady-state process model
• Lower layer: feedback control
⋄ Force the process system to operate at the optimal steady-state⋄ Tracking model predictive control (MPC) formulated with a quadratic stage
cost (DQ Mayne et al., Automatica, 2000)
• Disadvantages
⋄ Delay in optimization⋄ Inconsistent models used in each layer⋄ Next-generation (smart) manufacturing (PD Christofides et al., AIChE J., 2007)
▷ Tight integration between plant operations and process economicoptimization
▷ Real-time energy management
PROCESS ECONOMICS AND CONTROLBackground
• Traditional Paradigm
(Steady-state)Economic Optimization
Tracking MPC
J =∫ T
0
(|x(t)− x∗
SP |Qc+ |u(t)− u
∗
SP |Rc) dt
Process
x∗SP , u
∗SP
u∗(tk|tk)
• Steady-state operation
• Integration of economic optimizationand process control
• MPC with an economic stage cost oreconomic MPC (EMPC)
Economic MPC
J =∫
T
0
le(x(t), u(t))dt
Process
u∗(tk|tk)
• Dynamic/time-varying operation
Improve economic process performance via dynamic operation(M Diehl et al., IEEE TAC, 2011; R Huang et al., JPC, 2011; D Angeli et al., IEEE TAC, 2012; M Heidarinejad et al.,
AIChE J., 2012; M Ellis et al., JPC, 2014)
CENTRALIZED VS. DISTRIBUTED CONTROL
Process n
S
A
S
C
Process 2Process 1
Networked
control
system
S
A
S
C
S
A
S
C
Networked
control
system 2
Networked
control
system 1
Networked
control
system m
Process n
S
A
S
C
Process 2Process 1
S
A
S
C
S
A
S
C
• Centralized process control architecture
⋄ Computational complexity⋄ Organization and maintenance
• Move towards distributed process control architecture
⋄ Control inputs are evaluated in more than one distributed controllers⋄ Model Predictive Control (MPC): a natural framework for distributed control
design (PD Christofides et al., Comp. Chem. Engr., 2013)
DISTRIBUTED EMPC (DEMPC) RESULTS AND PRESENTWORK
• Despite the advances in computation, there is still motivation to use distributedEMPC architectures to deal with real-time computational constraints
• Previous work on DEMPC
⋄ Sequential DEMPC of a nonlinear chemical process network (X Chen et al., JPC, 2012)
⋄ Cooperative DEMPC▷ Linear systems (J Lee and D Angeli, Proc. of CDC-ECC, 2011)
▷ Nonlinear systems (J Lee and D Angeli, Proc. of MTNS, 2012)
⋄ DEMPC of interconnected nonlinear systems (PAA Driessen et al., Proc. of CDC, 2012)
⋄ DEMPC of interacting linear systems (M. Müller and F. Allgöwer, Proc. of IFAC, 2014)
• Present work
⋄ DEMPC of a nonlinear chemical process with time-varying operation
⋄ Design and evaluate: centralized EMPC, sequential distributed EMPC,iterative distributed EMPC
CATALYTIC OXIDATION OF ETHYLENE TO PRODUCEETHYLENE OXIDE
• Ethylene oxide production from ethylene in a nonisothermal continuous-stirredtank reactor (CSTR) (F. Özgülşen et al., CES, 1992; F. Alfani and J. J. Carberry, Chim. Ind., 1970)
• Reactions:C2H4 +
1
2O2
r1−−→ C2H4O
C2H4 + 3O2r2−−→ 2CO2 + 2H2O
C2H4O+5
2O2
r3−−→ 2CO2 + 2H2O
• Reaction rates:r1 = k1exp (−E1/RT )P 0.5
E
r2 = k2exp (−E2/RT )P 0.25E
r3 = k3exp (−E3/RT )P 0.5EO
• Dimensionless inputs
⋄ Feed volumetric flow rate u1
⋄ Feed ethylene concentration u2
⋄ Coolant temperature u3
• Dimensionless process state variables
⋄ Reactor gas density x1
⋄ Ethylene concentration x2
⋄ Ethylene oxide concentration x3
⋄ Reactor temperature x4
DYNAMIC PROCESS MODEL• Process state equations (F. Özgülşen et al., Chem. Eng. Sci., 1992)
dx1
dt= u1(1− x1x4)
dx2
dt= u1(u2 − x2x4)−A1e
γ1/x4(x2x4)0.5 −A2e
γ2/x4(x2x4)0.25
dx3
dt= −u1x3x4 +A1e
γ1/x4(x2x4)0.5 −A3e
γ3/x4(x3x4)0.5
dx4
dt=
u1
x1(1− x4) +
B1
x1eγ1/x4(x2x4)
0.5 +B2
x1eγ2/x4(x2x4)
0.25
+B3
x1eγ3/x4(x3x4)
0.5 − B4
x1(x4 − u3)
• States and inputs of the process:
x1 =ρ
ρref, x2 =
CE
Cref, x3 =
CEO
Cref, x4 =
T
Tref
u1 =Qf
Qref, u2 =
CE,f
Cref, u3 =
Tc
Tref
• Process model has the following general form:
x(t) = f(x(t), u1(t), u2(t), u3(t))
CONTROL OBJECTIVE• Operation goals
⋄ Maximize average ethylene oxide yield
Y (tf ) =
∫ tf
t0
u1(τ)x3(τ)x4(τ) dτ∫ tf
t0
u1(τ)u2(τ) dτ
⋄ Limit average ethylene fed based on feedstock limitations (integral constraint)1
tf − t0
∫ tf
t0
u1(τ)u2(τ) dτ = 0.175
⋄ Integral constraint imposed over operating periods tp to ensure that over thelength of operation the constraint is satisfied
• Economic stage cost to be maximized:∫ tk+N
tk
le(x(τ), u(τ)) dτ =
∫ tk+N
tk
u1(τ)x3(τ)x4(τ) dτ
CENTRALIZED EMPC ARCHITECTURE ANDFORMULATION
• Implemented with a shrinking horizon
• Account for the process economics andconstraint on the available average re-actant material
• EMPC Formulation:
EMPC Processu x
x
maximizeu∈S(∆)
∫ tk+Nk
tk
u1(τ)x3(τ)x4(τ) dτ
subject to ˙x(t) = f(x(t), u1(t), u2(t), u3(t))
x(tk) = x(tk)
u1 ∈ [0.0704, 0.7042], u2 ∈ [0.2465, 2.4648]
u3 ∈ [0.6, 1.1]
1
tp
∫ tk+Nk
tk
u1(τ)u2(τ)dτ
= 0.175− 1
tp
∫ tk
t0+jtp
u∗1(τ)u
∗2(τ)
• Maximize C2H4O yield
• Reactor state equations
• State measurement
• Input bounds
• Integral constraint on av-erage amount of ethylenefed to reactor
CENTRALIZED EMPC RESULTS
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
• Time-varying operation of inputs induces time-varying state trajectories
• Average ethylene oxide yield of 10.22% achieved under centralized EMPC
• Yield through steady-state operation: 6.38%
• Average yield under EMPC is 60.2% better than steady-state operation
SEQUENTIAL DISTRIBUTED EMPCSequential DEMPC 1-2
EMPC-1
EMPC-2 Process
u1, u2
u3
u1, u2
x
• EMPC-1 solves for the optimal u1 and u2 input trajectories
• The u1 and u2 input trajectories are sent to EMPC-2
• EMPC-2 solves for the optimal u3 input trajectory
SEQUENTIAL DISTRIBUTED EMPCSequential DEMPC 2-1
EMPC-2
EMPC-1 Process
u3
u1, u2
u3
x
• EMPC-1 solves for the optimal u3 input trajectory
• The u3 input trajectories are sent to EMPC-2
• EMPC-2 solves for the optimal u1 and u2 input trajectories
SEQUENTIAL DISTRIBUTED EMPC ALGORITHM
1. At time tk, all of the EMPCs receive state measurement x(tk) from the sensors
2. For j = 1 to 2
2.1 DEMPC j receives input trajectories of controllers preceding it
2.2 DEMPC j optimizes the objective function for the inputs it controls andassumes a profile ui = hi(x) for all i input trajectories of the controllersfollowing after it
2.2.1 DEMPC j communicates input trajectory to controller j + 1 if j < 2
2.2.2 If DEMPC j = 2, all controllers apply first step of their input trajectories tothe process
3. Set k + 1 → k and return to Step 1
• The controller hi(x) is an explicit controller
• In this case, hi(x) is a PI controller (i = 1, 2, 3)
SEQUENTIAL DISTRIBUTED EMPC 1-2
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Sequential DEMPC 1-2
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Centralized EMPC
• Average yield is 10.20% compared to 10.22% for centralized EMPC
SEQUENTIAL DISTRIBUTED EMPC 2-1
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Sequential DEMPC 2-1
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Centralized EMPC
• Average yield is 9.91% compared to 10.22% for centralized EMPC
ITERATIVE DEMPC
EMPC-1
EMPC-2 Process
u1, u2
u3
u1, u2 u3
x
• In parallel, EMPC-1 solves for the optimal u1 and u2 input trajectories whileEMPC-2 solves for the optimal u3 input trajectory
• The optimal input trajectories are sent to the other EMPC and EMPC-1 andEMPC-2 recomputes optimal input trajectories
• The iterative process repeats for a specified number of iterations
ITERATIVE DEMPC ALGORITHM
1. At time tk, all of the EMPCs receive the state measurement x(tk) from the sensors
2. For i = 1 to c and for all j ∈ {1, 2}
2.1 DEMPC j optimizes the objective function for the inputs it controls
2.2 DEMPC j communicates its input trajectory to the other distributed EMPCs
3. Of all iterations, apply the first step corresponding to the input trajectoriesleading to the best economic performance
4. Set k + 1 → k and return to Step 1
• c is the specified number of iterations
• Given the nonlinearities and non-convexity of the optimization problem, noguarantees that the iterations will improve the overall economic cost
• Solution: apply the control action corresponding to the best economicperformance over all the iterations
ITERATIVE DEMPC (1 ITERATION)
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Iterative DEMPC (1 iteration)
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Centralized EMPC
• Average yield is 10.05% compared to 10.22% for centralized EMPC
ITERATIVE DEMPC (4 ITERATIONS)
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Iterative DEMPC (4 iterations)
0 50 100 150 200 250 300 350 400 4500.9
1
1.1
x1
0 50 100 150 200 250 300 350 400 4500
1
2
x2
0 50 100 150 200 250 300 350 400 4500
0.2
0.4
x3
0 50 100 150 200 250 300 350 400 4500.5
1
Time
x4
0 50 100 150 200 250 300 350 400 4500
0.5
1
u1
0 50 100 150 200 250 300 350 400 4500
1
2
3
u2
0 50 100 150 200 250 300 350 400 450
0.6
0.8
1
1.2
Time
u3
Centralized EMPC
• Average yield is 10.06% compared to 10.22% for centralized EMPC
COMPARISON OF ETHYLENE OXIDE YIELDS
Strategy % Yield of ethylene oxide
1 Iteration 10.05
2 Iterations 10.06
3 Iterations 10.06
4 Iterations 10.06
Centralized 10.22
Sequential 1-2 10.20
Sequential 2-1 9.91
PI Controller 6.38
• Overall centralized EMPC achieved the highest yield
• Yields achieved demonstrate ability to distribute the control actions withoutsignificantly sacrificing the economic performance
COMPUTATION TIME OF ALL STRATEGIES
Strategy Average solution time (ms)
1 Iteration 832
2 Iterations 3530
3 Iterations 4704
4 Iterations 6104
Centralized 4244
Sequential 1-2 1039
Sequential 2-1 2969
• Iterative implementation of DEMPC has the lowest solution time
⋄ The number of decision variables per controller is smaller compared tocentralized EMPC
⋄ The calculations can be carried out in parallel processors
