Nonlinear control of power converters for HVDC applications

Transcription

1 Nonlinear control of power converters for HVDC applications Morten Hovd and Mohsen Vatani Workshop on Distributed Energy Management Systems Washington DC, April 22, 215

2 2 Table of Contents Modular Multilevel Converter (MMC) MMC structure MMC in HVDC systems MMC Control difficulties Finite Control Set Model Predictive Control (FCS-MPC) The controller highlights Simulation results Shortcomings of FCS-MPC Sum Of Squares decomposition method Controller design MMC model in the dq frame Simulation results Conclusion

3 3 Table of Contents Modular Multilevel Converter (MMC) MMC structure MMC in HVDC systems MMC Control difficulties Finite Control Set Model Predictive Control (FCS-MPC) The controller highlights Simulation results Shortcomings of FCS-MPC Sum Of Squares decomposition method Controller design MMC model in the dq frame Simulation results Conclusion

4 4 Introduction of MMC a large number of voltage cells connected in series by inserting desired number of cells, any voltage level can be produced less harmonics no need for AC filters redundancy is higher lower switching frequency and semiconductor loss reduced manufacturing cost due to similarity of cells v f S 1 + v S SM 2 - i v L c R c v + Cmi,j - arm C i u, a v t i l, a SMu1,a SMu2,a SMuN,a SM l1,a + v u, a - L i u, b L i u, c L R R R R R R i l, b i L L l, c L + SMl2,a v l, a SMu1,b SMu2,b SMuN,b SM l1,b SMl2,b leg SMu1,c SMu2,c SMuN,c SM l1,c SMl2,c i dc + V dc SM ln,a - SM ln,b SM ln,c -

5 5 MMC in HVDC system Besides other applications, MMC has become the most promising converter topology for HVDC stations MMC-HVDC projects: Project Trans Bay Nanhui Southwest Dalian France Zhoushan DC Voltage ± 2 kv ±3 kv ±3 kv ±32 kv ±32 kv ±2 kv Power 4 MW 2 MW 144 MW 1 MW 1 MW 4 MW Length 8 km 8.4 km 25 km 43 km 65 km 134 km operated by Siemens C-EPRI Alstom C-EPRI Siemens C-EPRI Year Location San Francisco Shanghai Sweden China France China type underwater offshore windfarm city connection under ground under ground multi terminal

6 6 MMC in HVDC system Tennet off-shore wind farm complex in North Sea near to the German coast :

7 7 MMC Control difficulties the control of the MMC converter is not as easy as other types of converters: Control of power transfer Balancing capacitor voltages Reducing circulating current Decrease switching frequency and loss Decrease communication load The control problem becomes a Multi-Input Multi-Output problem and classical PI controllers does not satisfy the objectives Advance control methods is introduced in recent years for MMC control: Repetitive control Model Predictive Control Proportional Resonant controller Optimization with Lagrange multipliers...

8 8 Table of Contents Modular Multilevel Converter (MMC) MMC structure MMC in HVDC systems MMC Control difficulties Finite Control Set Model Predictive Control (FCS-MPC) The controller highlights Simulation results Shortcomings of FCS-MPC Sum Of Squares decomposition method Controller design MMC model in the dq frame Simulation results Conclusion

9 9 The idea The calculation load of the MPC methods is high for real time control of MMCs. The idea: a pre-defined cost function is calculated one step ahead for all possible control actions and the best control action, which minimizes the cost function, is selected. ac-side current, circulating current, and summation of capacitor voltages are controlled in each arm Cost function: J j = c iv,j,ref 1 i v,j + c2 icir,ref i cir,j vdc,ref + c3 vu,j Σ vdc,ref + c 4 vl,j Σ.

10 1 The traditional FCS-MPC for MMCs 1. All possible control actions are evaluated, and the optimal action selected 2. Gives the switch settings directly 3. Complexity grows exponentially with number of SMs and with prediction horizon 4. Has been shown to work satisfactorily for low number of SMs, prohibitively complex for the number of SMs used in industrial applications

11 11 Our approach to FCS-MPC for MMCs 1. All SMs are treated as equal 2. Evaluate the cost function only to find the optimal number of inserted SM s (the optimal insertion index) 3. Individual switch settings determined by a sorting algorithm and the insertion index. The sorting algorithm is designed to balance capacitor voltages. 4. Complexity linear in the number of SM s, still exponential in horizon length

12 12 Control block diagram The following systems is simulated in PLECS/MATLAB. 3-phase MMC (Fig.1) Filter Transformer Utility Grid v dc 2 R c i v,abc L c L s V s v dc 2 v t,abc v f,abc i ref dc V dc v u,abc v l,abc PLL v f,abc θ Gating Signals i v f,dq cir,abc LPF dq f o =5Hz abc Sorting n u,abc FCS-MPC i v,abc Algorithm n l,abc Controller ref θ ref P i abc i Power ref abc dq dq Equations Q ref Local controller Central controller v f,abc

13 13 Simulation result Power reversal command, switching frequency=3.5 khz P&Q [MW&MVAR] 4 4 P Q (a) [kv] Σ v u,a Σ v u,a ref v dc (b) i cir,a [A] i cir,a ref i cir 3 (c) 8 i v,a [A] 8 i v,a ref i v,a (d) 3.1 v c,u,a [kv] (e) Time [s]

14 14 Simulation result By reducing switching frequency to 2 Hz P&Q [MW&MVAR] 4 4 P Q (a) [kv] Σ v u,a Σ v u,a ref v dc (b) i cir,a [A] i cir,a ref i cir 3 (c) 8 i v,a [A] 8 i v,a ref i v,a (d) 3.1 v c,u,a [kv] (e) Time [s]

15 15 Variations of the FCS-MPC Adding a constraint on the maximum change in insertion number for each timestep Works acceptably in simulations, but causes slower response to quick disturbances Reduces the exponential growth in complexity with horizon length

16 16 Shortcomings of FCS-MPC The computational complexity - in particular for the high number of SMs used in industrial applications No stability guarantee (common for finite horizon optimal control). Stability assessed by simulation

17 17 Table of Contents Modular Multilevel Converter (MMC) MMC structure MMC in HVDC systems MMC Control difficulties Finite Control Set Model Predictive Control (FCS-MPC) The controller highlights Simulation results Shortcomings of FCS-MPC Sum Of Squares decomposition method Controller design MMC model in the dq frame Simulation results Conclusion

18 18 The goal The MMC is modeled as a discrete-time bilinear system m x k+1 = Ax k + (B i x k + b i )u i,k = Ax k + (B x + B)u k i=1 the nonlinearity consists of products between the states and input To stabilize the system, the Lyapunov inequality should be fulfilled. V(x k ) V(x k+1 ) = x T k Px k x T k+1 Px k+1 > The SOS method designs the controller, by YALMIP package (in MATLAB) in the form of ratio of two polynomials as: u i (x) = c i(x k ) c (x k )

19 19 Controller design by SOS method in YALMIP Theorem Region of convergence: Given a quadratic function V(x) = x T Px, polynomials c i (x),i [1,...,m], and SOS polynomials c (x) and s 1 (x,z), a bilinear discrete time system in closed loop with the control law is stable x x T Px < γ, provided u i (x) = C(x)x ( c (x) + 1) [ ] T [ ] x x M(x) s z z 1 (x,z)(γ x T Px) > where [ ( c M(x) = (x) + 1)P (( c (x) + 1)A + (B x + B)C(x)) T ] P > P(( c (x) + 1)A + (B x + B)C(x)) ( c (x) + 1)P

20 2 List of variables states: i v,dq ac-side currents in dq reference frame i cir,dq circulating currents in dq reference frame i d dc component of the circulating current W the total stored energy in the converter W energy difference between the upper and lower arms inputs V u,dq upper arm voltage in the dq reference frame V l,dq lower arm voltage in the dq reference frame V d the dc component of arm voltages Other parameters ω rotating frequency of source voltage v f,dq the ac-side voltage of the converter

21 21 MMC model in the dq frame for the ac side current in dq reference frame: [ ] di v,dq R+2R c = L+2L ω c dt ω R+2R i c v,dq + v u,dq v l,dq L + 2L L+2L c c [ ] di Circulating current: cir,dq R dt = L ω ω R L dc component of circulating current: the stored energy dynamics: dw dt = dw u dt + dw l dt + 2v f,dq L + 2L c. i cir,dq 1 2L (v u,dq + v l,dq ), di d dt = R L i d 1 2L V d + 1 2L V dc = 3 4 v u,di v,d v u,di cir,d 3 4 v u,qi v,q v u,qi cir,q v l,di v,d v l,di cir,d v l,qi v,q v l,qi cir,q + 3V d i cir,, d W dt = dw u dt dw l dt = 3 4 v u,di v,d v u,di cir,d 3 4 v u,qi v,q v u,qi cir,q 3 4 v l,di v,d 3 2 v l,di cir,d 3 4 v l,qi v,q 3 2 v l,qi cir,q.

22 22 MMC model in the dq frame the bilinear model of MMC R+2Rc w L+2Lc w R+2Rc L+2Lc R w ẋ(t) = L w R x(t) + x(t)u L 1 R L }{{}}{{} B 1c Ac + x(t)u 2 + x(t)u }{{ 2 } }{{} 4 3 }{{ 2 B 2c B 3c B 4c + x(t)u } {{ } B 5c 1 L+2Lc 2L 1 } {{ } b 1c u L+2Lc 2L 1 } {{ } b 2c L+2Lc 1 u 2 + 2L 1 } {{ } b 3c 2v f,d 1 L+2Lc L+2Lc 2v f,q u L+2Lc u 2L u 5 + 2L V dc 2L }{{}}{{} b b 4c 4c } {{ } dc

23 23 Control block diagram The following system is simulated in PLECS/MATLAB 3-phase MMC (Fig.1) Filter Transformer Utility Grid v dc 2 R c i v,abc L c L s V s v dc 2 Gating Signals n u,abc Sorting n Algorithm l,abc Controller v t,abc Phase Disposition PWM v ref u,abc v ref l,abc θ PLL u ss x ss v f,abc v ref dq u,dq Input abc v ref Calculator dq l,dq (22) abc u~ SOS Control Steady State Calculator ~ x PU P ref Q ref x ss i v,dq i abc v,abc dq θ i abc icir,abc Deviation cir,dq dq State θ i Calculator dc (22) W Energy v c,mij Calculator W (14),(15)

24 24 Activation of controller Activation of controller at t =.15 s I cir,a (A) SOS CCSC Ref 25 (a) (kv) SOS CCSC Ref v leg,a Σ (b) Time [s]

25 25 Convergence to the operating point Before activation of the controller, the states are far from their references. I v,d (A) 6 3 (a) SOS CCSC Ref 6 I cir,a (A) 6 12 (b) SOS CCSC Ref I dc (A) SOS CCSC Ref (c)

26 26 Convergence to the operating point Before activation of the controller, the states are far from their references. (kv) v u,a Σ (d) SOS CCSC Ref (kv) v leg,a Σ SOS CCSC (e) Time [s]

27 27 Real power flow reversal command Initially, the MMC system is in a steady-state condition, transferring P = 4 MW to the ac grid. At t =.2 s, the real power flow is reversed to P = 4 MW. I v,d (A) 1 SOS CCSC Ref (a) 4 I cir,a (A) SOS CCSC Ref (b) 1 I dc (A) SOS CCSC 1 Ref (c)

28 28 Real power flow reversal command Initially, the MMC system is in a steady-state condition, transferring P = 4 MW to the ac grid. At t =.2 s, the real power flow is reversed to P = 4 MW. (kv) v u,a Σ SOS CCSC (d) (kv) v leg,a Σ SOS CCSC (e) Time [s]

29 29 Related work Application to buck-boost converters. Also a bilinear system for the averaged model with the duty cycle as input. Introducing integral action in the SOS design. Studying operating point changes. For non-linear systems, stability is not (necessarily) independent of the operating point. Studying robustness to parameter changes in the system

30 3 Ongoing and future work Application to MMC controllers in abc frame Handling of sinusoidal references Experimental verification

31 31 Table of Contents Modular Multilevel Converter (MMC) MMC structure MMC in HVDC systems MMC Control difficulties Finite Control Set Model Predictive Control (FCS-MPC) The controller highlights Simulation results Shortcomings of FCS-MPC Sum Of Squares decomposition method Controller design MMC model in the dq frame Simulation results Conclusion

32 32 Conclusion Modular Multilevel Converters have several advantages for AC/DC conversion These promising features made them the best converter topology for HVDC stations The main disadvantage is the control difficulty and the need for advanced control methods Finite Control Set Model Predictive Control method introduce a controller which is optimal for next sampling instant Using Sum of Squares decomposition method along with a Lyapunov function gives both a guarantee for stability of the converter and a good performance for the response

33 33 Acknowledgement Parts of this work has been performed in cooperation with prof. Maryam Saeedifard of Georgia Tech.

34 34 Questions? Thank you for your attention Questions?