Plug-and-Play Operation of Microgrids Florian D orfler ETH Z - PowerPoint PPT Presentation

Electric power networks & their conventional operation Plug-and-Play Operation of Microgrids Florian D¨ orfler ETH Z¨ urich electric energy is our lifeblood UC Louvain Seminar purpose of electric power grid : February 10, 2015 generate/transmit/distribute constraints : op, econ, & stab 1 / 32 Paradigm shifts & new problem scenarios . . . in a nutshell Microgrids Structure ◮ low-voltage distribution networks ◮ grid-connected or islanded ◮ autonomously managed Applications 1 2 8 9 4 ◮ hospitals, military, campuses, large vehicles, & isolated communities 1 controllable fossil fuel sources ⇒ stochastic renewable sources Benefits 2 centralized bulk generation ⇒ distributed low-voltage generation ◮ naturally distributed for renewables 3 synchronous generators ⇒ low/no inertia power electronics ◮ flexible, efficient, & reliable 4 generation follows load ⇒ controllable load follows generation 5 monopolistic energy markets ⇒ deregulated energy markets Operational challenges ◮ volatile dynamics & low inertia 6 centralized top-to-bottom control ⇒ distributed non-hierarchical control ◮ plug’n’play & no central authority 7 human in the loop & heuristics ⇒ “smart” real-time decision making 2 / 32 3 / 32

Conventional control architecture from bulk power ntwks A preview – plug-and-play operation architecture flat hierarchy, distributed, no time-scale separations, & model-free . . . 3. Tertiary control (offline) Goal: optimize operation source # 1 source # 2 source # n Strategy: centralized & forecast … … Tr ansceiver Tr ansceiver Tr ansceiver … 2. Secondary control (slower) Second ary S e cond ary S e cond ary Goal: maintain operating point Ter tiary T er tiary T er tiary Control Control Control Strategy: centralized Control Control Control 1. Primary control (fast) P r imary P r imary P r imary Goal: stabilization & load sharing Control Control Control Strategy: decentralized Microgrids : distributed, model-free, Power System online & without time-scale separation ⇒ break vertical & horizontal hierarchy 4 / 32 5 / 32 Outline Introduction Modeling Primary Control Tertiary Control modeling & assumptions Secondary Control Virtual Oscillator Control Conclusions we will illustrate all theorems with experiments

Modeling: a power system is a circuit Modeling: a power system is a circuit 1 synchronous AC circuit with 1 synchronous AC circuit with harmonic waveforms E i e i( θ i + ω ∗ t ) harmonic waveforms E i e i( θ i + ω ∗ t ) j j i i G ij + i B ij G ij + i B ij Z ∗ Z ∗ i i i i 2 ZIP loads : constant impedance, 2 ZIP loads : constant impedance, I ∗ I ∗ i i current, & power P ∗ i + i Q ∗ current, & power P ∗ i + i Q ∗ (today) (today) P ∗ i + i Q ∗ P ∗ i + i Q ∗ i i i i injection = � injection = � 3 coupling via Kirchhoff & Ohm 3 coupling via Kirchhoff & Ohm power flows power flows 4 identical lines G / B = const . 4 identical lines G / B = const . (equivalent to lossless case G / B = 0) (equivalent to lossless case G / B = 0) 5 decoupling: P i ≈ P i ( θ ) & Q i ≈ Q i ( E ) 5 decoupling: P i ≈ P i ( θ ) & Q i ≈ Q i ( E ) (near operating point) (near operating point) ◮ active power: ◮ trigonometric active power flow: P i = � j B ij E i E j sin( θ i − θ j ) + G ij E i E j cos( θ i − θ j ) P i ( θ ) = � j B ij sin( θ i − θ j ) ◮ reactive power: ◮ polynomial reactive power flow: Q i = − � j B ij E i E j cos( θ i − θ j ) + G ij E i E j sin( θ i − θ j ) Q i ( E ) = − � j B ij E i E j (not today) 6 / 32 6 / 32 Modeling the “essential” network dynamics & controls (models can be arbitrarily detailed) 1 synchronous machines (swing dynamics) me ch. electr. M i ¨ i + P c θ i = P ∗ i − P i ( θ ) torqu e torque primary control 2 DC & variable AC sources interfaced with voltage-source converters (droop characteristic) Ee i( θ + ω t ) P ∗ i + P c i = P i ( θ ) P i ( θ ) , Q i ( E ) 3 controllable loads (voltage- and frequency-responsive) P i + i Q i P ∗ i + P c i = P i ( θ ) Ee i( θ + ω t ) 7 / 32

Decentralized primary control of active power Putting the pieces together... differential-algebraic, nonlinear, large-scale closed loop Emulate physics of dissipative 50 49 51 n e twork physics 48 52 coupled synchronous machines : power c onsumed power supplied P mech i + P c power balance: = P ∗ i − P i ( θ ) M i ¨ θ + D i ˙ i θ i � j B ij si n( θ i − θ j ) power flow: P i ( θ ) = Hz � = P ∗ i − j B ij sin( θ i − θ j ) droop control ⇒ sum equations & set ˙ θ i = ω sync : Conventional wisdom: physics D i ˙ θ i = ( P ∗ i − P i ( θ )) i P ∗ ω sync = � i / � i D i are naturally stable & sync fre- quency reveals power imbalance 0 = P ∗ � passive loads: i − j B ij sin( θ i − θ j ) P / ˙ M i ¨ θ i + D i ˙ � θ droop control: θ i = P ∗ synchronous machines: i − j B ij sin( θ i − θ j ) ( ω i − ω ∗ ) ∝ ( P ∗ i − P i ( θ )) D i ˙ � θ i = P ∗ i − j B ij sin( θ i − θ j ) inverter sources: ω sync � D i ˙ θ i = P ∗ i − P i ( θ ) D i ˙ � θ i = P ∗ controllable loads: i − j B ij sin( θ i − θ j ) 8 / 32 9 / 32 A perspective from coupled oscillators Closed-loop stability under droop control Mechanical oscillator network Ω 1 Ω 2 Angles ( θ 1 , . . . , θ n ) evolve on T n as Theorem: stability of droop control [J. Simpson-Porco, FD, & F. Bullo, ’12] ∃ unique & exp. stable frequency sync ⇐ ⇒ active power flow is feasible M i ¨ θ i + D i ˙ θ i = Ω i − � j K ij sin( θ i − θ j ) • inertia constants M i > 0 Main proof ideas and some further results : Ω 3 • viscous damping D i > 0 sources P ∗ loads P ∗ � i + � ω sync = ω ∗ + • external torques Ω i ∈ R i • synchronization frequency: � sources D i • spring constants K ij ≥ 0 ( ∝ power balance) � P ∗ Droop-controlled power system (load # i ) i P ∗ P ∗ • steady-state power injections: P i = P ∗ i − D i ( ω sync − ω ∗ ) (source # i ) 1 2 0 = P ∗ � i − j B ij sin( θ i − θ j ) (depend on D i & P ∗ i ) D i ˙ P ∗ θ i = P ∗ � i − j B ij sin( θ i − θ j ) 3 • stability via incremental Chetaev energy function [C. Zhao, E. Mallada, & FD ’14] M i ¨ θ + D i ˙ � θ i = P ∗ i − j B ij sin( θ i − θ j ) 10 / 32 11 / 32

Tertiary control and energy management an offline resource allocation & scheduling problem tertiary control (energy management) 12 / 32 Tertiary control and energy management Objective I: decentralized proportional load sharing an offline resource allocation & scheduling problem � � 1) Sources have injection constraints : P i ( θ ) ∈ 0 , P i � � loads P ∗ 2) Load must be serviceable : 0 ≤ � � ≤ � sources P j � � j � 3) Fairness: load should be shared proportionally: P i ( θ ) / P i = P j ( θ ) / P j P 1 P 2 minimize { cost of generation, losses, . . . } P 1 P 2 subject to equality constraints: power balance equations inequality constraints: flow/injection/voltage constraints source # 2 source # 1 logic constraints: commit generators yes/no . load . . 12 / 32 13 / 32

Objective I: decentralized proportional load sharing Objective I: decentralized proportional load sharing P i ( θ ) ∈ � � � � 1) Sources have injection constraints : 0 , P i 1) Sources have injection constraints : P i ( θ ) ∈ 0 , P i � � � � loads P ∗ 2) Load must be serviceable : 0 ≤ � � ≤ � sources P j loads P ∗ � � 2) Load must be serviceable : 0 ≤ � � ≤ � sources P j � � j j � � 3) Fairness: load should be shared proportionally: P i ( θ ) / P i = P j ( θ ) / P j 3) Fairness: load should be shared proportionally: P i ( θ ) / P i = P j ( θ ) / P j Theorem: fair proportional load sharing [J. Simpson-Porco, FD, & F. Bullo, ’12] A little calculation reveals in steady state: Let the droop coefficients be selected proportionally : P ∗ j − ( D j ω sync − ω ∗ ) ! P ∗ i − ( D i ω sync − ω ∗ ) ! P i ( θ ) = P j ( θ ) D i / P i = D j / P j & P ∗ i / P i = P ∗ j / P j ⇒ = P i P j P i P i The the following statements hold: . . . so choose P ∗ P ∗ D i = D j (i) Proportional load sharing: P i ( θ ) / P i = P j ( θ ) / P j j i = and P i P j P i P j � � loads P ∗ � � (ii) Constraints met: 0 ≤ �� � ≤ � sources P j ⇔ P i ( θ ) ∈ 0 , P i � � j 13 / 32 13 / 32 Objective I: fair proportional load sharing Objective II: economic generation dispatch proportional load sharing is not always the right objective minimize the total accumulated generation (many variations possible) � sources α i u 2 minimize θ ∈ T n , u ∈ R nI f ( u ) = i subject to P ∗ source power balance: i + u i = P i ( θ ) P ∗ load power balance: i = P i ( θ ) source # 3 branch flow constraints: | θ i − θ j | ≤ γ ij < π/ 2 α i u ∗ i = α j u ∗ Unconstrained case: identical marginal costs at optimality j In conventional power system operation, the economic dispatch is solved offline , in a centralized way, & with a model & load forecast In a grid with distributed energy resources, the economic dispatch should be source # 2 source # 1 solved online , in a decentralized way, & without knowing a model load 14 / 32 15 / 32