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This project has received funding from the European Union’s Seventh Framework Programme for

research, technological development and demonstration under grant agreement no 611281.

Technical work in WP2

UNIZG-FER

Mato Baotić, Branimir Novoselnik,

Mladen Đalto, Jadranko Matuško,

Mario Vašak, Andrej Jokić

Nice, March 14, 2016

WP2 Work Plan (Task 2.4)

• Analysis of applicability and estimation of computational complexity of different uncertainty management strategiesfor SoS. (M15 – M20)

• Development of scenario-based robust dynamic management for SoS. (M18 – M24)

• Development of a stochastic methodology for the coordination of systems using SoS model subject to Gaussian distribution of uncertainties and disturbances. (M21 – M27)

• Development of a stochastic approach to the coordination of systems using SoS model subject to uncertainties and disturbances with arbitrary distributions, using both explicit and scenario-based approach. (M24 – M30)

• Development of methods for representation of the resulting SoS performance in stochastic sense, for merging with its neighboring systems. (M30 – M33)

2DYMASOS plenary meeting / Nice, March 14, 2016

Power distribution system

• Sources of uncertainty:

– Power demand prediction

– Uncontrollable generation prediction (e.g. wind, sun)

• Robust Optimal Power Flow (ROPF)

– Compute optimal power references such that constraints are

robustly satisfied for all possible realizations of uncertainty

• Objective

– E.g. nominal objective (total system losses)

• Constraints

– Must be satisfied for every realization of uncertainty


ROPF constraints (1)

• Power balance constraints

𝑃𝑖,𝑡𝐼 = 𝑃𝑖,𝑡

𝐺 − 𝑃𝑖,𝑡𝐷 𝑤𝑖,𝑡

𝑝=

𝑗∈𝑁 𝑖

𝑃𝑖𝑗,𝑡 , ∀𝑖 ∈ 𝒱, ∀𝑤𝑖,𝑡𝑝∈ 𝑊𝑖

𝑝

𝑄𝑖,𝑡𝐼 = 𝑄𝑖,𝑡

𝐺 − 𝑄𝑖,𝑡𝐷 𝑤𝑖,𝑡

𝑞=

𝑗∈𝑁 𝑖

𝑄𝑖𝑗,𝑡 , ∀𝑖 ∈ 𝒱, ∀𝑤𝑖,𝑡𝑞∈ 𝑊𝑖

𝑞

𝑃𝑖𝑗,𝑡 = 𝑔𝑖𝑗𝑉𝑖,𝑡2 − 𝑉𝑖,𝑡𝑉𝑗,𝑡 𝑔𝑖𝑗 cos 𝜃𝑖𝑗,𝑡 + 𝑏𝑖𝑗 sin 𝜃𝑖𝑗,𝑡

𝑄𝑖𝑗,𝑡 = −𝑏𝑖𝑗𝑉𝑖,𝑡2 + 𝑉𝑖,𝑡𝑉𝑗,𝑡 𝑏𝑖𝑗 cos 𝜃𝑖𝑗,𝑡 − 𝑔𝑖𝑗 sin 𝜃𝑖𝑗,𝑡


Constraints and target (2)

• Voltage constraints

𝑉 ≤ 𝑉𝑖,𝑡 ≤ 𝑉, ∀𝑖 ∈ 𝒱

• Current constraint

𝐼𝑖𝑗,𝑡2 = 𝛿𝑖𝑗,𝑡 𝑔𝑖𝑗

2 + 𝑏𝑖𝑗2 𝑉𝑖,𝑡

2 + 𝑉𝑗,𝑡2 − 2𝑉𝑖,𝑡𝑉𝑗,𝑡 cos 𝜃𝑖𝑗 ≤ 𝐼𝑖𝑗

2

• Generator (storage) constraints

𝑥𝑖,𝑡+1𝐺 = 𝐴𝑥𝑖,𝑡

𝐺 + 𝐵𝑢𝑖,𝑡𝐺

𝑦𝑖,𝑡𝐺 = 𝐶𝑥𝑖,𝑡

𝐺 + 𝐷𝑢𝑖,𝑡𝐺

𝑃𝑖𝐺 ≤ 𝑃𝑖,𝑡

𝐺 ≤ 𝑃𝑖𝐺

𝑄𝑖𝐺 ≤ 𝑄𝑖,𝑡

𝐺 ≤ 𝑄𝑖𝐺


Exponential growth of num. of constraints

• Let 𝑃𝑖𝐷 = 𝑃𝑖0

𝐷 + Δ𝑃𝑖𝐷, Δ𝑃𝑖

𝐷 ∈ Δ𝑃𝑖𝐷, Δ𝑃𝑖

𝐷

• Then 𝐏𝐃 = 𝑃1𝐷 𝑃2

𝐷 … 𝑃𝑛𝐷 𝑇 ∈ 𝕎𝑃

𝐷

– 𝕎𝑃𝐷 = co{𝐏𝟏

𝐃, 𝐏𝟐𝐃, … , 𝐏𝐦

𝐃}, where 𝐏𝒊𝐃 is one extreme realization of

demand uncertainty

• The same goes for reactive power demand 𝐐𝐃

• Constraints must be satisfied for every combination of

extreme realizations 𝐏𝐢𝐃 and 𝐐𝐢

𝐃!

• The number of constraints grows exponentially with

number of nodes in the network!


Tube MPC (1)

• System dynamics:

𝑥+ = 𝐴𝑥 + 𝐵𝑢 + 𝑤, 𝑥, 𝑢 ∈ 𝕏 × 𝕌, 𝑤 ∈ 𝕎

• Idea: Separate control into two parts

1. A portion that steers the nominal system to the origin

𝑧+ = 𝐴𝑧 + 𝐵𝑣

2. A portion that compensates for deviations from the nominal

system 𝑒+ = 𝐴 + 𝐵𝐾 𝑒 + 𝑤

• The linear feedback controller 𝐾 is fixed offline (such

that 𝐴 + 𝐵𝐾 is stable)

• Keep “real” trajectory close to the nominal

𝑢𝑖 = 𝐾 𝑥𝑖 − 𝑧𝑖 + 𝑣𝑖


Tube MPC (2)

• Bound maximum error (how far is the “real” trajectory from the nominal)– Compute minimum robust positive invariant set ℰ

– 𝑥𝑖 ∈ 𝑧𝑖 ⊕ℰ

• Compute tightened constraints on nominal system

– 𝑧𝑖 ∈ 𝕏⊖ ℰ

– 𝑣𝑖 ∈ 𝕌⊖𝐾ℰ

• Formulate as convex optimization problem– Optimize the nominal system with tightened constraints

– The cost is with respect to tube centers 𝑧𝑖– Choose terminal set and terminal objective function to ensure

robust feasibility and stability


Simple motivating example (1)

• One bus power system connected to the main grid

– Storage: 𝑥1 𝑘 + 1 = 𝑥1 𝑘 + 𝑢1 𝑘 , 𝑥1 𝑘 , 𝑢1 𝑘 ∈ 𝕏1 × 𝕌1

– Prosumer: 𝑥2 𝑘 + 1 = 𝑎2𝑥2 𝑘 + 𝑏2𝑢2 𝑘 , 𝑥2 𝑘 , 𝑢2 𝑘 ∈ 𝕏2 × 𝕌2

• Power balance constraint

– 𝑃𝑔𝑟𝑖𝑑 𝑘 = 𝑢1 𝑘 + 𝑥2 𝑘 + 𝑤 𝑘 , 𝑃𝑔𝑟𝑖𝑑 𝑘 ∈ ℙ, ∀𝑤 𝑘 ∈ 𝕎

• Trivial solution– Ignore the disturbance

– Main grid will cover every power imbalance (if ℙ is large enough)

– Uncertainty perceived by the main grid is 𝕎

• Can we do better?– Let the system itself compensate for the uncertainty locally as

much as possible

– Uncertainty perceived by the main grid is reduced



• Let 0 < 𝛽 < 1 be the part of disturbance compensated

by the main grid

– ത𝑃𝑔𝑟𝑖𝑑 𝑘 + 𝛽 𝑘 𝑤 𝑘 = 𝑢1 𝑘 + 𝑥2 𝑘 + 𝑤(𝑘)

– 𝑢1 𝑘 = ത𝑃𝑔𝑟𝑖𝑑 𝑘 − x2 k − 1 − 𝛽 𝑘 w k

– 𝑢1 𝑘 = ത𝑃𝑔𝑟𝑖𝑑 𝑘 − x2 k − 𝛼 𝑘 w k

𝑥1(𝑘 + 1)𝑥2(𝑘 + 1)

=1 −10 𝑎2

𝑥1(𝑘)𝑥2(𝑘)

+1 00 𝑏2

ത𝑃𝑔𝑟𝑖𝑑 𝑘𝑢2(𝑘)

+ 𝛼(𝑘)−10

𝑤(𝑘)

• Possible to use Tube MPC methodology

• Additional decision variable 𝛼(𝑘)



• With 𝛼(𝑘) we effectively scale the tube of trajectories!

• Computing the mRPI– Compute mRPI for 𝛼 = 1, i.e. ℛ ≔ σ𝑘=0

∞ 𝐴𝑐𝑘𝕎

– Then ℛ 𝛼 = 𝛼ℛ– For polytopic mRPI:

ℛ 𝛼 = 𝛼ℛ = 𝛼𝑒: 𝑒 ∈ ℛ = 𝛼𝑒: 𝐹𝑟𝑒 ≤ 𝑓𝑟 = { Ƹ𝑒 = 𝛼𝑒: 𝐹𝑟 Ƹ𝑒 ≤ 𝛼𝑓𝑟}

• Computing the tightened constraints

– ഥ𝕏 = 𝕏⊖ℛ(𝛼)– For polytopic 𝕏:

ഥ𝕏 = ҧ𝑥, 𝛼: 𝐹𝑥 ҧ𝑥 ≤ 𝑓𝑥 −max𝑟∈ℛ

𝐹𝑥𝛼𝑟 = ҧ𝑥, 𝛼: 𝐹𝑥 ҧ𝑥 ≤ 𝑓𝑥 − 𝛼max𝑟∈ℛ

𝐹𝑥𝑟

= ҧ𝑥, 𝛼: 𝐹𝑥 ҧ𝑥 ≤ 𝑓𝑥 − 𝛼 ℎℛ(𝐹𝑥)– Maximizations are all linear programs

– ℎℛ(𝐹𝑥) can be computed offline

– ഥ𝕏 is a set of linear constraints in ҧ𝑥(𝑘) and 𝛼(𝑘)

• Disturbance perceived by the main grid is 𝐾ℛ 𝛼 𝑘 ⊕ {(1 − 𝛼(𝑘))𝕎}


Numerical example (1)


Generalized microgrid

• „Microgrid” as one bus in the

overall distribution system

• „Microgrid” as one subgrid in the

overall distribution system

• „Microgrid” as the whole

distribution system

• Connected to the „main” grid or

in an islanded mode


Microgrid entities (1)

• 𝑀 generic entities

• Prosumer

– A device that can produce and/or consume electrical energy

• Uncontrollable prosumer

– Cannot be influenced by control signals

– 𝑝𝑘 = 𝑟𝑘 + 𝑤𝑘– 𝑟𝑘 is nominal power profile

– 𝑤𝑘 is uncertain but bounded disturbance whose value will be

discovered over the course of the time horizon

– e.g. random forecast error

– 𝑤𝑘 ∈ 𝕎 = {𝑤: 𝑤 ≤ 𝑤 ≤ 𝑤}


Microgrid entities (2)

• Controllable prosumer– LTI discrete-time dynamics

𝑥𝑖 𝑘 + 1 = 𝐴𝑖𝑥𝑖 𝑘 + 𝐵𝑖𝑢𝑖 𝑘𝑦𝑖 𝑘 = 𝐶𝑖𝑥𝑖(𝑘)

– In vector form:

𝒙𝒊 = መ𝐴𝑖𝑥𝑖 0 + 𝐵𝑖𝒖𝒊𝒚𝒊 = መ𝐶𝑖𝒙𝒊

መ𝐴𝑖 =

𝐴𝑖𝐴𝑖2

⋮𝐴𝑖𝑁

, 𝐵𝑖 =

𝐵𝑖 0 ⋯ 0𝐴𝑖𝐵𝑖 𝐵𝑖 ⋱ 0⋮ ⋱ ⋱ ⋮

𝐴𝑖𝑁−1𝐵𝑖 ⋯ 𝐴𝑖𝐵𝑖 𝐵𝑖

, መ𝐶𝑖 =

𝐶𝑖 0 ⋯ 00 𝐶𝑖 ⋱ 0⋮ ⋱ ⋱ ⋮0 ⋯ 0 𝐶𝑖

• Constraints– Polytopic state and input constraints: 𝕏𝑖 and 𝕌𝑖


Network model

• Linearized model of the power network– lossless lines

– constant voltage magnitudes

– line power flows proportional to the phase differences (assumed to be small) between nodal voltages

• Power balance constraint– Vector equality constraint (islanded mode)

𝒓 + 𝒘+

𝑖=0

𝑀

መ𝐶𝑖 𝒙𝒊 = 0, ∀𝑤 ∈ 𝕎

• Line constraints– Maximum allowed power flows through certain lines

– Linear inequalities in prosumer power injections


Input parametrization

• Disturbance feedback𝑢𝑖 𝑘 = 𝑣𝑖 𝑘 + 𝛼𝑖 𝑘 𝑤, 𝑤 ∈ 𝕎

– Nominal input 𝑣𝑖 𝑘 plus linear variation 𝛼𝑖 𝑘 with disturbance

𝑤

– Both 𝑣𝑖 𝑘 and 𝛼𝑖 𝑘 are decision variables

– 𝛼𝑖 𝑘 is a participation factor with which 𝑖-th prosumer

contributes to the compensation of disturbance

• Vector form𝒖𝒊 = 𝒗𝒊 + ො𝛼𝑖𝒘

ො𝛼𝑖 =

𝛼𝑖(0) 0 ⋯ 00 𝛼𝑖(1) ⋱ 0⋮ ⋱ ⋱ ⋮0 ⋯ 0 𝛼𝑖(𝑁 − 1)


Putting it all together (1)

• After input parametrization, each prosumer is

described by uncertain LTI discrete-time dynamics𝑥𝑖 𝑘 + 1 = 𝐴𝑖𝑥𝑖 𝑘 + 𝐵𝑖𝑣𝑖 𝑘 + 𝛼𝑖 𝑘 𝐵𝑖𝑤

– We can use Tube MPC framework

– Additional decision variables 𝛼𝑖(𝑘)

– We can „scale” the tubes of trajectories

– 𝑤 can be seen as an unforeseen imbalance between energy

production and consumption

• The nominal system is obtained by ignoring the

disturbanceҧ𝑥𝑖 𝑘 + 1 = 𝐴𝑖 ҧ𝑥𝑖 𝑘 + 𝐵𝑖 ҧ𝑣𝑖 𝑘

𝑣𝑖 𝑘 = ҧ𝑣𝑖 𝑘 + 𝐾 𝑥𝑖 𝑘 − ҧ𝑥𝑖 𝑘



• Error signal, i.e. the deviation of the real state

trajectory from the nominal one 𝑒𝑖 𝑘 = 𝑥𝑖 𝑘 − ҧ𝑥𝑖(𝑘)𝑒𝑖 𝑘 + 1 = 𝐴𝑐,𝑖𝑒𝑖 𝑘 + 𝛼𝑖 𝑘 𝐵𝑖𝑤

– 𝐴𝑐,𝑖 = 𝐴𝑖 + 𝐵𝑖𝐾𝑖, where 𝐾𝑖 is such that 𝐴𝑐,𝑖 is stable

• In vector form

– Nominal trajectory: ഥ𝒙𝒊 = መ𝐴𝑖 ҧ𝑥𝑖 0 + 𝐵𝑖ഥ𝒗𝒊

– Error signal trajectory: 𝒆𝒊 = መ𝐴𝑐,𝑖𝑒𝑖 0 + 𝐵𝑐,𝑖 ො𝛼𝑖𝒘

– 𝑒𝑖 0 = 𝑥𝑖 0 − ҧ𝑥𝑖 0

• Computing mRPI

– Compute mRPI for 𝛼𝑖 = 1, i.e. ℛ𝑖 ≔ ۩𝑘=0∞ 𝐴𝑐,𝑖

𝑘 𝐵𝑖𝕎

– ℛ𝑖 = {𝑒: 𝐹𝑟,𝑖𝑒 ≤ 𝑓𝑟,𝑖}



• Scaling mRPI for 𝛼𝑖 ≠ 1

– ℛ𝑖 𝛼𝑖 = 𝛼𝑖ℛ𝑖 = {𝑒: 𝐹𝑟,𝑖𝑒 ≤ 𝛼𝑖𝑓𝑟,𝑖}

– 𝛼𝑖 enters the inequality linearly

• Computing tightened state and input constraints

– ഥ𝕏𝑖 = { ҧ𝑥, 𝛼: 𝐹𝑥,𝑖 ҧ𝑥 ≤ 𝑓𝑥,𝑖 − 𝛼ℎℛ𝑖(𝐹𝑥,𝑖)}

– ഥ𝕌𝑖 = ҧ𝑣, 𝛼: 𝐹𝑢,𝑖 ҧ𝑣 ≤ 𝑓𝑢,𝑖 − 𝛼 ℎℛ𝑖 𝐹𝑢,𝑖 + ℎ𝕎 𝐹𝑢,𝑖

– Both are linear in 𝛼𝑖 𝑘

• Power balance constraint

𝒓 +

𝑖=0

𝑀

መ𝐶𝑖( መ𝐴𝑖𝑥𝑖 0 + 𝐵𝑖𝒗𝒊 + መ𝐴𝑐,𝑖(𝑥𝑖 0 − ҧ𝑥𝑖 0 )) = 0

𝐼𝑁 +

𝑖=0

𝑀

መ𝐶𝑖 𝐵𝑐,𝑖 ො𝛼𝑖 = 0



• Minimize cost function ത𝑉𝑁 ഥ𝒙, ഥ𝒗 ≔ σ𝑖=1𝑀 ത𝑉𝑁,𝑖 ഥ𝒙𝒊, ഥ𝒗𝒊

– ത𝑉𝑁,𝑖 ഥ𝒙𝒊, ഥ𝒗𝒊 ≔ σ𝑘=0𝑁−1 ℓ𝑖 ҧ𝑥𝑖 𝑘 , ҧ𝑣𝑖 𝑘 + 𝑉𝑓,𝑖( ҧ𝑥𝑖 𝑁 )

• Subject to

– ҧ𝑥𝑖 𝑘 + 1 = 𝐴𝑖 ҧ𝑥𝑖 𝑘 + 𝐵𝑖 ҧ𝑣𝑖 𝑘 , 𝑘 = 0,1,⋯ ,𝑁 − 1

– ҧ𝑥𝑖 𝑘 , 𝛼𝑖 𝑘 ∈ ഥ𝕏, 𝑘 = 0,1,⋯ ,𝑁 − 1

– ҧ𝑣𝑖 𝑘 , 𝛼𝑖 𝑘 ∈ ഥ𝕌, 𝑘 = 0,1,⋯ ,𝑁 − 1

– ҧ𝑥𝑖 𝑁 ∈ 𝕏𝑓,𝑖

– Power balance constraint

– ҧ𝑥𝑖 0 ∈ 𝑥𝑖 0 ⊕ℛ𝑖(𝛼𝑖(𝑘))



• 𝑀 = 12 prosumers

• 1 storage unit and 11 generic 2nd-order prosumers

• No line limits, e.g. one-bus network

• Prediction horizon length 𝑁 = 10

• Uncontrollable prosumers aggregated to one uncertain power injection– 𝑝 𝑘 = 𝑟 𝑘 + 𝑤(k)

– 𝑟 𝑘 = 1000(1 + 0.2 sin 𝑘)

– 𝑤 𝑘 ∈ 𝕎 = {𝑤:−20 ≤ 𝑤 ≤ 20}

• ℓ𝑖 ҧ𝑥𝑖 𝑘 , ҧ𝑣𝑖 𝑘 and 𝑉𝑓,𝑖( ҧ𝑥𝑖 𝑁 ) are quadratic functions

• 𝐾𝑖 is determined as an LQR with infinite horizon

• Closed-loop simulation with 100 steps

• Formulated with YALMIP and solved with CPLEX


Scenario-based approach (1)

• Taking into account every possible combination of

extreme uncertainty realization leads to an untractable

optimization problem

• Scenario approach: take into account only 𝑁uncertainty realization scenarios (instead of all possible

combinations of extreme uncertainty realizations)

• Constraint violation is tolerated, but with some

probability level 휀– E.g instead 𝑥 ∈ 𝕏 we have ℙ 𝑥 ∈ 𝕏 ≥ 1 − 휀

• Additionally, we can discard 𝑘 out of 𝑁 scenarios in

order to improve the objective value


Scenario-based approach (2)

• Theoretical guarantees that the solution obtained by inspecting 𝑁 scenarios only is a feasible solution for the chance-constrained optimization problem with high probability 1 − 𝛽, provided that 𝑁 and 𝑘 fulfill the following condition:

𝑘 + 𝑑 − 1𝑘

𝑖=0

𝑘+𝑑−1

𝑁𝑖

휀𝑖 1 − 휀 𝑁−𝑖 ≤ 𝛽

where 𝑑 is the number of optimization variables.

• Choose 𝑁 within the computational limit of the used solver, 휀 according to the acceptable level of risk, 𝛽 small enough to be negligible (e.g. 𝛽 = 10−10), and compute the largest 𝑘number of scenarios that can be discarded


Problem definition


Control goal


Control policy


Handling the probabilistic constraint (1)


Stochastic microgrid control


Acknowledgments

• This research has been supported by the European

Commission’s FP7-ICT project DYMASOS (contract no.

611281) and by the Croatian Science Foundation

(contract no. I-3473-2014).


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