Most applications discussed will be of the ‘decomposed’-type, as often there is only communication with a coordinator (TSO, DSO, retailer), not within a graph (‘networked’-type)
Distributed optimization • A distributed convex optimization problem:
• I.e. Two cars who want to charge as cheap as possible, respecting charger limitations AND without overloading a transformer • Can be solved by Dual decomposition: – ≈ price based: Iteratively update price (lagrange multiplier) depending on complicating constraint violation
• Can be solved by ADMM: – ≈ price based ADMM: add proximal regularization to price
Distributed optimization • A distributed convex optimization problem:
Complicating constraint
• I.e. Two cars who want to charge as cheap as possible, respecting charger limitations AND without overloading a transformer • Can be solved by Dual decomposition: – ≈ price based: Iteratively update price (lagrange multiplier) depending on complicating constraint violation
• Can be solved by ADMM, PCPM, … : – ≈ price based ADMM: add proximal regularization to price
Easy Example of Dual decomposition # customers
min xi
∑
−U i ( xi )
i =1
subj to
∑x
i
≤ Pmax
Easy Example of Dual decomposition # customers
min xi
∑ i =1
subj to Complicating constraint
−U i ( xi )
∑x
i
≤ Pmax
Easy Example of Dual decomposition Update price dependent on level of complicating constraint violation
Easy Example of Dual decomposition
Easy Example of Dual decomposition
Easy Example of Dual decomposition
Easy Example of Dual decomposition Not actual oscillations: Only when convergence vehicles will adapt their power, at other time instances this is only an information signal
Distributed optimization: application • Real time pricing with network constraints: – Sell energy, without going out of voltage limits
Customer
DSO price update
Energy provider price update
Distributed optimization: application • Real time pricing with network constraints: – Sell energy, without going out of voltage limits
Customer
DSO price update
Energy provider price update
Complicating constraint
Distributed optimization: application • Optimal Primary frequency support: – Restore power balance, guaranteeing maximal social welfare # customers
min xi
subj to
∑
−U i ( xi )
i =1
∑x +P i
baseload
= Pgenerator
– Frequency variation is a measure of power imbalance – Frequency variation is ‘universal communication signal’
Distributed optimization: application • Optimal Primary frequency support: – Restore power balance, guaranteeing maximal social welfare # customers
min xi
subj to
∑
−U i ( xi )
i =1
∑ xi + Pbaseload = Pgenerator
Complicating constraint violation ≈ ∆f
– Frequency variation is a measure of power imbalance – Frequency variation is ‘universal communication signal’
Optimal Primary frequency support: • Frequency price = running sum frequency deviation • = Integration of frequency deviation = Frequency add price component proportional with frequency
Type of problems Strict convex
Convex Nonconvex Integer
• Convergence depends on problem structure – Dual decomposition only for strict convex – ADMM makes problem locally strict convex • Some heuristic tricks for integer problems
Future research • • • • • •
Multi-stage stochastic optimization? Robust optimization? Risk minimization? New applications for distributed optimization? Thermal applications? Something different?: – Data mining?, forecasting? – Control theory?
Multi-stage stochastic optimization? • Stochastic MPC – i.e. charging of vehicles with uncertain prices 2012 Bemporad et al
• Risk minimization • Conditional value at risk minimization for example on grid constraints
Robust optimization • Classic robust optimization – i.e. Guarantee grid constraints when charging cars..
• Affine adjustable multistage optimization – Restrict to affine policies in multi-stage optimization – Affine recourse MPC (what is gain compared to certainty-equivalent MPC)
Thank you for your attention
Distributed optimization • What about the math? • Primal problem
• Each optimization problem has a ‘dual’ • If strong duality holds: – Solution of primal = solution of dual
• Dual decompostion is solving ‘the dual’
Distributed optimization • Lagrange dual function • Dual problem • Dual decomposition is (sub)gradient ascent method • Level of constrained violation = (sub)gradient of the dual
Distributed optimization • Lagrange dual function Separable
• Dual problem • Dual decomposition is (sub)gradient ascent method • Level of constrained violation = (sub)gradient of the dual
Distributed optimization • How to update the price? – Fixed step size, diminishing step size, polyak step size, …
• A lot of slight adaptations: – I.e. Randomized incremental (sub)gradient updates: • Send new price to a random selected customer and reupdate the price
– ADMM: add proximal regularization to price • Customers respond slowly to price changes
