Distributed quasi-newton method and its application to the optimal reactive power flow problem

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1 1 Distributed quasi-newton method and its application to the optimal reactive power flow problem Saverio Bolognani

2 Power distribution networks 2 Transmission Grid Distribution Grid up to ~150 MW Rural Network ~5 MW substations Low Voltage (50 kv) City Network ~2 MW City Power Plant ~400 kw Farm Industrial Customers Extra High High Medium Low Voltage Wind Farm Solar Farm

3 Reactive power compensators How to Own a Power Pla Innovative heating systems not only provide warmth but also satisfy of the electricity demand of an average four-person household. 3 Distributed Generation The electronic interface of every micro generator can act as a compensator: micro-hydroelectric, combined heat and power, wind, solar, waste thermal generation. Stirling engine Generator Burner Cold water Demand for resource-saving heat generation systems is growing. One driver of this development is the fact that well-insulated new buildings and renovated older structures have lower heating demand. In addition, high energy prices as well as insecurity on the part of consumers regarding the reliability of gas and oil supplies are also prompting researchers and developers to consider new heating methods. One such method is the simultaneous generation of heat and electricity by so-called CHP (combined heat and power) systems. These are among the most efficient methods of energy generation, because the fuel they use is transformed into electrical energy as well as heat usually in the form of steam and hot water. More than 90 percent of the energy contained in fuel can be utilized by these systems, compared with only about 38 percent for electrical generation by a conventional power plant. This high thermodynamic efficiency can make a major contribution to operating economy as well as environmental protection. Simultaneously, emissions of carbon dioxide and nitro- Households will soon be able to generate their own heat and electricity using a mini CHP device (left and above). Scientists are now fine tuning the technology. the distribution network is partially unknown and unmonitored these agents can connect and disconnect because of the stochastic character of the energy sources and the large number of DG units, a centralized dispatchment is too complex security of the energy supply may be jeopardized if a great amount of data is handled online by a single control center. plying it to single and multi-family homes is new; many manufacturers are already excited about exploiting this potential. Siemens Building Technologies (BT), for instance, has developed the electronics for a gas-fired micro heat and power cogeneration device (microchp). We see a clear line of development toward the use of personal small power plants in singlefamily homes in place of oil or gas-fired boilers, says Georges Van Puyenbroeck, director of sales and marketing at BT s OEM Boiler & Burner Equipment. With this goal in mind, BT specialists are working together with manufacturers of condensing boilers, including Viessmann, Vaillant, Remeha B.V., and the Baxi Group. How to Generate a Kilowatt. Until now, condensing boilers have produced only heat, but mounted boiler between the co is used to gene mentations per mum of one k which about 90 the home or fed grid. The device For consume their disposal th eration power p heat but also tw son household remaining elect grid to which th connected. Ope gas (LPG) is a readjustment of Siemens elec to keep the Stir ble operating r temperatures fo at the proper tim monitor the fee into the power u

4 Simplified model 4 Consider a tree describing the low-mid voltage distribution network. 1 f j j q j f i i q i user compensator N q i is the injected reactive power, f i is the reactive power flow.

5 Optimization problem 5 The optimization problem of having minimal power losses on the network corresponds to having minimal reactive power flows subject to i C U q i = 0 f i = f i (q 1,..., q N ) or in matricial form min F (f 2,..., f N ) = N i=2 f 2 i k i. 1 constraint - reactive power conservation N 1 constraints - power flow equations min subject to f T K 2 f f = Aq + B q 1 T N C q + 1 T N U q = 0

6 Optimization problem 6 By eliminating the second constraint, one obtains the quadratic problem min J(q) = q T M 2 q + qt m subject to 1 T N C q = c. which has the closed form solution ( ) c + 1 T q = M 1 N C M 1 m m 1 T N U M 1 1 NC 1 NC

7 Distributing the problem 7 Why this problem can still be interesting? unknown hessian M (depends on the topology) unknown constant c (depends on the demands) unknown vector m (depends on the demands).

8 Distributing the problem 7 Why this problem can still be interesting? unknown hessian M (depends on the topology) unknown constant c (depends on the demands) unknown vector m (depends on the demands). Gradient driven optimization algorithms Most of the algorithms for the solution of convex optimization problems are driven by the gradient, and assume that the gradient is available. q(t + 1) = q(t) Γg(q(t))

9 Distributed gradient estimation 8 The gradient can be rewritten as g = A T KAq + A T KB q = Mq + m = A T Kf = i E P i k i f i, g i g j = δ l (i, j)k l f l v i v j. l P ij j i The gradient can then be estimated element-wise and up to a constant from the steady state of the system: g i = v i + ξ.

10 Distributing the problem 9 q(t + 1) = q(t) Γg If a communication constraint is enforced via a graph G, then Γ cannot be a generic gain matrix. Sparse Γ The simplest approach consists in enforcing sparsity of Γ so that it is consistent with G. However, a sparse Γ is unlikely to solve the problem efficiently, because the global constraint couples the agents states non-separable cost functions couple the agents optimal choice nobody knows the whole system and can design Γ

11 Example: Newton descent 10 If the network topology is fully known and communication constraints are relaxed, it is possible to implement a constrained Newton algorithm that guarantees 1-step convergence: q(t + 1) = q(t) Γg = q(t) M 1 g + 1T M 1 g 1 T M 1 1 M 1 1 Γ = M 1 M 1 11 T M 1 1 T M 1 1

12 Example: Newton descent 10 If the network topology is fully known and communication constraints are relaxed, it is possible to implement a constrained Newton algorithm that guarantees 1-step convergence: q(t + 1) = q(t) Γg = q(t) M 1 g + 1T M 1 g 1 T M 1 1 M 1 1 Γ = M 1 M 1 11 T M 1 1 T M 1 1 Or, if an approximation for M 1 is available, one can implement an approximate Newton step q(t + 1) = q(t) Hg + 1T Hg 1 T H1 H1

13 Decomposition: descent + projection 11 In the approximate Newton descent step it is easy to recognize two parts q(t + 1) = q(t) Hg + 1T Hg 1 T H1 H1

14 Decomposition: descent + projection 11 In the approximate Newton descent step it is easy to recognize two parts q(t + 1) = q(t) Hg + 1T Hg 1 T H1 H1 an unconstrained descent step (requires knowledge of the system)

15 Decomposition: descent + projection 11 In the approximate Newton descent step it is easy to recognize two parts q(t + 1) = q(t) Hg + 1T Hg 1 T H1 H1 an unconstrained descent step (requires knowledge of the system) a projection step (requires knowledge of the others choice)

16 Sparse approximations + consensus 12 q(t + 1) = q(t) Hg+ 1T Hg 1 T H1 H1

17 Sparse approximations + consensus 12 Sparse H q(t + 1) = q(t) Hg+ 1T Hg 1 T H1 H1 By choosing a sparse approximation H for M 1, the computation of Hg and H1 depends only on neighbors data.

18 Sparse approximations + consensus 12 q(t + 1) = q(t) Hg+ 1T Hg 1 T H1 H1 Sparse H By choosing a sparse approximation H for M 1, the computation of Hg and H1 depends only on neighbors data. Consensus algorithm By running average consensus algorithms on the vectors [H i g H i 1] T, nodes agree on the projection step.

19 Sparse approximations + consensus 12 q(t + 1) = q(t) Hg+ 1T Hg 1 T H1 H1 Sparse H By choosing a sparse approximation H for M 1, the computation of Hg and H1 depends only on neighbors data. Consensus algorithm By running average consensus algorithms on the vectors [H i g H i 1] T, nodes agree on the projection step. This approach enables a whole class of methods in the form q i (t + 1) = q i (t) γ i (q j, g j (q), j N i ; η i, x)

20 Quasi-Newton method 13 Quasi-Newton methods In these methods an estimate of the hessian s inverse is updated at every step so that it satisfies the secant condition H(t + 1) g(t) = q(t) (where H is the estimate of the inverse of the hessian, and d is the projection of the gradient of the constraint) it minimizes H(t + 1) H(t).

21 Quasi-Newton method 13 Quasi-Newton methods In these methods an estimate of the hessian s inverse is updated at every step so that it satisfies the secant condition H(t + 1) g(t) = q(t) (where H is the estimate of the inverse of the hessian, and d is the projection of the gradient of the constraint) it minimizes H(t + 1) H(t). A quasi-newton method (Broyden s method) can be applied to our constrained optimization problem.

22 Quasi-Newton method 14 where d = Ωg. q(t + 1) = q(t) Gd(t) G(t + 1) = G(t) + [ q G d] dt d T d

23 Quasi-Newton method 14 q(t + 1) = q(t) Gd(t) G(t + 1) = G(t) + where d = Ωg. Update equation for the single node: [ q G d] dt d T d q i (t + 1) = q i (t) G i d(t) G i (t + 1) = G i (t) + [ q i G i d] d T d T d.

24 Quasi-Newton method 14 q(t + 1) = q(t) Gd(t) G(t + 1) = G(t) + [ q G d] dt d T d where d = Ωg. Update equation for the single node: q i (t + 1) = q i (t) G i d(t) G i (t + 1) = G i (t) + [ q i G i d] d T d T d. Finite time convergence We proved that this method converges in at most 2N steps.

25 Distributed quasi-newton method 15 Communication constraints Suppose that communication constraints are now enforced: the update equation must keep the estimate H sparse. H(t + 1) = H(t) + P E [D + ( q H g) g T ], where and { Aij if (i, j) E (P E (A)) ij = 0 otherwise ( D + ) ij = 1/g (i)t g (i) if g (i) 0 0 if g (i) = 0.

26 Distributed quasi-newton method 16 Let s complete the algorithm by introducing the projection step: obtaining q(t + 1) = q(t) H(t)g(t) + 1T H(t)g(t) 1 T H(t)1 H1 Distributed quasi-newton method q i (t + 1) = q i (t) H i (t) T g (i) (t) + xh i (t) T 1 (i) H i (t + 1) = H i (t) + [ ] q i H i (t) T g (i) g (i) (t) (t) g (i) (t) T g (i) (t) where x = z 1 / z 2, z being the result of consensus algorithm on z (i) (0) = [ ] Hi (t) T g (i) (t) H i (t) T 1 (i).

27 Numerical simulations M H complete H sparse sparsity constraint

28 Numerical simulations 18 cost function F(q) iteration Newton (solid), quasi-newton (dashed), distribute quasi-newton (dot-dashed), steepest descent (dotted).

29 19 Bolognani, S., and Zampieri, S. (2010). Distributed Quasi-Netwon Method and its Application to the Optimal Reactive Power Flow Problem. In Proceedings of NECSYS 2010,, France. Thanks! Saverio Bolognani Department of Information Engineering University of Padova (Italy) sbologna This work is licensed under the Creative Commons BY-NC-SA 2.5 Italy License