A networked control strategy for reactive power compensation in a - PowerPoint PPT Presentation

A networked control strategy for reactive power compensation in a smart power distribution network Saverio Bolognani Information and Control in Networks Focus Period at LCCC, Lund University October 26, 2012

REACTIVE POWER COMPENSATION

Power distribution networks

Smart power distribution grid Smart microgenerators We consider a portion of the electrical power distribution network populated by a number of microgeneration devices (solar panels, ...), each of them equipped with sensing and communication capabilities. The power electronics of these microgenerators can be exploited for providing useful ancillary services. We focus on the problem of optimal reactive power compensation for the minimization of distribution losses.

Power distribution grid We assume that voltages and currents are sinusoidal signals, at the same frequency, and thus described by their amplitude and phase. ξ e Electric network z e u v i v

Reactive power Reactive power flows Whenever a device in the grid injects (is supplied with) a current that is out of phase with the voltage, we have injection (delivery) of reactive power. Adopting the phasorial notation for voltages + and currents, we define the complex power i v ↑ u v s v = p v + jq v := u v ¯ − i v

Reactive power “facts” ◮ Loads in the microgrid require reactive power ◮ reactive power can be obtained from the transmission grid or produced by the microgenerators in the grid ◮ producing reactive power has no fuel cost ◮ larger flows of reactive power correspond to quadratically larger power losses on the cables. Optimal reactive power compensation problem Injecting reactive power in the grid as close as possible to the loads that need it, in order to minimize power distribution losses.

MICROGRID MODEL

Graph model ξ e z e u v i v Nodes of the graph represent loads (in white) that cannot be con- trolled, and microgenerators (in black) which can be commanded, can sense the grid, and can communicate. Nodes are connected by a tree T , representing the electrical con- nection (power lines) among them.

Graph model ξ e Node 0 z e + i 0 → u 0 u v − i v Node 0 represents the point of connection of the microgrid to the transmission grid. Its voltage u 0 corresponds in amplitude to the nominal voltage U N of the microgrid: u 0 = U N e j φ 0 .

Graph model Nodes v � = 0 ξ e z e + i v ↑ u v u v − i v Node voltage u v and node current i v satisfy u v ¯ i v = s v for microgenerators and loads (can be extended to exponential / ZIP model).

Graph model ξ e Edges e u τ ( e ) u σ ( e ) z e u v → ξ e i v Voltage drop u τ ( e ) − u σ ( e ) and the current ξ e flowing on the edge e satisfy u τ ( e ) − u σ ( e ) = z e ξ e where z e is the impedance of the power line e .

Microgrid nonlinear equations The voltages u v and the currents i v of the microgrid are therefore implicitely defined by the system of nonlinear equations  Lu = i    u v ¯ i v = s v v � = 0  u 0 = U N e j φ 0 ,   where L is the weighted Laplacian of the graph L = A T Z − 1 A and A is the incidence matrix of the graph.

OPTIMIZATION PROBLEM

Optimization problem The optimization problem consists in deciding the reactive power injection at the microgenerators that minimizes power distribution losses. Decision variables q v , v ∈ C Losses Grid state Microgrid ¯ i T ℜ ( X ) i u , i Cost nonlinear function Problem parameters equations s v , v ∈ U p v , v ∈ C U N , φ 0 , L In order to design an algorithm we need an explicit expression for the grid state as a function of the decision variables.

Explicit grid solution Approximate solution We constructed the Taylor expansion of the system state for large nominal voltage U N . i v ( U N ) = ¯ s v + c v ( U N ) U 2 U N N u v ( U N ) = U N + [ X ¯ s ] v + d v ( U N ) . U 2 U N N This model extends the DC power flow model, by relaxing the assumption of zero losses (i.e. purely inductive lines).

Approximate problem The approximate solution of the grid equations allows us to rewrite the cost function (losses) as a quadratic function of the decision variables. 1 p T ℜ ( X ) p + 1 q T ℜ ( X ) q + 1 ˜ J = J ( p , q , U N ) U 2 U 2 U 3 N N N where ˜ J is bounded for large U N , and q satisfies 1 T q = 0. Quadratic cost function We approximated the original problem as a convex quadratic optimization problem subject to a linear equality constraint.

DISTRIBUTED ALGORITHM

Motivation for a distributed algorithm Implementing a centralized solver for the quadratic (linearly constrained) optimization problem is impossible: ◮ complete knowledge of the system parameters { p v , v ∈ C} , { s v , v ∈ U} L , and state { q v , v ∈ C} is required ◮ coordination and communication among all nodes U ∪ C is required ◮ compensators ◮ are in large number ◮ can connect and disconnect ◮ have limited communication capabilities.

Distributed architecture C j Consider the family of subsets of C C i C k {C 1 , . . . , C ℓ } Control such that � ℓ i = 1 C i = C . Let each cluster be managed by an intelligent unit (possibly, one of the compensators), which Graph model 0 σ ( e ) τ ( e ) ◮ knows the relative position of e the compensators v ◮ collects data from the ξ e compensators Electric network ◮ processes the collected data z e ◮ commands the compensators. u v i v

Iterative algorithm At (possibly uneven) time step 1) a cluster C i activates; 2) the supervisor of C i determines the optimal update step that minimizes the global cost function; 3) compensators in C i actuate the system by updating their state q v , v ∈ C i , while other compensators keep their state constant. ✬✩ C j C i C k ✫✪

Iterative algorithm At (possibly uneven) time step 1) a cluster C i activates; 2) the supervisor of C i determines the optimal update step that minimizes the global cost function; 3) compensators in C i actuate the system by updating their state q v , v ∈ C i , while other compensators keep their state constant. ✬✩ C j ✫✪ C i C k

Iterative algorithm At (possibly uneven) time step 1) a cluster C i activates; 2) the supervisor of C i determines the optimal update step that minimizes the global cost function; 3) compensators in C i actuate the system by updating their state q v , v ∈ C i , while other compensators keep their state constant. ✬✩ C j C i C k ✫✪

Computation of the optimal step for C i The optimal update that has to be performed by cluster C i is given by the (constrained) Newton step: q opt, i  = q h for each h �∈ C i ,  h  q opt, i Γ ( i ) � = q h − hk ∇ for each h ∈ C i , J k h   k ∈C i where ◮ Γ ( i ) is function of the Hessian ℜ ( X ) , ◮ ∇ J is the gradient. In general, these are global quantities. However, according to the approximate model for the power system state, both Γ ( i ) hk and ∇ J k can be obtained from local data.

Computation of the optimal step for C i Hessian estimation Γ ( i ) is a function of the electric distances (mutual effective impendances) between the microgenerators belonging to the cluster C i . Gradient estimation ∇ J k , k ∈ C i , can be estimated from voltage measurements performed by the microgenerators that belong to C i . To solve the subproblem faced by the supervisor of the cluster C i , only parameters and measurements from the microgenerators belonging to C i are needed.

Resulting algorithm We therefore obtained the following distributed control algorithm. Offline initialization Each supervisor computes Γ ( i ) according to the electric distance among compensators in the cluster. Online iterative algorithm 1. a cluster C i activates; 2. agents not in C i hold their state constant; 3. agents in C i 3.1 measure their voltage and estimate ∇ J ; 3.2 compute the optimal update step − Γ ( i ) ∇ J ; 3.3 update their state;

Resulting feedback law ∇ J ( i ) 2 cos θ (Ω r R eff Ω r ) ♯ − Γ ( i ) ∇ J ( i ) K r ( u C ) Discrete time 1 integrator z − 1 q C u C Power distribution network u V\C q V\C p

Remark Feedback signals over the physical system Other applications of distributed optimization share this same feature (radio power control, congestion avoidance protocols in data networks). In these applications, the iterative tuning of the decision variables associated to each agent (radio power, transmission rate) depends on congestion indices that are function of the entire state of the system. However, these indices can be detected locally by each agent by measuring some feedback signals: error rates, delays, signal-to-noise ratios, etc.

Radio power control p i ≥ SIR min , SIR i = ∀ i j � = i p j w ij + n i G Ai � Distributed radio power control algorithms consist in update laws for the transmitting power p i in the form p + i = f i ( SIR j , j ∈ N i ∪ { i } ) .

Data network congestion avoidance protocol � max U r ( x r ) subject to Ax ≤ C x > 0 r Congestion on a route r depends on the transmission rates of all routes which share a link with r , and which are generally unknown. Typical protocols adjust the rate x r as a function of a feedback signal (e.g. delay, packet losses).