is used to model actuators operational constraints such as limits on the maximum apparent power for the inverter of a BESS. The term Edi models the perturbation of external quantities (outside
temperature, solar radiation, etc.) on the state dynamics. Finally, as it is quite common [107, 121], the quantities η, ξ ∈ Rm are conversion factors that translate the input to the active and reactive power injection at the resource (expressed in MW, Mvar).
With these definitions, we can the introduce the set of admissible input trajectories, which comply with the system dynamics and constraints, over the considered prediction horizon, N :
U (ˆx, ˆd) = u xi+1= Axi+ Bui+ E ˆdi yi = Cxi si = [η ξ]T ui ui ∈ U, yi∈ Y, si ∈ S x0 = ˆx ∀i = 1, . . . , N (4.2)
In the definition of the set U , both the initial state,x, as well as the vector of external perturbations,ˆ d, along the prediction horizon, are considered as parameters and are, therefore, fixed. Regarding the initial state we assume it to be either directly measurable or that it can be estimated by means of a global observer relying on state estimation techniques (e.g. Discrete Kalman filter, least square estimation, etc.). For d we assume the availability of reliable predictions, denoted as ˆd, that can be obtained, e.g., from external entities (weather stations).
Given that there might be more than one controllable resource in the aggregator portfolio, we further introduce the notation, uj, to denote the input trajectories for the j-th resource together
with the corresponding admissible set, Uj(ˆxj, ˆdj).
We refer the reader to Section 4.7 for an example of the model for the case of a BESS.
4.4
Power Network Models and Constraints
We consider a balanced distribution network with n buses with only one slack bus, with index 0, which is assumed to be the Point of Common Coupling (PCC) with the upstream network. Thus, all other buses are indexed sequentially starting from the PCC. The network can be modelled as a linear circuit represented by a connected weighted graph O(V, E, W) where V = {0, · · · , n − 1} is the set of vertices (buses) and E ⊆ V × V is the set of edges (branches).
Letzht= rht+jxht∈ C be the impedance between bus h and t, where rht∈ R>0is the resistance and
xht ∈ R>0 is the inductive reactance. The edge weights of the associated graph are the associated
and bht = −xht/(r2ht+ x2ht) ∈ R<0 the susceptance. The network is represented by symmetric
admittance matrix Y ∈ Cn×n, where the off-diagonal elements are given by Yht = −yht for each
branch {h, t} ∈ E (0 if h, t /∈ E), and the diagonal elements are given by ysh
h +
Pn−1
t=0,t6=hyht, where
ysh
h is the shunt admittance at bus h. We represent Y = G + jB, where G and B respectively are
the conductance and susceptance matrices. It should be noted thatGhh= gshh +Pn−1t=0,t6=hght > 0
and Ght= −ght< 0.
Remark 6. For the sake of simplicity, we assume in the following that no voltage regulating trans- formers are present in the branches. Under this assumption, the matrices G and B are symmetric. Moreover, G is a Laplacian matrix (loopy Laplacian if the shunt conductances are not equal to zero) [32]. We highlight however that such an assumption is not strictly necessary for the control framework proposed in the following.
As we are interested in the scheduling problem, we consider the network to be in steady state and operating in perfect sinusoidal conditions. Thus, to each node h ∈ V, we can associate a phasor voltageϑh := vheθh
∈ C, with vh, θh ∈ R and complex apparent power sh := ph+ jqh ∈ C, with
ph, qh ∈ R. Considering the aforementioned notation, it is possible to concisely write the AC power
flow equations relating voltages and power injections. The power flow equations descend directly from Kirchhoff’ s and Ohm’ s laws [70] as
s = diag(ϑ) ◦ (Y ϑ). (4.3)
where ◦ denotes element-wise vector multiplication, and the overline notation denotes complex conjugation. By defining the grid state
z := [v, θ, p, q]T ∈ R4n
one can rewrite the PF equation (4.3) implicitly as F(z) = 0 where the map is defined as F: R4n→
R2n and it is obtained by rearranging the power flow equation after expressing it in rectangular or polar coordinates.
4.4.1 Partition of the grid state
In the following section, we consolidate the notation that will be used in the remainder of the manuscript. In particular, we provide herein a partition of the grid state,z, that will be helpful for later derivations.
Let us define by zcont ∈ Rnc the components ofz that are directly controllable2 by the aggregator, and by zw ∈ Rnw the uncontrollable and uncertain components of the grid state. Together, these two sets of components form the vector of exogenous variables. We define by
2
Without loss of generality, we are implicitly assuming that the controllable resources do not have errors in the deployed control setpoints. In fact, if this is not true, the uncertainty related to the discrepancy between the setpoints and the implemented power injections could be treated as all other sources of uncertainty.
4.4 Power Network Models and Constraints 57 zex:= " zcont zw # ∈ R2n
the vector formed by stacking all the exogenous variables at each time instant.
The remaining2n components are, in turn, dependent variables which can be derived, through the power flow map, F(z), from the exogenous components. For this reason, we define by zend ∈ R2n
the vector containing all dependent variables to which we refer to as endogenous variables.
Finally, by utilizing the newly introduced notation, we can rewrite the set of power system states that satisfy equation (4.3) as:
M := {(zex, zend) | F(zex, zend) = 0} (4.4)
which, as it was shown in [16], describes a 2n-dimensional smooth manifold in R4n to which the grid state, z, belongs at each time instant, i.
Remark 7. For the sake of simplicity and clarity of explanation, in the remainder of the chapter, we make the assumption that every bus is a PQ bus. Under this assumption, the state partition is such that for every node,h, the exogenous variables are represented by active, ph and reactive, qh power
injections that can be either controllable or uncertain depending on the particular configuration. In turn, the endogenous variables are represented by the voltage magnitude, vh and angle, θh.
We emphasize that all the results presented hereafter can be easily extended to accommodate other configurations.
Remark 8. In what follows, we will assume the voltage at the slack bus to be fixed to a known constant level. Therefore, we are neglecting the possibility, often used in distribution networks to perform voltage control, to modulate the slack voltage by means of On Load Tap Changers (OLTC). This is done principally for two reasons 1) we focus on a scenario in which the network experiences overvoltages at very specific locations which motivated the deployment (or retrofitting) of DERs, 2) optimizing over the transformer tap’s positions would result in a mixed integer problem which is difficult to handle in an uncertain setting like the one considered in this chapter. We highlight, however, that the method detailed in the following sections could be extended to account for OLTC as well by considering, for instance, the tap’s positions as pseudo-continuous variables [27].
4.4.2 Grid constraints
Equation (4.4) enforces the grid state, z to lay on the PF manifold, M, which describes the physical relation between voltages and apparent power. In a multi-OPF setting, this map needs to be satisfied at each time instant, i. To compactly write this, we denote the PF constraints along the prediction horizon,N , as:
F(zex, zend) := F1(zex0 , z0end) .. . FN(zNex, zNend) = 0 (4.5)
which is obtained by simply stacking all the constraints of the form (4.4) along the horizon. A similar definition is assigned to the variables,zex and zend.
We assume that the network is subject to additional constraints which define the safe region of operation for the system. Thus, at each bus, h, and for each time instant, i, we impose:
v ≤ vhi ≤ ¯v ihti ≤ ¯Iht
where v, v, denote upper and lower limits on the voltage magnitudes, i¯ ht
i the magnitude of the
current flowing in the branch fromh to t, and ¯Iht the current flow limit for the branch.
Finally, by condensing all the constraints along the horizon, we can define the feasible region for the endogenous variables:
Z := zend v ≤ vhi ≤ ¯v h = {1, . . . , n} iht i ≤ ¯Iht ∀(h, t) ∈ E ∀i = {1, . . . , N } (4.6)
4.5
Problem Formulation
At the time of planning, the most important objective for the aggregator is to allocate enough controllable power so as to guarantee the satisfaction of all the physical constraints of the network. This should be ensured against all load’s and generation’s forecasts, and even in presence of distur- bances forcing the system to deviate from its nominal (or predicted) mode of operation. According to the previously defined notation, we capture this source of uncertainty with the termzw, which,
represents the uncontrollable and uncertain components of zex. In particular, we assume zw to
belong to a compact uncertainty set Zw ⊆ RN ·nw.
Concurrently, the aggregator aims at maximizing its economic return by offering additional ancillary services to the TSO in order to fully exploit all the available controllable power. To this aim, the aggregator is required to compute an appropriate schedule for the controllable resources, in order to ensure a satisfactory level of tracking of the signal, r, received by the TSO. In particular, the tracking should be ensured for all values of the uncertainty in a set R ⊆ RN.