Problem formulation

Consider a power network defined by an undirected connected graph on n nodes (or buses)

N ={1,2, . . . , n}. For two nodes k and l inN, let k ∼l denote that k is connected to l in G by a transmission line with admittanceykl.

Time is discrete and is indexed by t. Akin to Chapter 3 Section 3.5 and Chapter 4, we define the following notation.

• pD

k(t) +iqDk(t) is the apparent power demand at bus k ∈ N and time t, which are

assumed to be known. Demand profiles often show diurnal variations [196], i.e., they exhibit cyclic behavior with each day being the time period of the cycle. Let T time- steps denote the cycle length of the variation. In particular, for all k ∈ N, t ≥ 0, assume

pD_k(t+T) +iqD_k(t+T) = pD_k(t) +iq_kD(t).

• pG

k(t)+iqkG(t) is the apparent power generation at busk ∈ N and timet. These decision

as pG k ≤p G k(t)≤pGk, and qG_k ≤qkG(t)≤qGk. (6.1) • ck pGk

denotes the cost of generating power pG_k at bus k∈ N. The cost of generation is assumed to be independent of timetand depends only on the generation technology at bus k. Also, suppose that the function ck : R+ → R+ is non-decreasing and

convex. These assumptions apply to commonly used cost functions in the literature [8, 12, 36, 192], e.g., convex and nondecreasing piecewise linear or quadratic ones.

• Vk(t) be the complex voltage at bus k ∈ N and time t. Voltage magnitudes at nodes

are bounded as

V_k ≤ |Vk(t)| ≤Vk. (6.2)

• Fork ∼l in G, pkl(t) +iqkl(t) be the apparent power flow from bus k to busl at time

t which satisfies

pkl(t) +iqkl(t) =Vk(t) (Vk(t)−Vl(t)) H

y_klH, (6.3)

pkl(t) is constrained by capacity limit fkl. Thus we have

|pkl(t)| ≤fkl. (6.4)

Note that in this study, we chose to constrain the real power flow pkl as opposed to

the apparent power flow on the line joining busesk and l.

• γk(t) and δk(t) are the average charging and discharging powers of the storage unit at

bus k∈ N at timet, respectively. The energy transacted over a time-step is converted to power units by dividing it by the length of the time-step. This transformation conveniently allows us to formulate the problem in units of power [137]. Let 0 <

αγ, αδ ≤1 denote the charging and discharging efficiencies, respectively of the storage

technology used, i.e., the power flowing in and out of the storage device at nodek ∈ N

at timetisαγγk(t) and _α1

δδk(t), respectively [189,197]. The roundtrip efficiency of this

storage technology is α =αγαδ ≤1. Note that we assume that the storage units only

transact in real power.

• sk(t) denotes the storage level at node k ∈ N at time t and s0k is the storage level at

nodek at timet = 0. From the definitions above, we have that

sk(t) =s0k+ t X τ=1 αγγk(τ)− 1 αδ δk(τ) . (6.5)

For each k ∈ N, assume s0

k = 0, so that the storage units are empty at installation

time.

• bk ≥0 is the storage capacity at bus k. Thus, sk(t) for all t satisfies the following:

0≤sk(t)≤bk. (6.6)

• his the available storage budget and denotes the total amount of storage capacity that can be installed in the network. Our optimization algorithm decides the allocation of storage capacity bk at each nodek ∈ N and thus, we have

X

k∈N

bk≤h. (6.7)

• Charging and discharging rates of each storage device are assumed to be upper-bounded by ramp limits. These limits are proportional to the capacity of the corresponding device, i.e., for allk ∈ N,

0≤γk(t)≤γbk, (6.8a)

where γ ∈(0,_α1_γ] and δ ∈(0, αδ] are fixed constants. storage device

⇠

·

··

(t)

⇥

(t)

(t)

(t)

p

(t)

p

(t)

p

(t)

Figure 6.1: Real power balance at node k∈ N.

Balancing real power that flows in and out of bus k ∈ N at time t, as shown in Figure 6.1, we have:

pG_k(t)−pD_k(t)−γk(t) +δk(t) =

X

l∼k

pkl(t). (6.9)

Also, maintaining reactive power balance, we have

q_kG(t)−q_kD(t) = X

l∼k

qkl(t). (6.10)

Now, optimally placing storage over an infinite horizon is equivalent to solving this prob- lem over a singe cycle, provided the state of the storage levels at the end of a cycle is the same as its initial condition [137]. Thus, for each k ∈ N, we have

T X t=1 αγγk(t)− 1 αδ δk(t) = 0. (6.11)

For convenience, denote [T] := {1,2, . . . , T}. Using the above notation, we define the fol- lowing optimization problem.

Storage placement problem P: minimize X k∈N T X t=1 ck pGk(t) over (pG_k(t), qG_k(t), γk(t), δk(t), V(t), pkl(t), bk), k ∈ N, k∼l, t∈[T], subject to (6.1),(6.2),(6.3),(6.4),(6.5),(6.6),(6.7),(6.8),(6.9),(6.11),

where, (6.1) represents generation constraints, (6.2) represents voltage magnitude constraints, (6.3) links the power flows to the voltages, (6.6), (6.7),(6.8),(6.11) represent the constraints imposed on the charging/discharging policy of the energy storage devices, (6.9) represents the power balance constraints at each bus of the network and (6.7) represents the constraint on the sum of the capacities of all storage devices being no greater than the available storage budget. With the demand profiles and network parameters as input, P defines the optimal investment decision strategy for sizing storage units at different buses, the economic dispatch of the various generators and the optimal control policy of the installed storage units. For any variable z, definez∗ as its value at optimum.

6.2.1

Network models

As with the market power problem in Chapter 4, we investigateP with two network models. First in Section 6.3, we solveP using the conic relaxation of the AC power flow model. Thus, we represent (6.2) – (6.3) in terms of W(t) =V(t)[V(t)]H_{, which is a positive semidefinite}

matrix of rank 1 for each t ∈ [T]. Then the resultant nonconvex program is replaced by a semidefinite program (SDP) by relaxing the rank constraint as discussed in Chapters 2 and 3. We call this problem PAC. Note that any of the conic relaxations based on chordal SDP or SOCP can also be used for studying the optimal solution of PAC_{. However, in Section}

6.3, we use the SDP approach to study PAC for some IEEE benchmark systems.

In Section 6.3, we make a few observations about storage placement in networks. How- ever, the SDP formulation is not amenable to characterize any of these properties analytically.

Hence, we simplify the formulation with the DC approximation [32, 33], first presented in Chapter 4. Thus, the voltage limits in (6.2) are dropped (since voltage magnitudes are as- sumed to be at nominal value in this approximation) and the relation in (6.3) is modified to

pkl = (θk−θl)/xkl,

where for each node k ∈ N, θk represents the voltage angle at bus k and xkl is the purely

reactive impedance of line k ∼ l. In essence, the admittance ykl = (ixkl)−1 is purely imag-

inary as losses are neglected under DC approximation. For the storage placement problem (denote by PDC_{) with this simplified linearized version of the constraints, we prove a result}

characterizing the optimal placement of the storage resources in Section 6.4. We further prove some results on the placement for networks with specific topologies.