Bounds for Lognormal HVAC Perturbations

Note toward a probabilistic model of seasonal load shape distortion.

Author: J. Landers

Context

We model the canonical (non‑HVAC) seasonal envelope for a building type \(T\) by a smooth sinusoid and then introduce a lognormal multiplicative HVAC bump that can strongly distort the seasonal shape for a small subset of buildings.

S_0(t) := S_T^{\text{nonHVAC}}(t) = \] \[ 1 + a_T \sin\!\Bigl( \frac{2\pi}{365} \bigl( \mathrm{doy}(t) - \phi_T \bigr) \Bigr),

Normalizing over a year gives a probability density \(\,p(t)\) on the time axis:

p(t) = \frac{ S_0(t) }{ \displaystyle \int_{0}^{365} S_0(u)\,du }, \] \[ S_0(t) > 0.

To capture HVAC‑driven summer behavior, each service point \(i\) of type \(T\) draws a random lognormal HVAC scaling \(F_i\) with \(\mathbb{E}[F_i]=1\) and applies a canonical bump \(\Delta_T\) through a smooth HVAC envelope \(w(t)\in[0,1]\):

S_i(t) = S_0(t)\Bigl[ 1 + \Delta_T F_i w(t) \Bigr].

After renormalization this yields a perturbed density \(q_i\) representing the building's seasonal profile. We measure the deviation from the canonical profile \(p\) using total variation distance.

Theorem: Total Variation Bound

Theorem

TV bound for a lognormal HVAC bump.

Let \(p\) be the canonical seasonal density defined above and let \(w:[0,365]\to[0,1]\) be the HVAC envelope. Define the mean HVAC weight

m := \int_{0}^{365} w(t)\,p(t)\,dt,

and the type‑dependent constant

C_T := \frac{1}{2} \int_{0}^{365} p(t)\, \bigl|w(t) - m\bigr|\,dt.

For a service point \(i\) of type \(T\), let \(F_i > 0\) be a random variable and set

\varepsilon_i := \Delta_T\bigl(F_i - 1\bigr), \] \[ S_i(t) := S_0(t)\bigl[ 1 + \varepsilon_i w(t) \bigr], \] \[ q_i(t) := \frac{ S_i(t) }{ \displaystyle \int_{0}^{365} S_i(u)\,du }.

Assume there exists a constant \(0 < \eta < 1\) such that

\bigl|\varepsilon_i m\bigr| \le \eta \quad \text{almost surely.}

Then the total variation distance satisfies

\mathrm{TV}(p,q_i) := \frac{1}{2} \int_{0}^{365} \bigl|p(t) - q_i(t)\bigr|\,dt \le \] \[ \frac{C_T}{1-\eta}\, \Delta_T\, \bigl|F_i - 1\bigr|.

In particular, if

F_i \sim \mathrm{LogNormal} \bigl(\mu_T,\sigma_T^{2}\bigr), \] \[ \mu_T = -\tfrac{1}{2}\sigma_T^{2} \quad (\mathbb{E}[F_i]=1),

then

\mathbb{E}\bigl[\mathrm{TV}(p,q_i)\bigr] \le \frac{C_T}{1-\eta}\, \Delta_T\, \sqrt{ e^{\sigma_T^{2}} - 1 }.

Proof (sketch)

Perturbative normalization argument.

By construction,

S_i(t) = S_0(t)\bigl[ 1 + \varepsilon_i w(t) \bigr], \] \[ \varepsilon_i = \Delta_T\bigl(F_i - 1\bigr).

Using \(p(t) = S_0(t) / \int_0^{365} S_0(u)\,du\) and \(\int p = 1\), we have

\int_{0}^{365} S_i(u)\,du = \] \[ \Bigl( \int_{0}^{365} S_0(u)\,du \Bigr) \bigl[1 + \varepsilon_i m\bigr],

where \(m = \int w(u)\,p(u)\,du\). Hence

q_i(t) = p(t)\, \frac{ 1 + \varepsilon_i w(t) }{ 1 + \varepsilon_i m }.

Subtracting and simplifying,

p(t) - q_i(t) = \] \[ p(t)\, \frac{ \varepsilon_i\bigl(m - w(t)\bigr) }{ 1 + \varepsilon_i m }.

Taking absolute values and using \(\bigl|\varepsilon_i m\bigr|\le\eta\) gives

\bigl|p(t) - q_i(t)\bigr| \le \] \[ p(t)\, \frac{ \bigl|\varepsilon_i\bigr|\, \bigl|w(t)-m\bigr| }{ 1 - \eta }.

Integrating and dividing by two yields

\mathrm{TV}(p,q_i) \le \] \[ \frac{\bigl|\varepsilon_i\bigr|}{2(1-\eta)} \int_{0}^{365} p(t)\,\bigl|w(t)-m\bigr|\,dt = \] \[ \frac{C_T}{1-\eta}\, \bigl|\varepsilon_i\bigr|.

Substituting \(\varepsilon_i = \Delta_T(F_i-1)\) gives the first inequality. For the lognormal case, Cauchy–Schwarz and \(\operatorname{Var}(F_i) = e^{\sigma_T^{2}} - 1\) imply

\mathbb{E}\bigl[|F_i-1|\bigr] \le \sqrt{ e^{\sigma_T^{2}} - 1 },

which yields the stated bound on \(\mathbb{E}[\mathrm{TV}(p,q_i)]\). \(\square\)

Discussion and Interpretation

The constant \(C_T\) depends only on the canonical seasonal shape and the HVAC envelope \(w(t)\): it quantifies how unevenly HVAC effort is distributed across the year. If \(w(t)\) is nearly constant, then \(|w(t)-m|\) is small and \(C_T\) is small.

The parameter \(\Delta_T\) is the canonical shoulder → summer bump for type \(T\), while \(\sigma_T\) controls the spread of building‑ specific HVAC behavior via the lognormal factor \(F_i\). Larger \(\sigma_T\) increases the chance that a building becomes a rare outlier with a highly distorted seasonal profile.

The factor \((1-\eta)^{-1}\) is a safety margin: it prevents the normalization \(1 + \varepsilon_i m\) from approaching zero and keeps the perturbative analysis valid.

Sketch: Rare-Event Tail Bound

The total variation bound can be used to control the probability of rare, structurally abnormal seasonal shapes. For a threshold \(\tau > 0\), define the event \(\mathcal{E}_\tau := \{\mathrm{TV}(p,q_i) > \tau\}\).

Theorem (sketch)

Tail bound for large seasonal distortion.

Under the assumptions of the theorem above, define

K_T := \frac{C_T}{1-\eta}\,\Delta_T.

Then

\mathbb{P} \bigl( \mathrm{TV}(p,q_i) > \tau \bigr) \le \] \[ \mathbb{P} \Bigl( \bigl|F_i - 1\bigr| > \frac{\tau}{K_T} \Bigr).

If in addition \(F_i \sim \mathrm{LogNormal} \bigl(\mu_T,\sigma_T^{2}\bigr)\) with \(\mu_T=-\tfrac12\sigma_T^{2}\), then for large thresholds \(\tau\) one has the lognormal tail estimate

\mathbb{P} \bigl( \mathrm{TV}(p,q_i) > \tau \bigr) \;\lesssim\; \] \[ \exp\!\Biggl( -\, \frac{ \bigl( \ln(\tau/K_T) \bigr)^{2} }{ 2\sigma_T^{2} } \Biggr),

where the symbol \(\lesssim\) hides lower‑order terms in the lognormal tail.

The rare‑event geometry is thus governed by the trio \((\Delta_T,\sigma_T,C_T)\): a larger canonical bump, heavier lognormal spread, or a more concentrated HVAC envelope all increase the probability that a building's seasonal shape deviates dramatically from the canonical profile.