Demand Side Management

Demand side management allows for the shifting of demands in time. It thus gives the model the possibility to divert from the strict restriction that all demands have to be fulfilled at all timesteps. Demand side management adds two variables to an urbs problem and the variable vector then reads:

\[x^{\text{T}}=(\zeta, \underbrace{\rho_{yvct}}_{\text{commodity variables}}, \underbrace{\kappa_{yvp}, \widehat{\kappa}_{yvp}, \tau_{yvpt}, \epsilon^{\text{in}}_{yvcpt}, \epsilon^{\text{out}}_{yvcpt}}_{\text{process variables}}, \underbrace{\kappa_{yaf}, \widehat{\kappa}_{yaf}, \pi^{\text{in}}_{yaft}, \pi^{\text{out}}_{yaft}}_{\text{transmission variables}},\underbrace{ \delta^{\text{up}}_{yvct}, \delta^{\text{down}}_{yvct(tt)}}_ {\text{DSM variables}}).\]

The new variable \(\delta^{\text{up}}_{yvct}\) represent the upshift of the momentary demand at time \(t\) and \(\delta^{\text{down}}_{yvct(tt)}\) the corresponding downshifts. The downshifts need two time indices as they are referencing to the corresponding upshift with the first index \(t\) and the timesteps they actually occur via the second time index \(tt\). The latter is then restricted to an interval around the reference upshift since loads cannot in general be shifted indefinitely. As it is modeled in urbs, DSM does not introduce any costs. To clarify the terms used for the DSM feature the following illustrative example is helpful.

Example of a DSM Process

An example scenario with parameters below can be used to clarify the mathematical structure of a DSM process.

Site	Commodity	delay	eff	recov	cap-max-do	cap-max-up
South	Elec	3	1	1	2000	2000

First, an series of three upshifts, i.e. demand increases, indexed with the modeled timesteps 3,4 and 5 occurs in the example.

DSM upshift process

\(t\)
1	0
2	0
3	1445
4	1580
5	2000
6	0

The corresponding downshifts can then be visualized using a matrix, where the row index \(t\) corresponds to the upshifts above, that have to be compensated by downshifts. The modeled timesteps where the downshifts actually occur are labeled by \(tt\) and represent the column indices.

DSM downshift process

\(t\) \ \(tt\)	1	2	3	4	5	6
1	0	0	0	0
2	0	0	0	0	0
3	1445	0	0	0	0	0
4	555	0	555	0	0	470
5		2000	0	0	0	0
6			0	0	0	0

The DSM upshift process is relatively easy to understand, for every time step \(t\) one upshift is made and it can not exceed 2000. The table for DSM downshift process shows, that the sum over all elements for every row index \(t\), is equal to the upshift made at time step \(t\). The blank spaces in the table are because of delay time restriction. For instance, an upshift in \(t = 1\) may not be compensated with a downshift in \(tt = 5\), as delay time is equal to 3 in our example. The restriction of the total DSM downshifts is given by the sum of all column elements for every index \(tt\). This sum may not exceed 2000 as well, due to given parameters.

Commodity Dispatch Constraints

Demand side management changes the vertex rule. Every upshift \(\delta^{\text{up}}_{yvct}\) leads to an additional demand, i.e., to an additional required output of the system, and vice versa for the downshifts. Effectively this changes the vertex rule (Kirchhoff’s current law) for demand commodities with DSM to:

\[\begin{split}&\forall y\in Y,~v\in V,~c \in C_{\text{dem}},~ t \in T_m:\\\\ &-d_{yvct}-\delta^{\text{up}}_{yvct} \geq \text{CB}(y,v,c,t)\\ &-d_{yvct}+\sum_{tt\in [(t - y_{yvc})/\Delta t,(t + y_{yvc})/\Delta t]} \delta^{\text{down}}_{yvc(tt)t} \geq \text{CB}(y,v,c,t).\end{split}\]

The downshift equation requires a little elaboration. Here, the total downshift occurring at a timestep \(t\) can be caused by downshifts linked to different upshifts, which in the notation above occur at times \(tt\). All downshift contributions within the delay time \(y_{yvc}\) of their respective upshifts are then summed up.

DSM Variables Rule

This central constraint rule for DSM in urbs links the up- and down shifts of DSM events. An upshift (multiplied with the DSM efficiency) at time \(t\) must be compensated with multiple downshifts during a certain time interval. The lower and upper bounds of this time interval are given by \(t - y_{yvc}\) and \(t + y_{yvc}\), where \(y_{yvc}\) is the delay time parameter specifying the allowed duration of a DSM event. Inside this time interval, another time index \(tt\) is required. It is used to index the downshift processes that are always linked to one upshift. Of course, the intervals of several upshifts can overlap. Mathematically, this rule can be noted like this:

\[\begin{split}&\forall y\in Y,~v\in V,~c\in C^{\text{DSM}}_{dem},~t\in T_m:\\\\ &e_{yvc}\delta^{\text{up}}_{yvct}=\sum_{tt\in [(t - y_{yvc})/\Delta t,(t + y_{yvc})/\Delta t]} \delta^{\text{down}}_{yvct(tt)},\end{split}\]

where \(e_{yvc}\) is the DSM efficiency. Note here, that the summation is over the timesteps where the downshifts are occurring as opposed to the vertex rule above, where the summation is over the timesteps of the corresponding upshifts.

DSM Shift Limitations

DSM shifts are limited in size in both directions. This is modeled by

\[\begin{split}&\forall y\in Y,~v\in V,~c\in C^{\text{DSM}}_{\text{dem}}, t\in T_m:\\\\ &\delta^{\text{up}}_{yvct}\leq \Delta t \cdot \overline{K}^{\text{up}}_{yvc}\\\\ &\sum_{tt\in [(t - y_{yvc})/\Delta t,(t + y_{yvc})/\Delta t]}\delta^{\text{down}}_{yvc(tt)t}\leq \Delta t \cdot \overline{K}^{\text{down}}_{yvc},\end{split}\]

where again the downshifts are summed over the corresponding upshifts, making sure that at no time there is a total downshift sum larger than the set maximum value.

In addition to these limitations on the single shift directions, the total sum of shifts is also limited in an urbs model via:

\[\begin{split}&\forall y\in Y,~v\in V,~c\in C^{\text{DSM}}_{\text{dem}}, t\in T_m:\\\\ &\delta^{\text{up}}_{yvct}+ \sum_{tt\in [(t - y_{yvc})/\Delta t,(t + y_{yvc})/\Delta t]}\delta^{\text{down}}_{yvc(tt)t} \leq \text{max} \{\overline{K}^{\text{up}}_{yvc},\overline{K}^{\text{down}}_{yvc}\}.\end{split}\]

DSM Recovery

Assuming that DSM is linked to some real physical devices, it is necessary to allow these devices to have some minimal time between DSM events, where, e.g., the ability to perform DSM is recovered. This is modeled in the following way:

\[\begin{split}&\forall y\in Y,~v\in V,~c\in C^{\text{DSM}}_{\text{dem}}, t\in T_m:\\\\ & \sum_{tt=t}^{o_{yvc}/\Delta t-1}\delta^{\text{up}}_{yvc(tt)}\leq \overline{K}^{\text{up}}_{yvc}\cdot y_{yvc},\end{split}\]

where \(o_{yvc}\) is the recovery time in hours. This constraint limits the total amount of upshifted energy within the recovery period (lhs) to the maximum allowed energy shift retained for the maximum amount of allowed shifting time for one shifting event. This means that only one full shifting event can occur within the recovery period.

This concludes the demand side management constraints.