Example Extensions

Matpower includes three OPF extensions implementing via callbacks, respectively, the co-optimization of energy and reserves, interface ﬂow limits and dispatchable DC transmission lines.

7.6.1 Fixed Zonal Reserves

This extension is a more complete version of the example of ﬁxed zonal reserve requirements used for illustration above in Sections 7.3 and 7.4. The details of the extensions to the standard OPF problem are given in equations (7.2)–(7.5) and a description of the relevant input and output data structures is summarized in Tables 7-5 and 7-6, respectively.

Table 7-5:Input Data Structures for Fixed Zonal Reserves

ngr — Table 7-5:Input Data Structures for Fixed Zonal Reserves

Table 7-6:Output Data Structures for Fixed Zonal Reserves

The code for implementing the callbacks can be found in toggle_reserves. A wrapper around runopf that turns on this extension before running the OPF is provided in runopf_w_res, allowing you to run a case with an appropriate reserves ﬁeld, such as t_case30_userfcns, as follows.

See help runopf_w_res and help toggle_reserves for more information. Examples of using this extension and a case ﬁle deﬁning the necessary input data can be found in t_opf_userfcns and t_case30_userfcns, respectively. Additional tests for runopf_w_res are included in t_runopf_w_res.

7.6.2 Interface Flow Limits

This extension adds interface ﬂow limits based on ﬂows computed from a DC network model. It is implemented in toggle_iflims. A ﬂow interface $k$ is deﬁned as a set $ℬk$ of branch indices $i$ and a direction for each branch. If $pi$ represents the real power ﬂow (“from” bus $→$ “to” bus) in branch $i$ and $di$ is equal to 1 or $− 1$ to indicate the direction,⁵³ then the interface ﬂow $f k$ for interface $k$ is deﬁned as

where each branch ﬂow $pi$ is an approximation calculated as a linear function of the bus voltage angles based on the DC power ﬂow model from equation (3.29).

This extension adds to the OPF problem a set of $nif$ doubly-bounded constraints on these ﬂows.

where $Fmin k$ and $F max k$ are the speciﬁed lower and upper bounds on the interface ﬂow, and $ℐf$ is a the set indices of interfaces whose ﬂow limits are to be enforced.

The data for the problem is speciﬁed in an additional if ﬁeld in the Matpower case struct mpc. This ﬁeld is itself a struct with two sub-ﬁelds, map and lims, used for input data, and two others, P and mu, used for output data. The format of this data is described in detail in Tables 7-7 and 7-8.

∑
( k nk)× 2 — Table 7-7:Input Data Structures for Interface Flow Limits

Table 7-7:Input Data Structures for Interface Flow Limits

nif × 1 — Table 7-8:Output Data Structures for Interface Flow Limits

See help toggle_iflims for more information. Examples of using this extension and a case ﬁle deﬁning the necessary input data for it can be found in t_opf_userfcns and t_case30_userfcns, respectively. Note that, while this extension can be used for AC OPF problems, the actual AC interface ﬂows will not necessarily be limited to the speciﬁed values, since it is a DC ﬂow approximation that is used for the constraint.

Running a case that includes the interface ﬂow limits is as simple as loading the case, turning on the extension and running it. Unlike with the reserves extension, Matpower does not currently have a wrapper function to automate this.

7.6.3 DC Transmission Lines

Beginning with version 4.1, Matpower also includes a simple model for dispatchable DC transmission lines. While the implementation is based on the extensible OPF architecture described above, it can be used for simple power ﬂow problems as well, in which the case the (OPF only) formulation callback is skipped.

A DC line in Matpower is modeled as two linked “dummy” generators, as shown in Figures 7-2 and 7-3, one with negative capacity extracting real power from the network at the “from” end of the line and another with positive capacity injecting power into the network at the “to” end. These dummy generators are added by the ext2int callback and removed by the int2ext callback. The real power ﬂow $pf$ on the DC line at the “from” end is deﬁned to be equal to the negative of the injection of corresponding dummy generator. The ﬂow at the “to” end $pt$ is deﬁned to be equal to the injection of the corresponding generator.

Matpower links the values of $pf$ and $pt$ using the following relationship, which includes a linear approximation of the real power loss in the line.

Here the linear coeﬃcient $l1$ is assumed to be a small ( $≪ 1$ ) positive number. Obviously, this is not applicable for bi-directional lines, where the ﬂow could go either direction, resulting in decreasing losses for increasing ﬂow in the “to” $→$ “from” direction. There are currently two options for handling bi-directional lines. The ﬁrst is to use a constant loss model by setting $l = 0 1$ . The second option is to create two separate but identical lines oriented in opposite directions. In this case, it is important that the lower limit on the ﬂow and the constant term of the loss model $l0$ be set to zero to ensure that only one of the two lines has non-zero ﬂow at a time.⁵⁴

Upper and lower bounds on the value of the ﬂow can be speciﬁed for each DC line, along with an optional operating cost. It is also assumed that the terminals of the line have a range of reactive power capability that can be used to maintain a voltage setpoint. Just as with a normal generator, the voltage setpoint is only used for simple power ﬂow; the OPF dispatches the voltage anywhere between the lower and upper bounds speciﬁed for the bus. Similarly, in a simple power ﬂow the input value for $p f$ and the corresponding value for $pt$ , computed from (7.8), are used to specify the ﬂow in the line.

Most of the data for DC lines is stored in a dcline ﬁeld in the Matpower case struct mpc. This ﬁeld is a matrix similar to the branch matrix, where each row corresponds to a particular DC line. The columns of the matrix are deﬁned in Table B-5 and include connection bus indices, line status, ﬂows, terminal reactive injections, voltage setpoints, limits on power ﬂow and VAr injections, and loss parameters. Also, similar to the branch or gen matrices, some of the columns are used for input values, some for results, and some, such as PF can be either input or output, depending on whether the problem is a simple power ﬂow or an optimal power ﬂow. The idx_dcline function deﬁnes a set of constants for use as named column indices for the dcline matrix.

An optional dclinecost matrix, in the same form as gencost, can be used to specify a cost to be applied to $pf$ in the OPF. If the dclinecost ﬁeld is not present, the cost is assumed to be zero.

Matpower’s DC line handling is implemented in toggle_dcline and examples of using it can be found in t_dcline. The case ﬁle t_case9_dcline includes some example DC line data. See help toggle_dcline for more information.

Running a case that includes DC lines is as simple as loading the case, turning on the extension and running it. Unlike with the reserves extension, Matpower does not currently have a wrapper function to automate this.

7.6.4 OPF Soft Limits

Matpower includes an extension that replaces limits (such as branch ﬂow limits, voltage magnitude bounds, generator limits, etc.) in an optimal power ﬂow with soft limits, that is, limits that can be violated with some linear penalty cost. This can be useful in identifying the cause of infeasibility in some optimal power ﬂow problems. The extension is implemented in toggle_softlims. A limited version for branch ﬂow constraints in a DC model only, was introduced in Matpower 5.0, with a generalization to all of the standard OPF limits added in Matpower 7.0. The soft AC branch ﬂow limit implemention provides an example of user-deﬁned nonlinear constraints.

with a soft limit involves introducing a new non-negative variable $si$ to represent the violation of the original constraint. The original hard constraint (7.9) is replaced by one in which this violation variable is used to “relax” the constraint by adding it to the right-hand.

Finally, a penalty cost on violations is implemented via a linear cost coeﬃcient, $civ$ , applied to each violation variable $si$ , so that the additional user deﬁned cost term from (6.47) for the set of soft limits looks like

Take, as an example, branch ﬂow limits. The ﬂow constraints in (6.9) and (6.10) are rewritten by replacing the right hand side with a ﬂow violation variable $sirate a$

In the case of AC ﬂow constraints, the ﬂow functions $fif$ and $fit$ are nonlinear and can be either apparent power, active power, or current as shown in (6.11). In the case of DC ﬂow constraints, where there are no losses, these ﬂows can be implemented in terms of the ﬂow at the from end of the line and its negative, both linear functions of the bus voltage angles, based on (6.30) and (6.31).

Matpower implements soft limits for each of the types of constraints listed in Table 7-9, where the formulation for each constraint type is shown.

vim ≥ vi,mmin — Table 7-9:Soft Limit Formulation

vim ≤ vi,mmax — Table 7-9:Soft Limit Formulation

The parameters for the problem are speciﬁed in an additional softlims ﬁeld in the Matpower case struct mpc. This ﬁeld is itself a struct where each sub-ﬁeld corresponds to the name of one of the limits listed in Table 7-9. For each of the limits, the input parameters are speciﬁed in the ﬁelds detailed in Table 7-10, with defaults summarized in Table 7-11.

nsl × 1 — Table 7-10:Input Data Structures for OPF Soft Limits

⋅VbasekV ) — Table 7-11:Default Soft Limit Values

P i,gmin≥ 0 — Table 7-11:Default Soft Limit Values

Depending on whether the original constraint in (7.10) is an upper bound or a lower bound, it can be written as one of the following

where $hi,lim$ is the actual value of the hard limit. When introducing a soft limit, this original hard limit can be removed completely or modiﬁed to have a new value. Matpower provides a several ways of modifying the original hard limit based on ﬁelds hl_mod (“hard limit modiﬁcation”) and hl_val (“hard limit value”) in the respective limit structures. This new hard limit is implemented via an upper bound on the corresponding violation variable $si$ . Table 7-12 summarizes the new hard limits and the corresponding bounds on the violation variable as functions of the values of hl_mod and hl_val.

hi,nliewm — Table 7-12:Possible Hard-Limit Modiﬁcations

Table 7-12:Possible Hard-Limit Modiﬁcations

Unspeciﬁed limits in the mpc.softlims input parameters are handled diﬀerently, depending on the value of the Matpower option 'opf.softlims.default' (see Table C-5).

Soft limit outputs, summarized in Table 7-13, include the amount and cost of any overloads, that is, violations of the original hard limits. These can be found as additional ﬁelds, overload and ov_cost, under each limit in results.softlims. The Kuhn-Tucker multipliers on the soft limit constraints are also included in the usual columns of the corresponding matrix in the results, e.g. bus(:, MU_VMAX), branch(:, MU_SF), etc. When there are no violations, these shadow prices are the same as if hard limits were imposed. When there is a violation, they are equal to the cost $cv$ associated with the violation variable, except in the case where a modiﬁed hard limit is also binding, in which case it also reﬂects the shadow price on this modiﬁed hard constraint.

nb∕l∕g × 1 — Table 7-13:Output Data Structures for OPF Soft Limits

n × 1
b∕l∕g — Table 7-13:Output Data Structures for OPF Soft Limits

See help toggle_softlims for more information on this extension. Examples of using this extension can be found in t_opf_softlims. Running a case that includes the default soft limits is as simple as loading the case, turning on the extension and running it. Unlike with the reserves extension, Matpower does not currently have a wrapper function to automate this.

name	description
mpc	Matpower case struct
reserves	additional ﬁeld in mpc containing input parameters for zonal reserves in the following sub-ﬁelds:
cost	( $ng$ or $ngr$ ^† ) $×1$ vector of reserve costs in $/MW, $c$ from (7.3)
qty	( $ng$ or $ngr$ ^† ) $× 1$ vector of reserve quantity upper bounds in MW, $ith$ element is $rimax$
zones	$nrz × ng$ matrix of reserve zone deﬁnitions
	zones(k,j) = ${ 1if gen j belongs to reserve zone k (j ∈ Zk) 0otherwise (j∈∕Zk)$
req	$nrz × 1$ vector of zonal reserve requirements in MW, $th k$ element is $Rk$ from (7.5)

name	description
results	OPF results struct, superset of mpc with additional ﬁelds for output data
reserves	zonal reserve data, with results in the following sub-ﬁelds:^‡
R	reserve allocation $ri$ , in MW
Rmin	lower bound of $ri$ in MW from (7.2), i.e. all zeros
Rmax	upper bound of $ri$ in MW from (7.2), i.e. $max min(ri ,Δi)$
mu	shadow prices on constraints in $/MW
l	shadow price on lower bound of $ri$ in (7.2)
u	shadow price on upper bound of $ri$ in (7.2)
Pmax	shadow price on capacity constraint (7.4)
prc	reserve price in $/MW, computed for unit $i$ as the sum of the shadow prices of constraints $k$ in (7.5) in which unit $i$ participates ( $k \| i ∈ Zk$ )

name	description
results	OPF results struct, superset of mpc with additional ﬁelds for output data
if	additional ﬁeld in results containing output parameters for interface ﬂow limits in the following sub-ﬁelds:
P	$nif × 1$ vector of actual ﬂow in MW across the corresponding interface (as measured at the “from” end of associated branches)
mu.l	$nif × 1$ vector of shadow prices on lower ﬂow limits ( $u$ /MW)^†
mu.u	$nif × 1$ vector of shadow prices on upper ﬂow limits ( $u$ /MW)^†

name	hard-limit formulation	soft-limit formulation
VMIN, VMAX^†	$vim ≥ vi,mmin$ , $vim ≤ vi,mmax$	$vim + sivmin ≥ vim,min$ , $vim − sivmax ≤ vi,mmax$
RATE_A^‡	$\| \| \|\|fi(Θ, V )\|\| ≤ fi f m max$	$\| \| \|\|f i(Θ,V )\|\|− s - ≤ f i f m ratea max$
	$\|\| i \|\| i ft(Θ, Vm) ≤ fmax$	$\|\| i \|\| i ft(Θ,Vm ) − srate-a ≤ fmax$
PMIN, PMAX	$i i,min pg ≥ pg$ , $i i,max pg ≤ pg$	$i i,min pg + spmin ≥ pg$ , $i i,max pg − spmax ≤ pg$
QMIN, QMAX^†	$i i,min qg ≥ qg$ , $i i,max qg ≤ qg$	$i i,min qg +sqmin ≥ qg$ , $i i,max qg − sqmax ≤ qg$
ANGMIN	$i i i,min 𝜃f − 𝜃t ≥ Δ 𝜃$	$i i i,min 𝜃f − 𝜃t + sangmin ≥ Δ 𝜃$
ANGMAX	$i i i,max 𝜃f − 𝜃t ≤ Δ 𝜃$	$i i i,max 𝜃f − 𝜃t − sangmax ≤ Δ 𝜃$

name	description
VMIN	voltage magnitude lower bound
idx	all buses
cost	$100,000/p.u.^†
hl_mod	'replace'
hl_val	0
VMAX	voltage magnitude upper bound
idx	all buses
cost	$100,000/p.u.^†
hl_mod	'remove'
RATE_A	branch ﬂow limit
idx	all on-line branches with bounded ﬂow limits
cost	$1000/MVA, $1000/MW, or $1000/(kA $⋅VbasekV )$ ^† where the units depend on 'opf.flow_lim'
hl_mod	'remove'
PMIN	generator active power lower bound
idx	all on-line generators, excluding dispatchable loads
cost	$1000/MW^†
hl_mod	'replace'
hl_val	0 if $P i,gmin≥ 0$ , $− ∞$ otherwise
PMAX, QMIN, QMAX	generator active power upper bound, reactive power lower/upper bounds
idx	all on-line generators, excluding dispatchable loads
cost	$1000/MW or $1000/MVAr^†
hl_mod	'remove'
ANGMIN, ANGMAX	branch angle diﬀerence lower/upper bounds
idx	all on-line branches with angle diﬀerence limits, $Δ 𝜃$ , where $∘ \|Δ𝜃\| < 360$
cost	$1000/deg^†
hl_mod	'remove'

h_mod	new hard limit $hi,nliewm$	upper bound on violation variable $si$
'remove'	$±∞$ ^†	$∞$
'replace'	$hl val$	$\|\| i,lim\|\| \|hl val − horig \|$
'scale'	$hl val ⋅hi,orliigm$	$\| \| \|\|(hl val − 1)⋅hi,orliigm\|\|$
'shift'	$hi,lim ± hl val orig$ ^†	$hl-val$

7.6 Example Extensions

7.6.1 Fixed Zonal Reserves

7.6.2 Interface Flow Limits

7.6.3 DC Transmission Lines

7.6.4 OPF Soft Limits