pscopf_comp.gms

$title pscopf
$ontext
PSCOPF: Preventive Security-Constrained Optimal Power Flow

The model is an ACOPF with preventive security constraints

Broadly the OPF (optimal power flow) model
involves choosing real power output for a set of generators
so as to meet specified real power demand.
Generator cost functions specify the cost of each generator
producing a given amount of real power.
The objective of OPF is to minimize the total generation cost.

The AC (alternating current) OPF model requires that power
flow from generator buses to load buses through a network of buses,
lines, transformers, and shunt elements according to a set of
power flow equations.
The power flow equations define constraints on the net injections
of real and reactive power at each bus,
the voltage magnitude and angle at each bus,
and the real and reactive power flow along lines and transformers.
Net injection is generation minus demand minus shunt consumption.
Engineering bounds are placed on bus voltage magnitude,
on power and current flows over lines and transformers,
and on generator real and reactive power outputs.

Security constraints further constrain the generator real power
outputs and other decision variables by requiring that under
certain prespecified contingencies, the system will react in
a way that continues to meet engineering limits.
For example, one such contingency might be defined by the
loss of a generator and all its power output.
Another contingency might be defined by the loss
of a line, in which case the power and current flow along that
line is lost, and also the constraints defining those flows are
removed.

In the event of a security contingency,
power injections and flows and bus voltages may change.
The contingency flows and voltages are subject to all the
original engineering bounds.
Voltage magnitudes at buses containing generators are subject
to further constraints under a contingency.
Specifically, the voltage at any generator bus should
remain the same under a contingency as the value assigned
in the base case.
In practice this constraint may be violated,
but all generators at such a bus receive control signals from
the bus voltage, and therefore they produce as much
reactive power as possible as long as the bus voltage is too low,
or conversely consume as much reactive power as possible
as long as the bus voltage is too high.
This generator behavior is known as PV/PQ switching since,
while the bus voltage can be maintained, the bus is a PV bus,
i.e. having fixed net real power (P) injection and fixed voltage
magnitude (V), but if the bus voltage cannot be maintained,
it becomes a PQ bus, i.e. with both fixed values of both
real and reactive net power injection.
This PV/PQ switching constraint on generator behavior is,
algebraically, a pair of complementarity constraints:

if V_i_k < V_i_k0 then Q_gen_g_k = Q_gen_max_g
if V_i_k > V_i_k0 then Q_gen_g_k = Q_gen_min_g

for all buses i, generators g at bus i, security contingencies k,

where k0 represents the base case, and

V_i_k = voltage magnitude at bus i in contingency k
V_i_k0 = voltage magnitude at bus i in base case (k0)
Q_gen_g_k = reactive power generation of generator g in contingency k
Q_gen_max_g = maximum reactive power generation of generator g
Q_gen_min_g = minimum reactive power generation of generator g

The security concept modeled is preventive security,
as opposed to corrective security,
meaning that the actions that may be taken by generators
and other controlled elements of the power system
in order to meet the security constraints in the event of
a security contingency are very limited.
For our purposes,
generator real power output can change only according to a prespecified
participation factor. That is,

P_gen_g_k = P_gen_g_k0 + alpha_g_k * P_delta_k

for all contingencies k and all generators g active in scenario k

where k0 represents the base case, and

P_gen_g_k = real power output of generator g in contingency k
P_gen_g_k0 = real power output of generator g in the base case (k0)
alpha_g_k = participation factor for generator g in contingency k
P_delta_k = total real power power output change by active generators in contingency k

This GAMS code file (GMS) implements a method of solving
the PSCOPF problem.
All input data is described in sets and parameters.
The algebraic model is specified by variables and constraints,
with appropriate bounds on the variables.
The AC power flow is modeled by representing bus voltage in
polar form and power injections and flows in rectangular form.
The model is a nonconvex nonlinear program (NLP).
Specifically, no discrete variables are needed,
but the AC power flow equations make the model nonconvex.
The NLP solver Knitro is called to solve the model,
but other solvers, such as IPOPT, would also be appropriate.

Algebraically, the most challenging aspect of this model is the
complementarity constraints enforcing the PV/PQ switching behavior.
Other SCOPF implementations, both academic and commercial, have
modeled these constraints in various ways. A few of these are:
(1) smooth algebraic approximation of complementarity,
(2) penalization of voltage deviation from base case to security contingency
(3) fixing PV/PQ to one side of the complementarity or the other.
These approaches may be used in succession.
The model given here first uses method (2) to find an initial
solution. This solution is used to select some bus voltage magnitude
and reactive power output variables to fix for a second solve by method (3).
This approach can fail. The second solve with method (3) may be infeasible
even if the overall problem is feasible. The second solve can give
a nonoptimal solution to the overall problem. The solution to the
first solve may present an ambiguous choice of variables to fix for
the second solve.
$offtext

* data gms file
$if not set ingdx $set ingdx pscopf_data.gdx

* voltage magnitude deviation penalty
$if not set voltage_penalty $set voltage_penalty 1000000
$if not set voltage_tolerance $set voltage_tolerance 1e-6

* solution
$if not set solutionname $set solutionname solution

set baseCase /baseCase/;

sets
  circuits c
  branches i
  buses j
  cases k
  generators l
  polynomialCostTerms m
  units u;

alias(circuits,c,c0,c1,c2,c3);
alias(branches,i,i0,i1,i2,i3);
alias(buses,j,j0,j1,j2,j3);
alias(cases,k,k0,k1,k2,k3);
alias(generators,l,l0,l1,l2,l3)
alias(polynomialCostTerms,m,m0,m1,m2,m3);
alias(units,u,u0,u1,u2,u3);

* network topology
set
  ijOrigin(i,j)
  ijDestination(i,j)
  ijjOriginDestination(i,j1,j2)
  icMap(i,c)
  ljMap(l,j)
  luMap(l,u)
  ikInactive(i,k)
  lkInactive(l,k)
  ikActive(i,k)
  lkActive(l,k)
  kBase(k);

* branches with zero series impedance are treated specially
set
  iSeriesImpedanceZero(i)
  iSeriesImpedanceNonzero(i);

* system technical characteristics
parameter
  baseMVA;

* bus technical characteristics
parameters
  jBaseKV(j)
  jShuntConductance(j) g^sh
  jShuntSusceptance(j) b^sh;

* bus performance limits
parameters
  jVoltageMagnitudeMin(j) v^min
  jVoltageMagnitudeMax(j) v^max;

* bus power demand
parameters
  jRealPowerDemand(j) p^dem
  jReactivePowerDemand(j) q^dem;

* branch technical characteristics
parameters
  iSeriesResistance(i) r^s
  iSeriesReactance(i) x^s
  iSeriesConductance(i) g^s
  iSeriesSusceptance(i) b^s
  iChargingSusceptance(i) b^c
  iTapRatio(i) tau^tr
  iPhaseShift(i) theta^tr;

* branch performance limits
parameters
  iPowerMagnitudeMax(i) s^max;

* generator performance limits
parameters
  lRealPowerMin(l) "p^gen,min"
  lRealPowerMax(l) "p^gen,max"
  lReactivePowerMin(l) "q^gen,min"
  lReactivePowerMax(l) "q^gen,max";

* generator cost characteristics
parameters
  lmRealPowerCostCoefficient(l,m)
  lmRealPowerCostExponent(l,m);

* contingency modeling parameters
parameters
  penaltyCoeff /%voltage_penalty%/
  lParticipationFactor(l);

* load data from GDX input file
$gdxin '%ingdx%'
$loaddc circuits
$loaddc branches
$loaddc buses
$loaddc cases
$loaddc generators
$loaddc polynomialCostTerms
$loaddc units
$loaddc ijOrigin
$loaddc ijDestination
$loaddc icMap
$loaddc ljMap
$loaddc luMap
$loaddc ikInactive
$loaddc lkInactive
$loaddc kBase
$loaddc baseMVA
$loaddc jBaseKV
$loaddc jShuntConductance
$loaddc jShuntSusceptance
$loaddc jVoltageMagnitudeMin
$loaddc jVoltageMagnitudeMax
$loaddc jRealPowerDemand
$loaddc jReactivePowerDemand
$loaddc iSeriesResistance
$loaddc iSeriesReactance
$loaddc iChargingSusceptance
$loaddc iTapRatio
$loaddc iPhaseShift
$loaddc iPowerMagnitudeMax
$loaddc lRealPowerMin
$loaddc lRealPowerMax
$loaddc lReactivePowerMin
$loaddc lReactivePowerMax
$loaddc lmRealPowerCostCoefficient
$loaddc lmRealPowerCostExponent
$loaddc lParticipationFactor
$gdxin

* solution
parameter
  modelStatus /0/
  solveStatus /0/
  lkReactivePowerSlackLo(l,k)
  lkReactivePowerSlackUp(l,k)
  jkReactivePowerGenSlackLo(j,k)
  jkReactivePowerGenSlackUp(j,k)
  jkVoltMagDevLo(j,k)
  jkVoltMagDevUp(j,k)
  jkVoltMagDevLoReactivePowerGenSlackUpCompViol(j,k)
  jkVoltMagDevUpReactivePowerGenSlackLoCompViol(j,k);

* technical variables
variables
  lkRealPower(l,k)
  lkReactivePower(l,k)
  jkVoltageMagnitude(j,k)
  jkVoltageAngle(j,k)
  jkShuntRealPower(j,k) bus to shunt
  jkShuntReactivePower(j,k) bus to shunt
  ikRealPowerOrigin(i,k) bus to branch
  ikReactivePowerOrigin(i,k) bus to branch
  ikRealPowerDestination(i,k) bus to branch
  ikReactivePowerDestination(i,k) bus to branch
  kRealPowerShortfall(k) missing real power that must be made up by increased generation;

* violation variables for use in PV/PQ switching penalty approach
positive variables
  jkVoltageMagnitudeViolationPos(j,k)
  jkVoltageMagnitudeViolationNeg(j,k);

* cost variables
variables
  obj
  penalty
  cost;

* equations
equations
  objDef
  penaltyDef
  costDef
  jkRealPowerBalance(j,k)
  jkReactivePowerBalance(j,k)
  ikPowerMagnitudeOriginBound(i,k)
  ikPowerMagnitudeDestinationBound(i,k)
  jkShuntRealPowerDef(j,k)
  jkShuntReactivePowerDef(j,k)
  ijjkRealPowerOriginDef(i,j1,j2,k)
  ijjkReactivePowerOriginDef(i,j1,j2,k)
  ijjkRealPowerDestinationDef(i,j1,j2,k)
  ijjkReactivePowerDestinationDef(i,j1,j2,k)
  ijjkVoltageMagnitudeSeriesImpedanceZeroEq(i,j1,j2,k)
  ijjkVoltageAngleSeriesImpedanceZeroEq(i,j1,j2,k)
  ikRealPowerSeriesImpedanceZeroEq(i,k)
  ijjkReactivePowerSeriesImpedanceZeroEq(i,j1,j2,k)
  lkRealPowerRecoveryDef(l,k)
  jkVoltageMagnitudeMaintenance(j,k)
  jkVoltageMagnitudeMaintenanceViolationDef(j,k);

* model
model
  pscopf /
    objDef
    penaltyDef
    costDef
    jkRealPowerBalance
    jkReactivePowerBalance
    ikPowerMagnitudeOriginBound
    ikPowerMagnitudeDestinationBound
    jkShuntRealPowerDef
    jkShuntReactivePowerDef
    ijjkRealPowerOriginDef
    ijjkReactivePowerOriginDef
    ijjkRealPowerDestinationDef
    ijjkReactivePowerDestinationDef
    ijjkVoltageMagnitudeSeriesImpedanceZeroEq
    ijjkVoltageAngleSeriesImpedanceZeroEq
    ikRealPowerSeriesImpedanceZeroEq
    ijjkReactivePowerSeriesImpedanceZeroEq
    lkRealPowerRecoveryDef
    jkVoltageMagnitudeMaintenanceViolationDef
/;

* process into per unit for optimization model
jShuntConductance(j) = jShuntConductance(j) / baseMVA;
jShuntSusceptance(j) = jShuntSusceptance(j) / baseMVA;
*jVoltageMagnitudeMin(j)
*jVoltageMagnitudeMax(j)
jRealPowerDemand(j) = jRealPowerDemand(j) / baseMVA;
jReactivePowerDemand(j) = jReactivePowerDemand(j) / baseMVA;
*iSeriesResistance(i)
*iSeriesReactance(i)
*iChargingSusceptance(i)
*iTapRatio(i)
*iPhaseShift(i)
iPowerMagnitudeMax(i) = iPowerMagnitudeMax(i) / baseMVA;
lRealPowerMin(l) = lRealPowerMin(l) / baseMVA;
lRealPowerMax(l) = lRealPowerMax(l) / baseMVA;
lReactivePowerMin(l) = lReactivePowerMin(l) / baseMVA;
lReactivePowerMax(l) = lReactivePowerMax(l) / baseMVA;
lmRealPowerCostCoefficient(l,m) = lmRealPowerCostCoefficient(l,m) * power(baseMVA,lmRealPowerCostExponent(l,m));
*lmRealPowerCostExponent(l,m);

* setup some utility sets
ijjOriginDestination(i,j1,j2)$(ijOrigin(i,j1) and ijDestination(i,j2)) = yes;
ikActive(i,k) = not ikInactive(i,k);
lkActive(l,k) = not lkInactive(l,k);

* data validity checks
loop(i,
  if(sum(j$ijOrigin(i,j),1) > 1,
    abort 'branch with multiple origins';);
  if(sum(j$ijOrigin(i,j),1) < 1,
    abort 'branch with no origin';);
  if(sum(j$ijDestination(i,j),1) > 1,
    abort 'branch with multiple destinations';);
  if(sum(j$ijDestination(i,j),1) < 1,
    abort 'branch with no destination';);
  if(sum(c$icMap(i,c),1) > 1,
    abort 'branch with multiple circuit ids';);
  if(sum(c$icMap(i,c),1) < 1,
    abort 'branch with no circuit ids';);
);
loop(l,
  if(sum(j$ljMap(l,j),1) > 1,
    abort 'generator with multiple connection buses';
  );
  if(sum(j$ljMap(l,j),1) < 1,
    abort 'generator with no connection bus';
  );
  if(sum(u$luMap(l,u),1) > 1,
    abort 'generator with multiple unit ids';
  );
  if(sum(u$luMap(l,u),1) < 1,
    abort 'generator with no unit id';
  );
);
if(card(kBase) > 1,
  abort 'more than one base case';);
if(card(kBase) < 1,
  abort 'less than one base case';);
loop(k$(not kBase(k)),
  if(sum(l$lkActive(l,k),abs(lParticipationFactor(l))) = 0,
    abort 'contingency with no active participating generators';);
);

*lParticipationFactor(l) = lParticipationFactor(l) / sum(l0,lParticipationFactor(l0));

* compute line parameters
iSeriesImpedanceNonzero(i) = (
  abs(iSeriesResistance(i)) gt 0 or
  abs(iSeriesReactance(i)) gt 0);
iSeriesImpedanceZero(i) = (
  not iSeriesImpedanceNonzero(i));
iSeriesConductance(i)$iSeriesImpedanceNonzero(i)
  = iSeriesResistance(i)
  / (sqr(iSeriesResistance(i)) + sqr(iSeriesReactance(i)));
iSeriesSusceptance(i)$iSeriesImpedanceNonzero(i)
  = -iSeriesReactance(i)
  / (sqr(iSeriesResistance(i)) + sqr(iSeriesReactance(i)));

* bounds
lkRealPower.lo(l,k)$lkActive(l,k) = lRealPowerMin(l);
lkReactivePower.lo(l,k)$lkActive(l,k) = lReactivePowerMin(l);
jkVoltageMagnitude.lo(j,k) = jVoltageMagnitudeMin(j);
lkRealPower.up(l,k)$lkActive(l,k) = lRealPowerMax(l);
lkReactivePower.up(l,k)$lkActive(l,k) = lReactivePowerMax(l);
jkVoltageMagnitude.up(j,k) = jVoltageMagnitudeMax(j);

* equation definitions

* general objective
objDef..
      obj
  =e= cost
   +  penaltyCoeff * penalty;

* penalty
penaltyDef..
      penalty
  =e= sum((j,k)$(not kBase(k) and sum(l$(lkActive(l,k) and ljMap(l,j)),1)),
          jkVoltageMagnitudeViolationPos(j,k)
        + jkVoltageMagnitudeViolationNeg(j,k));

* generation cost
costDef..
      cost
  =e= sum(k$kBase(k),
        sum(l$lkActive(l,k),
              sum(m$(abs(lmRealPowerCostExponent(l,m)) > 0),
                  lmRealPowerCostCoefficient(l,m)
                * power(lkRealPower(l,k),lmRealPowerCostExponent(l,m)))
            + sum(m$(abs(lmRealPowerCostExponent(l,m)) = 0),
                  lmRealPowerCostCoefficient(l,m))));

* power in = power out
jkRealPowerBalance(j,k)..
      sum(l$(lkActive(l,k) and ljMap(l,j)),lkRealPower(l,k))
  =e= jkShuntRealPower(j,k)
   +  sum(i$(ikActive(i,k) and ijOrigin(i,j)),ikRealPowerOrigin(i,k))
   +  sum(i$(ikActive(i,k) and ijDestination(i,j)),ikRealPowerDestination(i,k))
   +  jRealPowerDemand(j);

* power in = power out
jkReactivePowerBalance(j,k)..
      sum(l$(lkActive(l,k) and ljMap(l,j)),lkReactivePower(l,k))
  =e= jkShuntReactivePower(j,k)
   +  sum(i$(ikActive(i,k) and ijOrigin(i,j)),ikReactivePowerOrigin(i,k))
   +  sum(i$(ikActive(i,k) and ijDestination(i,j)),ikReactivePowerDestination(i,k))
   +  jReactivePowerDemand(j);

ikPowerMagnitudeOriginBound(i,k)$ikActive(i,k)..
      sqrt(1 + sqr(ikRealPowerOrigin(i,k)) + sqr(ikReactivePowerOrigin(i,k)))
  =l= sqrt(1 + sqr(iPowerMagnitudeMax(i)));

ikPowerMagnitudeDestinationBound(i,k)$ikActive(i,k)..
      sqrt(1 + sqr(ikRealPowerDestination(i,k)) + sqr(ikReactivePowerDestination(i,k)))
  =l= sqrt(1 + sqr(iPowerMagnitudeMax(i)));

jkShuntRealPowerDef(j,k)..
      jkShuntRealPower(j,k)
  =e= jShuntConductance(j)*sqr(jkVoltageMagnitude(j,k));

jkShuntReactivePowerDef(j,k)..
      jkShuntReactivePower(j,k)
  =e= -jShuntSusceptance(j)*sqr(jkVoltageMagnitude(j,k));

ijjkRealPowerOriginDef(i,j1,j2,k)$(ijjOriginDestination(i,j1,j2) and iSeriesImpedanceNonzero(i) and ikActive(i,k))..
      ikRealPowerOrigin(i,k)
  =e= (iSeriesConductance(i)/sqr(iTapRatio(i)))*sqr(jkVoltageMagnitude(j1,k))
   +  (-iSeriesConductance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*cos(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i))
   +  (iSeriesSusceptance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*sin(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i));

ijjkReactivePowerOriginDef(i,j1,j2,k)$(ijjOriginDestination(i,j1,j2) and iSeriesImpedanceNonzero(i) and ikActive(i,k))..
      ikReactivePowerOrigin(i,k)
  =e= ((-iSeriesSusceptance(i)-0.5*iChargingSusceptance(i))/sqr(iTapRatio(i)))*sqr(jkVoltageMagnitude(j1,k))
   +  (iSeriesSusceptance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*cos(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i))
   +  (iSeriesConductance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*sin(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i));

ijjkRealPowerDestinationDef(i,j1,j2,k)$(ijjOriginDestination(i,j1,j2) and iSeriesImpedanceNonzero(i) and ikActive(i,k))..
      ikRealPowerDestination(i,k)
  =e= iSeriesConductance(i)*sqr(jkVoltageMagnitude(j2,k))
   +  (-iSeriesConductance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*cos(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i))
   +  (-iSeriesSusceptance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*sin(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i));

ijjkReactivePowerDestinationDef(i,j1,j2,k)$(ijjOriginDestination(i,j1,j2) and iSeriesImpedanceNonzero(i) and ikActive(i,k))..
      ikReactivePowerDestination(i,k)
  =e= (-iSeriesSusceptance(i)-0.5*iChargingSusceptance(i))*sqr(jkVoltageMagnitude(j2,k))
   +  (iSeriesSusceptance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*cos(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i))
   +  (-iSeriesConductance(i)/iTapRatio(i))*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)*sin(jkVoltageAngle(j2,k) - jkVoltageAngle(j1,k) + iPhaseShift(i));

ijjkVoltageMagnitudeSeriesImpedanceZeroEq(i,j1,j2,k)$(ijjOriginDestination(i,j1,j2) and iSeriesImpedanceZero(i) and ikActive(i,k))..
      jkVoltageMagnitude(j2,k)
   -  jkVoltageMagnitude(j1,k)/iTapRatio(i)
  =e= 0;

ijjkVoltageAngleSeriesImpedanceZeroEq(i,j1,j2,k)$(ijjOriginDestination(i,j1,j2) and iSeriesImpedanceZero(i) and ikActive(i,k))..
      jkVoltageAngle(j2,k)
   -  jkVoltageAngle(j1,k)
   +  iPhaseShift(i)
  =e= 0;

ikRealPowerSeriesImpedanceZeroEq(i,k)$(iSeriesImpedanceZero(i) and ikActive(i,k))..
      ikRealPowerOrigin(i,k)
   +  ikRealPowerDestination(i,k)
  =e= 0;

ijjkReactivePowerSeriesImpedanceZeroEq(i,j1,j2,k)$(iSeriesImpedanceZero(i) and ijjOriginDestination(i,j1,j2) and ikActive(i,k))..
      ikReactivePowerOrigin(i,k)
   +  ikReactivePowerDestination(i,k)
   +  iChargingSusceptance(i)*jkVoltageMagnitude(j1,k)*jkVoltageMagnitude(j2,k)/iTapRatio(i)
  =e= 0;

lkRealPowerRecoveryDef(l,k)$(lkActive(l,k) and not kBase(k))..
      lkRealPower(l,k)
  =e= sum(k0$kBase(k0),lkRealPower(l,k0))
   +  lParticipationFactor(l)*kRealPowerShortfall(k)
   /  sum(l1$lkActive(l1,k),lParticipationFactor(l1))
;

jkVoltageMagnitudeMaintenance(j,k)$(not kBase(k) and sum(l$(lkActive(l,k) and ljMap(l,j)),1))..
      jkVoltageMagnitude(j,k)
  =e= sum(k0$kBase(k0),jkVoltageMagnitude(j,k0));

jkVoltageMagnitudeMaintenanceViolationDef(j,k)$(not kBase(k) and sum(l$(lkActive(l,k) and ljMap(l,j)),1))..
      jkVoltageMagnitude(j,k)
   -  sum(k0$kBase(k0),jkVoltageMagnitude(j,k0))
  =e= jkVoltageMagnitudeViolationPos(j,k)
   -  jkVoltageMagnitudeViolationNeg(j,k);

* set a start point
$ontext
* random start point
lkRealPower.l(l,k)$lkActive(l,k) = uniform(lRealPowerMin(l),lRealPowerMax(l));
lkReactivePower.l(l,k)$lkActive(l,k) = uniform(lReactivePowerMin(l),lReactivePowerMax(l));
jkVoltageMagnitude.l(j,k) = uniform(jVoltageMagnitudeMin(j),jVoltageMagnitudeMax(j));
jkVoltageAngle.l(j,k) = normal(0,1);
jkShuntRealPower.l(j,k) = normal(0,1);
jkShuntReactivePower.l(j,k) = normal(0,1);
ikRealPowerOrigin.l(i,k)$ikActive(i,k) = normal(0,1);
ikReactivePowerOrigin.l(i,k)$ikActive(i,k) = normal(0,1);
ikRealPowerDestination.l(i,k)$ikActive(i,k) = normal(0,1);
ikReactivePowerDestination.l(i,k)$ikActive(i,k) = normal(0,1);
$offtext

*scaling
pscopf.scaleopt=0;
jkVoltageMagnitudeViolationPos.scale(j,k)$(not kBase(k) and sum(l$(lkActive(l,k) and ljMap(l,j)),1)) = 1;
jkVoltageMagnitudeViolationNeg.scale(j,k)$(not kBase(k) and sum(l$(lkActive(l,k) and ljMap(l,j)),1)) = 1;
jkVoltageMagnitudeMaintenanceViolationDef.scale(j,k)$(not kBase(k) and sum(l$(lkActive(l,k) and ljMap(l,j)),1)) = 1e3;

* solver options

option nlp=examiner;

$onecho > knitro.opt
feastol 2.25e-9
*feastol_abs 1e-2
opttol 1e-4
maxcgit 10
ftol 1e-4
ftol_iters 3
maxtime_real 300
$offecho

$onecho > examiner.opt
examinesolupoint 1
examinesolvpoint 1
examinegamspoint 0
examineinitpoint 0
subsolver knitro.1
$offecho

pscopf.optfile=1;
pscopf.tolproj=1e-12;

* solve penalty formulation
solve pscopf using nlp minimizing obj;
modelStatus = pscopf.modelstat;
solveStatus = pscopf.solvestat;

* assess slacks and deviations
lkReactivePowerSlackLo(l,k)$lkActive(l,k)
  = max(0,lkReactivePower.l(l,k)-lReactivePowerMin(l));
lkReactivePowerSlackUp(l,k)$lkActive(l,k)
  = max(0,lReactivePowerMax(l)-lkReactivePower.l(l,k));
jkReactivePowerGenSlackLo(j,k)
  = sum(l$(ljMap(l,j) and lkActive(l,k)),lkReactivePowerSlackLo(l,k));
jkReactivePowerGenSlackUp(j,k)
  = sum(l$(ljMap(l,j) and lkActive(l,k)),lkReactivePowerSlackUp(l,k));
jkVoltMagDevLo(j,k)
  = max(0,sum(k1$kBase(k1),jkVoltageMagnitude.l(j,k1))-jkVoltageMagnitude.l(j,k))
    $(sum(l$(lkActive(l,k) and ljMap(l,j)),1) > 0);
jkVoltMagDevUp(j,k)
  = max(0,jkVoltageMagnitude.l(j,k)-sum(k1$kBase(k1),jkVoltageMagnitude.l(j,k1)))
    $(sum(l$(lkActive(l,k) and ljMap(l,j)),1) > 0);
jkVoltMagDevLoReactivePowerGenSlackUpCompViol(j,k)
  = jkVoltMagDevLo(j,k)*jkReactivePowerGenSlackUp(j,k);
jkVoltMagDevUpReactivePowerGenSlackLoCompViol(j,k)
  = jkVoltMagDevUp(j,k)*jkReactivePowerGenSlackLo(j,k);

display
  jkVoltMagDevLo
  jkVoltMagDevUp;

* set bounds to achieve complementarity based on current point
* if v < v* and q = qmax then fix q = qmax
* i.e. if vDevLo > qSlackUp then fix q = qmax
*
* if v > v* and q = qmin then fix q = qmin
* i.e. if vDevUp > qSlackLo then fix q = qmin
*
* if qmin < q < qmax and v = v* then fix v = v*
* i.e. if vDevLo <= qSlackUp and vDevUp <= qSlackDown then fix v = f*
parameter
  voltageMagnitudeDeviationTolerance /%voltage_tolerance%/;
loop((j,k)$(not kBase(k) and sum(l$(lkActive(l,k) and ljMap(l,j)),1) > 0),
$ontext
  if(jkVoltMagDevLo(j,k) > jkReactivePowerGenSlackUp(j,k),
    lkReactivePower.lo(l,k)$(lkActive(l,k) and ljMap(l,j)) = lReactivePowerMax(l);
  );
  if(jkVoltMagDevUp(j,k) > jkReactivePowerGenSlackLo(j,k),
    lkReactivePower.up(l,k)$(lkActive(l,k) and ljMap(l,j)) = lReactivePowerMin(l);
  );
  if(jkVoltMagDevLo(j,k) le jkReactivePowerGenSlackUp(j,k) and jkVoltMagDevUp(j,k) le jkReactivePowerGenSlackLo(j,k),
    jkVoltageMagnitudeViolationPos.fx(j,k) = 0;
    jkVoltageMagnitudeViolationNeg.fx(j,k) = 0;
  );
$offtext
*$ontext
  if(jkVoltMagDevLo(j,k) > voltageMagnitudeDeviationTolerance,
    lkReactivePower.fx(l,k)$(lkActive(l,k) and ljMap(l,j)) = lReactivePowerMax(l);
    jkVoltageMagnitudeViolationPos.fx(j,k) = 0;
  elseif jkVoltMagDevUp(j,k) > voltageMagnitudeDeviationTolerance,
    lkReactivePower.fx(l,k)$(lkActive(l,k) and ljMap(l,j)) = lReactivePowerMin(l);
    jkVoltageMagnitudeViolationNeg.fx(j,k) = 0;
  else
    jkVoltageMagnitudeViolationPos.fx(j,k) = 0;
    jkVoltageMagnitudeViolationNeg.fx(j,k) = 0;
  );
*$offtext
);

* resolve with fixed complementarity and no penalties
*$ontext
if(modelStatus = 2,
penaltyCoeff = 0;
pscopf.holdfixed = 1;
solve pscopf using nlp minimizing obj;
modelStatus = pscopf.modelstat;
solveStatus = pscopf.solvestat;
);
*$offtext

* resolve with fixed variables
$ontext
jkVoltageAngle.fx(j,k)$(sum(l$(ljMap(l,j) and lkActive(l,k)),lRealPowerMax(l) - lRealPowerMin(l)) > 0) = jkVoltageAngle.l(j,k);
jkVoltageMagnitude.fx(j,k)$(sum(l$(ljMap(l,j) and lkActive(l,k)),lReactivePowerMax(l) - lReactivePowerMin(l)) > 0) = jkVoltageMagnitude.l(j,k);
solve pscopf using nlp minimizing obj;
modelStatus = pscopf.modelstat;
solveStatus = pscopf.solvestat;
$offtext

*$ontext
$onecho > examiner.op2
examineGamsPoint 1
examineInitPoint 0
returnGamsPoint 1
$offecho
* further solve with GAMS examiner
option nlp = examiner;
pscopf.optfile = 2;
solve pscopf using nlp minimizing obj;
*$offtext

* compute solution from variable values
$include pscopf_compute_solution.gms

* assess solution
lkReactivePowerSlackLo(l,k)$lkActive(l,k)
  = max(0,lkReactivePower.l(l,k)-lReactivePowerMin(l));
lkReactivePowerSlackUp(l,k)$lkActive(l,k)
  = max(0,lReactivePowerMax(l)-lkReactivePower.l(l,k));
jkReactivePowerGenSlackLo(j,k)
  = sum(l$(ljMap(l,j) and lkActive(l,k)),lkReactivePowerSlackLo(l,k));
jkReactivePowerGenSlackUp(j,k)
  = sum(l$(ljMap(l,j) and lkActive(l,k)),lkReactivePowerSlackUp(l,k));
jkVoltMagDevLo(j,k)
  =  max(0,sum(k1$kBase(k1),jkVoltageMagnitude.l(j,k1))-jkVoltageMagnitude.l(j,k))
    $(sum(l$(lkActive(l,k) and ljMap(l,j)),1) > 0);
jkVoltMagDevUp(j,k)
  = max(0,jkVoltageMagnitude.l(j,k)-sum(k1$kBase(k1),jkVoltageMagnitude.l(j,k1)))
    $(sum(l$(lkActive(l,k) and ljMap(l,j)),1) > 0);
jkVoltMagDevLoReactivePowerGenSlackUpCompViol(j,k)
  = jkVoltMagDevLo(j,k)*jkReactivePowerGenSlackUp(j,k);
jkVoltMagDevUpReactivePowerGenSlackLoCompViol(j,k)
  = jkVoltMagDevUp(j,k)*jkReactivePowerGenSlackLo(j,k);

* translate back to data units
lkReactivePowerSlackLo(l,k)$lkActive(l,k)
  = baseMVA*lkReactivePowerSlackLo(l,k);
lkReactivePowerSlackUp(l,k)$lkActive(l,k)
  = baseMVA*lkReactivePowerSlackUp(l,k);
jkReactivePowerGenSlackLo(j,k)
  = baseMVA*jkReactivePowerGenSlackLo(j,k);
jkReactivePowerGenSlackUp(j,k)
  = baseMVA*jkReactivePowerGenSlackUp(j,k);
*jkVoltMagDevLo(j,k)
*jkVoltMagDevUp(j,k)
jkVoltMagDevLoReactivePowerGenSlackUpCompViol(j,k)
  = baseMVA*jkVoltMagDevLoReactivePowerGenSlackUpCompViol(j,k);
jkVoltMagDevUpReactivePowerGenSlackLoCompViol(j,k)
  = baseMVA*jkVoltMagDevUpReactivePowerGenSlackLoCompViol(j,k);
iPowerMagnitudeMax(i) = baseMVA * iPowerMagnitudeMax(i);
ikRealPowerOrigin.l(i,k)$ikActive(i,k) = baseMVA * ikRealPowerOrigin.l(i,k);
ikReactivePowerOrigin.l(i,k)$ikActive(i,k) = baseMVA * ikReactivePowerOrigin.l(i,k);
ikRealPowerDestination.l(i,k)$ikActive(i,k) = baseMVA * ikRealPowerDestination.l(i,k);
ikReactivePowerDestination.l(i,k)$ikActive(i,k) = baseMVA * ikReactivePowerDestination.l(i,k);
jkShuntRealPower.l(j,k) = baseMVA * jkShuntRealPower.l(j,k);
jkShuntReactivePower.l(j,k) = baseMVA * jkShuntReactivePower.l(j,k);
lRealPowerMin(l) = baseMVA * lRealPowerMin(l);
lRealPowerMax(l) = baseMVA * lRealPowerMax(l);
lReactivePowerMin(l) = baseMVA * lReactivePowerMin(l);
lReactivePowerMax(l) = baseMVA * lReactivePowerMax(l);
jRealPowerDemand(j) = baseMVA * jRealPowerDemand(j);
jReactivePowerDemand(j) = baseMVA * jReactivePowerDemand(j);
*jkVoltageMagnitude.l(j,k) = jBaseKV(j) * jkVoltageMagnitude.l(j,k);
jkVoltageAngle.l(j,k) = 180 * jkVoltageAngle.l(j,k) / pi;
lkRealPower.l(l,k)$lkActive(l,k) = baseMVA * lkRealPower.l(l,k);
lkReactivePower.l(l,k)$lkActive(l,k) = baseMVA * lkReactivePower.l(l,k);
kRealPowerShortfall.l(k)
  = baseMVA * kRealPowerShortfall.l(k)
  / sum(l1$lkActive(l1,k),lParticipationFactor(l1));
*kRealPowerShortfall.l(k) = 0;

* output
$set outputtype 0
$include pscopf_write_solution.gms
$set outputtype 1
$include pscopf_write_solution.gms
$set outputtype 2
$include pscopf_write_solution.gms