Subproblem sampling vs. scenario reduction: efficacy comparison for stochastic programs in power systems applications

Abstract

In this paper, we present a two-stage stochastic programming and simulation-based framework for tackling large-scale planning and operational problems that arise in power systems with significant renewable generation. Traditional algorithms (the L-shaped method, for example) used to solve the sample average approximation of the true problem suffer from computational difficulties when the number of scenarios or the size of the subproblem increases. To address this, we develop a cutting plane method that uses sampling internally within optimization to select only a random subset of subproblems to solve in any iteration. We analyze the convergence property of the subproblem sampling-based method and demonstrate its computational advantages on two alternative formulations of the stochastic unit commitment-economic dispatch problem. We conduct the numerical experiments on modified IEEE-30 and IEEE-118 test systems. We also present detailed steps for assessing the quality of solutions obtained from sampling-based stochastic programming methods and determining a solution to prescribe to the system operators.

Appendices

A nomenclature

Sets
${\mathcal {T}}$	Time scale decision epochs
${{\mathcal {B}}}$	Buses
${{\mathcal {L}}}$	Transmission lines
${{\mathcal {D}}}$	Demand
${\mathcal {G}}$	Conventional generators
${{\mathcal {R}}}$	Renewable generators

Generator Parameters
$\underline{C}_g/\overline{C}_g$	Minimum/maximum required generatio when g is operational
$\overline{R}_g/\underline{R}_g$	Ramp-up/ ramp-down limit at generator g
$\overline{S}_g/\underline{S}_g$	Start-up/ shut down limit at generator g
$UT_g/DT_g$	Minimum uptime/ downtime limit at generator g
$f_{g,t}^{SU}/f_{g,t}^{SD}$	Incurred startup/shutdown cost for generator g at time t
$c^0_g$	No-load cost at generator g
$c^{\kappa }_g$	Variable cost of the $\kappa$th piece for generator g
$d_g^{gs}$	Shedding penalty at generator g
$\gamma _g^{\kappa }$	Production amount of the $\kappa$th piece at generator g, $\kappa =\{1,..,\kappa ^{max}\}$
$\hat{V}(\gamma _g^{\kappa })$	Aggregated cost of generating $\gamma _g^{\kappa }$ units of output

Bus and Line Parameters
V	Bus voltage
${\theta ^{min}},{\theta ^{max}}$	Angle limits of the connected buses
${p^{min}} ,{p^{max}}$	Line capacity limits
X	Line reactance

Other Parameters
$D_{i,t}$	Demand at demand node i at time t
$\rho _t$	The required reserve amount at time t
$d_i^{ls}$	Value of lost load at demand node i
$\breve{G}_{g,t}$	Total amount of available renewable energy at the renewable generator g at time t

UC MLR Decision Variables
$x_{g,t}$	1 if g is online at period t, 0 otherwise
$s_{g,t}/z_{g,t}$	1 if g is turned on/off, 0 otherwise
$G^{\prime }_{g,t}$	Day-ahead production amount beyond $\underline{C}_g$ provided by generator g at time t
$r_{g,t}$	Spinning reserve provided by unit g at time t

UC ALS Decision Variables
$\hat{x}_{g,t}$	1 if g remains operational at period t, 0 otherwise
$\hat{s}_{g,t}/\hat{z}_{g,t}$	1 if g is turned on/off, 0 otherwise
$G^{\prime }_{g,t}$	Day-ahead production amount beyond $\underline{C}_g$ provided by generator g at time t
$\bar{G}_{g,t}$	Maximum day-ahead generation amount that g can supply at time t

ED Decision Variables
$G_{g,t}$	Hour-ahead generation level at generator g at time t
$p_{(i,j),t}$	Power flow on line (i, j) at time t
${r^{ls}_{it}}$	Load curtailment at demand node i
${r^{gs}_{gt}}$	Generation curtailment at generator g at time t
$\theta _{it}$	Angle of the node i at time t
$v_{g,t}$	Day-ahead generation cost amount of generator g at time t

B Detailed optimization model for S-UCED

We present the two UC models and then the ED model. These formulations are modified to suit the 2-SP model structure that is studied in this paper. Finally, we present the two S-UCED model in the form of a multi-period 2-SP model.

1.1 B.1 Unit commitment model

The UC problem determines the operating schedule of the generation units and their production amount in order to meet the demand over a the 24 h horizon. The objective of this model is to minimize the startup, shutdown, and day-ahead generation cost for the operational generators. We begin by presenting the constraints for the UC problem.

• (a) Logical / state-transition constraints: In order to ensure appropriate values when each generator is turned on (off) as stated in Garver [9], constraint (12) links the binary commitment decisions together in the MLR formulation.

  $$\begin{aligned} x_{g,t} - x_{g,t-1} = {s_{g,t}} - z_{g,t} \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}. \end{aligned}$$

  (12)

  The state transition variables $(\hat{x},\hat{s},\hat{z})$ introduced in Atakan et al. [1], represent the transition of generator states between two consecutive time periods. Constraint (13) relates the three variables in ALS.

  $$\begin{aligned} {\hat{s}_{g,t-1}}+\hat{x}_{g,t-1}=\hat{z}_{g,t}+\hat{x}_{g,t} \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}. \end{aligned}$$

  (13)

• (b) Minimum up/downtime constraints: As studied in Rajan et al. [30], we present the restrictions on up/downtime in MLR below:

  $$\begin{aligned}&\sum \limits ^{t} _{i=t-UT_g+1} s_{g,i}\le x_{g,t} \quad \forall g \in {\mathcal {G}} ,\,t \in {\mathcal {T}}, \end{aligned}$$

  (14a)
  $$\begin{aligned}&\sum \limits ^{t} _{i=t-DT_g+1} z_{g,i}\le 1-x_{g,t} \quad \forall g \in {\mathcal {G}} ,\,t \in {\mathcal {T}}. \end{aligned}$$

  (14b)

  Considering the transition nature of the variables in ALS, constraints (15a) and (15b) are based on a more recent formulation in Atakan et al. [1] used in the ALS formulation.

  $$\begin{aligned}&\sum \limits ^{t-1} _{i=t-UT_g+1} \hat{s}_{g,i}\le \hat{x}_{g,t} \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}, \end{aligned}$$

  (15a)
  $$\begin{aligned}&\sum \limits ^{t} _{i=t-DT_g} \hat{s}_{g,i}\le 1-\hat{x}_{g,t-DT_g} \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}. \end{aligned}$$

  (15b)

  Constraint (15a) reflects the restrictions on a remain-on generator that it could have been turned on at most once in the previous $UT_g-1$ time periods. If the generator does not remain online, it could not have been turned on in these time periods. In a similar manner, constraint (15b) ensures that a generator that remains operational cannot be switched on again in the current time period t or the next $DT_g$ time periods.

• (c) Reserve requirements: According to Morales-España et al. [24], constraint (16) in the MLR formulation ensures that the overall spinning reserves exceed the required reserve amount $\rho$.

  $$\begin{aligned} \sum _{g \in {\mathcal {G}}} r_{g,t} \ge \rho _t \quad \forall t \in {\mathcal {T}}. \end{aligned}$$

  (16)

  Alternatively, constraint (17) involves the total maximum generation amount for the conventional generators and the net demand to fulfill the reserve requirement in the ALS formulation and is presented below.

  $$\begin{aligned} \sum _{g \in {\mathcal {G}}} \overline{G}_{g,t} \ge \left( \sum _{i \in {\mathcal {D}}} D_{i,t} - \sum _{i \in {\mathcal {R}}} \breve{G}_{i,t} \right) + \rho _t \quad \forall t \in {\mathcal {T}} . \end{aligned}$$

  (17)

• (d) Production limits: The following production constraints are proposed in Morales-España et al. [24] utilizing the start up and shut down limits for $t \in {\mathcal {T}}$ for the MLR formulation:

  $$\begin{aligned} G^{\prime }_{gt} + r_{g,t}&\le (\overline{C}_g - \underline{C}_g)x_{g,t} \nonumber \\&\quad - (\overline{C}_g - \overline{S}_g)s_{g,t} -(\overline{C}_g - \underline{S}_g)z_{g,t+1} \quad \forall g \not \in {\mathcal {G}}^1, \end{aligned}$$

  (18a)
  $$\begin{aligned} G^{\prime }_{gt} + r_{g,t}&\le (\overline{C}_g - \underline{C}_g)x_{g,t} \nonumber \\&\quad -(\overline{C}_g - \overline{S}_g)s_{g,t} \quad \forall g \in {\mathcal {G}}^1, \end{aligned}$$

  (18b)
  $$\begin{aligned} G^{\prime }_{gt} + r_{g,t}&\le (\overline{C}_g - \underline{C}_g)x_{g,t} \nonumber \\&\quad - (\overline{C}_g - \underline{S}_g)z_{g,t+1} \quad \forall g \in {\mathcal {G}}^1. \end{aligned}$$

  (18c)

  In the state-transition formulation, the on/off status of a generator is obtained by $(\hat{s}_{g,t}+\hat{x}_{g,t})$. Therefore, the production limits in the ALS formulation are proposed as follows:

  $$\begin{aligned}&\overline{G}_{g,t} \ge {G}^{\prime }_{g,t}+\underline{C}_g (\hat{s}_{g,t}+\hat{x}_{g,t}) \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}, \end{aligned}$$

  (19a)
  $$\begin{aligned}&\overline{G}_{g,t}\le \overline{C}_g(\hat{s}_{g,t} +\hat{x}_{g,t})+(\underline{S}_g-\overline{C}_g)\hat{z}_{g,t+1} \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}. \end{aligned}$$

  (19b)

• (e) Demand constraints: The following constraints in the MLR model impose limits on production levels in order to meet the net demand when the unit is online:

  $$\begin{aligned} \sum _{g \in {\mathcal {G}}} (G^{\prime }_{g,t}+ \underline{C}_gx_{g,t}) \ge \left( \sum _{i \in {\mathcal {D}}} D_{i,t} - \sum _{i \in {\mathcal {R}}} \breve{G}_{i,t} \right) \quad \forall t \in {\mathcal {T}}. \end{aligned}$$

  (20)

  The use of state-transition variables necessitates the demand constraints in the ALS model to be updated as follows:

  $$\begin{aligned} \sum _{g \in {\mathcal {G}}} (G^{\prime }_{g,t} + \underline{C}_g (\hat{s}_{g,t} + \hat{x}_{g,t})) \ge \left( \sum _{i \in {\mathcal {D}}} D_{i,t} - \sum _{i \in {\mathcal {R}}} \breve{G}_{i,t} \right) \quad \forall t \in {\mathcal {T}}. \end{aligned}$$

  (21)

• (f) Ramping constraints: in the MLR model, Morales-España et al. [24] demonstrate utilization of the spinning reserves and the production amount beyond the minimum in the ramping constraints:

  $$\begin{aligned}&G^{\prime }_{g,t} + r_{g,t} - G^{\prime }_{g,t-1} \le \overline{R}_g \quad \forall g \in {\mathcal {G}},\, t \in {\mathcal {T}} , \end{aligned}$$

  (22a)
  $$\begin{aligned}&G^{\prime }_{g,t-1} - G^{\prime }_{g,t} \le \underline{R}_g \quad \forall g \in {\mathcal {G}},\, t \in {\mathcal {T}}. \end{aligned}$$

  (22b)

  Considering both the transition state and the production variables, the ramping restrictions of the generators in the ALS formulation are updated as follows:

  $$\begin{aligned}&\overline{G}_{g,t}-{G}^{\prime }_{g,t-1}\le \overline{S}_g \hat{s}_{g,t}+(\overline{R}_g+\underline{C}_g)\hat{x}_{g,t} \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}, \end{aligned}$$

  (23a)
  $$\begin{aligned}&{G}^{\prime }_{g,t-1}-{G}^{\prime }_{g,t}\le (\underline{S}_g -\underline{C}_g)\hat{z}_{g,t}+\underline{R}_g\hat{x}_{g,t} \quad \forall g \in {\mathcal {G}},\,t \in {\mathcal {T}}. \end{aligned}$$

  (23b)

1.2 Economic dispatch model

The presented economic dispatch model is based on Gangammanavar et al. [7] and it concerns with minimizing the total cost of power production and the shedding penalty at the generators as well as the renewable and the load curtailment as a multi-time period optimization problem solved in response to renewable energy uncertainty under different constraints and restrictions that are explained subsequently. In the ED model, the UC commitment variables ($x_g$, $s_g$, $z_g$ ) in MLR, and ($\hat{x}_g$, $\hat{s}_g$, $\hat{z}_g$) and the maximum generation amount $\overline{G}_g$ in ALS are considered as the input. The variable $G_g$ in ED is synonymous to $G^{\prime }_g + \underline{C}_g$ in the UC model. The objective function of the ED model can be written as

$$\begin{aligned} \min \quad \sum _{t \in {\mathcal {T}}} \left( \sum _{g \in {\mathcal {G}}} v_{g,t} +\sum _{g \in {\mathcal {G}} \cup {\mathcal {R}}} d^{gs}_{g}r^{gs}_{g,t} +\sum _{i \in {\mathcal {D}}} d^{ls}_{i}r^{ls}_{i,t} \right) \end{aligned}$$

(24)

Next, we present the constraints associated with this problem.

• (g) Generator capacity and limits on actual production: Constraints (25) in the MLR formulation set the limits on the production amount at every online generator unit g.

  $$\begin{aligned} G_{gt}&\le \overline{C}_g{x_{g,t}} \nonumber \\&\quad - (\overline{C}_g - \overline{S}_g){s_{g,t}} -(\overline{C}_g- \underline{S}_g){z_{g,t+1}} \quad \forall g \not \in {\mathcal {G}}^1,\, t \in {\mathcal {T}}, \end{aligned}$$

  (25a)
  $$\begin{aligned} G_{gt}&\le \overline{C}_g x_{g,t} \nonumber \\&\quad - (\overline{C}_g - \overline{S}_g){s_{g,t}} \quad \forall g \in {\mathcal {G}}^1,\, t \in {\mathcal {T}}, \end{aligned}$$

  (25b)
  $$\begin{aligned} G_{gt}&\le \overline{C}_g{x_{g,t}} \nonumber \\&\quad - (\overline{C}_g - \underline{S}_g){z_{g,t+1}} \quad \forall g \in {\mathcal {G}}^1,\, t \in {\mathcal {T}}. \end{aligned}$$

  (25c)

  On the other hand, engaging the maximum production that a remain-on generator g supplies in the capacity constraint in the ALS model yields

  $$\begin{aligned} \underline{C}_g ({\hat{x}_{g,t}} + {\hat{s}_{g,t}}) \le G_{g,t} \le {\overline{G}_{g,t}} \quad \forall g \in {\mathcal {G}},\, t \in {\mathcal {T}} . \end{aligned}$$

  (26)

• (h) Ramping Constraints: These constraints are similar to the UC ramping constraints discussed in the previous section and are brought below. In MLR we have

  $$\begin{aligned}&G_{g,t} - G_{g,t-1} \le \overline{R}_g \quad \forall g \in {\mathcal {G}}, t \in {\mathcal {T}}, \end{aligned}$$

  (27a)
  $$\begin{aligned}&G_{g,t-1}- G_{g,t} \le \underline{R}_g \quad \forall g \in {\mathcal {G}}, t \in {\mathcal {T}} . \end{aligned}$$

  (27b)

  Ramping constraints for the ALS model are presented below.

  $$\begin{aligned}&G_{g,t} - G_{g,t-1} \le \overline{S}_g \hat{s}_{g,t} + \overline{R}_g \hat{x}_{g,t} \quad \forall g \in {\mathcal {G}},\, t \in {\mathcal {T}}, \end{aligned}$$

  (28a)
  $$\begin{aligned}&G_{g,t-1} - G_{g,t} \le \underline{S}\hat{z}_{g,t}+ \underline{R}_g \hat{x}_{g,t} \quad \forall g \in {\mathcal {G}},\, t \in {\mathcal {T}}. \end{aligned}$$

  (28b)

• (i) Power flow balance equation: The following flow conservation constraints are suggested in Gangammanavar et al. [7]:

  $$\begin{aligned}&\sum _{j:(j,i) \in {\mathcal {L}}} p_{ji,t} - \sum _{j:(i,j) \in {\mathcal {L}}} p_{ij,t} + \sum _{j \in {\mathcal {G}}_i} \left( G_{j,t} - r^{gs}_{j,t} \right) +\sum _{j \in {\mathcal {R}}_i} \left( G_{j,t}(\tilde{\xi }) -r^{gs}_{j,t}\right) \nonumber \\&\quad = \sum _{j \in {\mathcal {D}}_i} \left( D_{j,t} - r^{ls}_{j,t}\right) \quad \forall i \in {\mathcal {B}},\, t \in {\mathcal {T}}. \end{aligned}$$

  (29)

  Equation (29) ensures that the flow balance between the supply and the demand is satisfied at every bus.

• (j) Line flow equation: Since the power transmission and power loss follow a non-linear function of the difference between the angles at the buses, second-order linear approximation techniques are used to address the values which update the line flow equations as follows:

  $$\begin{aligned} p_{ij,t}=\frac{V_iV_j}{X_{ij}}(\theta _{i,t}-\theta _{j,t}) \quad \forall (i,j) \in {\mathcal {L}}, t \in {\mathcal {T}}. \end{aligned}$$

  (30)

• (k) Production-cost restrictions: In this paper, we assume the production costs are convex and piece-wise linear. Thus, as presented in Atakan et al. [1], we can write the production cost constraints as follows:

  $$\begin{aligned}&v_{g,t} \ge c^{\kappa }_{g} (G_{g,t} - \gamma _g^{\kappa -1}) +\hat{V}_g(\gamma _g^{\kappa -1}) - \hat{V}(\underline{C}_g) \nonumber \\&\qquad \forall g \in {\mathcal {G}}, \, t \in {\mathcal {T}} , \quad \kappa = 1,\ldots ,\kappa ^{max} \end{aligned}$$

  (31)

  where $\kappa$ is the level of production. In the special case of affine production cost, the production cost constraint reduces to

  $$\begin{aligned} v_{g,t} \ge c^{1}_{g} G_{g,t} - \hat{V}(\underline{C}_g) \quad \forall g \in {\mathcal {G}}, \, t \in {\mathcal {T}}. \end{aligned}$$

  (32)

• (l) Bounds: Constraints (33a) and (33b) are the limit on the line capacities and the bus angles respectively. Additionally, restrictions on the generation and load curtailment variables are given in (33c) and (33d).

  $$\begin{aligned}&p^{min}_{ij} \le p_{ij,t} \le p^{max}_{ij} \quad \forall (i,j) \in {\mathcal {L}},\, t \in {\mathcal {T}}, \end{aligned}$$

  (33a)
  $$\begin{aligned}&\theta ^{min}_{i} \le \theta _{i,t} \le \theta ^{max}_{i} \quad \forall i \in {\mathcal {B}}, t \in {\mathcal {T}}, \end{aligned}$$

  (33b)
  $$\begin{aligned}&0 \le r^{ls}_{i,t} \le D_{i,t} \quad \forall i \in {\mathcal {D}},\, t \in {\mathcal {T}}, \end{aligned}$$

  (33c)
  $$\begin{aligned}&0 \le r^{gs}_{i,t} \le G_{i,t} \quad \forall i \in {\mathcal {G}} \cup {\mathcal {R}},\, t \in {\mathcal {T}}. \end{aligned}$$

  (33d)

1.3 Decomposed multi period two-stage S-UCED model

The complete formulation of the decomposed multi period two-stage S-UCED models are presented below. The MLR formulation for the model can be written as

$$\begin{aligned} \begin{array}{rl} \min \quad &{}\sum \limits_{t \in {\mathcal {T}}} \left( \sum \limits_{g \in {\mathcal {G}}} \left( f^{SU}_{g,t}s_{g,t} + f^{SD}_{g,t}z_{g,t} + \hat{V}(\underline{C}_g) x_{g,t}\right) \right) + {\mathbb {E}}\{Q(x,s,z,\tilde{\xi })\}\\ &{}\text {s.t} \ (12), (14), (16), (18), (20), (22)\\ &{}(x,s,z,)\in \{0,1\}^{3|{\mathcal {G}}||{\mathcal {T}}|}, \ (G^\prime ,r)\ge 0 \end{array} \end{aligned}$$

(UC MLR)

where,

$$\begin{aligned} \begin{array}{rl} Q(x,s,z, {\xi })=\min \quad &{}\sum \limits_{t \in {\mathcal {T}}} \left( \sum \limits_{g \in {\mathcal {G}}} v_{g,t} +\sum \limits_{g \in {\mathcal {G}} \cup {\mathcal {R}}} d^{gs}_{g}r^{gs}_{g,t} + \sum \limits_{i \in {\mathcal {D}}} d^{ls}_{i}r^{ls}_{i,t}\right) \\ \text {s.t} \quad &{} (25), (27), (29), (30), (32), (33), (G,v)\ge 0. \end{array} \end{aligned}$$

(ED MLR)

Second, the ALS formulation is presented.

$$\begin{aligned} \begin{array}{rl} \min \ {} &{}\sum \limits _{t \in {\mathcal {T}}} \left( \sum \limits _{g \in {\mathcal {G}}}\left( f^{SU}_{g,t} \hat{s}_{g,t} + f^{SD}_{g,t}\hat{z}_{g,t}+ \hat{V} (\underline{C}_g)(\hat{s}_{g,t}+\hat{x}_{g,t})\right) \right) +{\mathbb {E}}\{Q(\hat{x},\hat{s},\hat{z},\overline{G}, \tilde{\xi })\}\\ \text {s.t} \ {} &{} (13), (15), (17), (19), (21), (23)\\ &{} (\hat{x},\hat{s},\hat{z})\in \{0,1\}^{3|{\mathcal {G}}|| {\mathcal {T}}|}, \ (G^\prime ,\overline{G})\ge 0 \end{array} \end{aligned}$$

(UC ALS)

where,

$$\begin{aligned} \begin{array}{rl} Q(\hat{x},\hat{s},\hat{z},\overline{G},{\xi })= \min \ {} &{} \sum \limits_{t \in {\mathcal {T}}}\left( \sum \limits_{g \in {\mathcal {G}}} v_{g,t} +\sum \limits_{g \in {\mathcal {G}} \cup {\mathcal {R}}} d^{gs}_{g}r^{gs}_{g,t} +\sum \limits_{i \in {\mathcal {D}}} d^{ls}_{i}r^{ls}_{i,t} \right) \\ \text {s.t} \ {} &{} (26), (28) -(30), (32), (33), (G,v)\ge 0. \end{array} \end{aligned}$$

(ED ALS)