# Self-scheduling of Wind-Thermal Systems Using a Stochastic MILP Approach

Rui Laia, Isaias Gomes, Hugo Pousinho, Rui Melício, Victor Mendes

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Rui Laia, Isaias Gomes, Hugo Pousinho, Rui Melício, Victor Mendes. Self-scheduling of Wind-Thermal Systems Using a Stochastic MILP Approach. 8th Doctoral Conference on Computing, Electrical and Industrial Systems (DoCEIS), May 2017, Costa de Caparica, Portugal. pp.242-250, 10.1007/978-3-319-56077-9_23. [https://inria.hal.science/hal-01629587](https://inria.hal.science/hal-01629587)

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# Self-scheduling of Wind-thermal Systems Using a Stochastic MILP Approach

Rui Laia¹,², Isaias L.R. Gomes¹,², Hugo M.I. Pousinho¹, Rui Melício¹,² and
Victor M.F. Mendes²,³

¹Universidade de Lisboa, Instituto Superior Técnico, IDMEC, Portugal

²Universidade de Évora, Escola de Ciências e Tecnologia, Departamento de Física, Portugal

³Instituto Superior de Engenharia de Lisboa, Portugal
ruimelicio@gmail.com

**Abstract.** In this work a stochastic (Stoc) mixed-integer linear programming (MILP) approach for the coordinated trading of a price-taker thermal (Ther) and wind power (WP) producer taking part in a day-ahead market (DAM) electricity market (EMar) is presented. Uncertainty (Uncer) on electricity price (EPr) and WP is considered through established scenarios. Thermal units (TU) are modelled by variable costs, start-up (ST-UP) technical operating constraints and costs, such as: forbidden operating zones, minimum (Min) up/down time limits and ramp up/down limits. The goal is to obtain the optimal bidding strategy (OBS) and the maximization of profit (MPro). The wind-Ther coordinated configuration (CoConf) is modelled and compared with the unCoConf. The CoConf and unCoConf are compared and relevant conclusions are drawn from a case study.

**Keywords:** Wind thermal coordination; Stochastic programming; MILP.

## 1 Introduction

The emissions derived of the use of nonrenewable fuels and the aspiration to attain independence of energy [1] lead a considerable European countries to promote generation of the electricity from renewable (Rnew) resources by adopting some instruments of support for Rnew energy production, namely investments incentives, green certificates, soft balancing costs and feed-in-tariffs [2].

At the end of 2014, 43.70% of all novel Rnew farms were based on WP and was the 7th year consecutively that over 55.0% of added capacity of power in the EU was Rnew [3]. In the face of the increasing Rnew energy incorporation in the last years, supply of energy still depending on nonrenewable fuels since more than 60% of the electricity generated all over the world in 2012 was based on nonrenewable fuel Ther plants [4].

In a restructured EMar, power resources owners' operate under competition level due to the nodal variations of EPr [5] in order to obtain the best revenue bidding in the DAM [6]. For the WP producers (WPP), WP and the market-clearing EPr Uncer are
to be addressed in order to know the amount of energy to produce in order to present
optimal offers. In absence of conformity, i.e., there is a deviation (Dev), economic
penalizations is due to happen [7]. For Ther power producers, only market-clearing
EPr Uncer has to be addressed.

## 2 Relationship to Smart Systems

A smart system can be stated as an embedded system that incorporates advanced
systems and provide the inhabitants with sophisticated monitoring and control over
how something happens in the system [8], for example a wind farm or a TU. Smart
systems are capable of sensing, making diagnosis, describing, qualifying and
managing how something happens in the system, incorporating both technical
intelligence and cognitive functions. In smart systems, electronic devices will be
communicating with software base system, allowing the user to access information
about the functionality of the system [8]. These systems are highly reliable, often
miniaturized, networked, predictive and energy autonomous [9]. Future power
systems should ensure security, reliability and efficiency in energy management.
Using the abilities of smart systems to monitoring the energy demand and the energy
production of other units can play a vital role in what regards the unit commitment of
TU. Particularly, monitoring and high quality real-time data of the exploitation of
Rnew energy sources, namely WP, that usually requires a certain amount of spinning
reserve due to their intermittent nature may represent additional information at the
moment of unit commitment of TU. With this information, the Wind-Ther Power
Producer (WTPP) can make a more accurate decision concerning the participation in
EMars and therefore foremost revenue [10,11]. Also, benefits of environmental are
predictable with the increase in the capability of discovery offers able to be satisfied
with a high level of being pleased and less needed of spinning reserve, less TU are
needed and less nonrenewable fuel is used.

## 3 State of the Art

For Ther conversion of energy into electricity, several methods of optimization to
resolve the problem of unit commitment (UC) have been used in the literature,
including a technique of primacies list, classical mathematical programming
techniques, like Lagrangian Relaxation (LR) and dynamic programming (DP) and
more newly artificial intelligence (AI) techniques [12]. Although, requiring small
computation time and easy to implement, the priority list technique does not
guarantee an opportune resolution near the global optimal one, which implies an
operation of higher cost [13,14]. DP methods are flexible but these methods are
characterized by a known limitation by the "curse of dimensionality". Although the
LR can overcome the previous limitation, does not necessarily lead to a viable
resolution, implying further processing for satisfying the infringed constraints in order
to find a viable resolution, which does not guarantee solution optimal. Although, AI
techniques based on simulating annealing and ANN have been applied, the major
limitation of the AI techniques concerning with the possibility to obtain a resolution near the global optimum one is a disadvantage. The MILP method has been useful with success for solving the problem of UC [15]. MILP is suitable for the formulation of bidding strategies due to its rigorousness and extensive capability of modeling [16]. WPP usually have significant difficulties to predict their power output accurately. In addition, WPP have to face Uncer on EPr. These Uncer have to be expediently considered, i.e., treated into the variables of the problems [17] to be addressed by a WPP in order to know how much to produce and the price for bidding. The technical literature presents methods for WP bidding strategies solving using different approaches: the first one is the use of WP with technologies of storage of energy [18]; the use of economic options as a tool for WPP to hedge against WP Uncer [19]; another approach is the design of Stoc models in order to obtain OBS for WPP participating in an EMar [20], without the aforementioned policies. The 3rd line of action is a Stoc formulation explicitly modelling the Uncer faced by a WPP [21], using indeterminate measures and an established of scenarios built by WP forecast and market-clearing EPr forecast [22] requests.

Hence, this paper provides an effective approach based on Stoc MILP to find out the optimal bidding strategies of a single entity having to manage a coordinated wind-Ther system, so as to maximize the expected revenue in the Iberian day-ahead EMar.

# 4 Problem Formulation

## WPP

Considering the variability and intermittent nature of WP the physical delivering usually differs from the offer submitted by WPP to the DAM. The revenue $RV_h$ of a WPP proposing a power of $E_h^{offer}$, but actually producing $E_h^{act}$ for period $h$ is stated as:

$$RV_h = \pi_h^D E_h^{offer} + IC_h \quad (1)$$

In (1), the DAM price is $\pi_h^D$, the imbalance (Imb) cost is $IC_h$. The total Dev for period $h$ is stated as:

$$D_h = E_h^{act} - E_h^{offer} \quad (2)$$

The price that WPP will pay for excess of production is $\pi_h^+$, the price to be charged for deficit of production is $\pi_t^-$. The Imb prices can be given by means of price ratios stated as:

$$pr_h^+ = \frac{\pi_h^+}{\pi_h^D}, pr_h^+ \le 1 \quad \text{and} \quad pr_h^- = \frac{\pi_h^-}{\pi_h^D}, pr_h^- \ge 1 \quad (3)$$

In (3), the $\pi_h^+$ is never greater than 1. The $\pi_t^-$ is never lower than 1.

Ther power producer

The operating cost, $T_{sih}$, for a TU can is stated as:
$$T_{sih} = B_i b_{sih} + g_{sih} + u_{sih} + A_i z_{sih} \quad \forall s, \forall i, \forall h \quad (4)$$

In (4), the fixed production cost is $B_i$, the added variable cost is $g_{sih}$, the ST-UP and shut-down (Sh-Down) costs are $u_{sih}$ and $A_i$, of the unit. The last three costs are in general described by nonlinear function (Func) and worse than that some of the functions are non-convex and non-differentiable functions, but some kind of smoothness is expected and required to use MILP, for instance, as being sub differentiable functions.

The ST-UP and Sh-Down costs of units in (4) are considered to be such that is possible to approximate those Func by a piecewise linear. Hence, the $g_{sih}$, is:

$$g_{sih} = \sum_{l=1}^{L} T_{i}^{l} \Delta_{sih}^{l} \quad \forall s, \forall i, \forall h \quad (5)$$

$$E_{sih} = E_{i}^{\min} b_{sih} + \sum_{l=1}^{L} \Delta_{sih}^{l} \quad \forall s, \forall i, \forall h \quad (6)$$

$$(M_i^1 - E_i^{\min}) j_{sih}^l \le \Delta_{sih}^l \quad \forall s, \forall i, \forall h \quad (7)$$

$$\Delta_{sih}^{l} \le (M_i^1 - E_i^{\min}) b_{sih} \quad \forall s, \forall i, \forall h \quad (8)$$

$$(M_i^l - M_i^{l-1}) j_{sih}^l \le \Delta_{sih}^l \quad \forall s, \forall i, \forall h, \forall l = 2, \dots, L-1 \quad (9)$$

$$\Delta_{sih}^{l} \le (M_i^l - M_i^{l-1}) j_{sih}^{l-1} \quad \forall s, \forall i, \forall h, \forall l = 2, \dots, L-1 \quad (10)$$

$$0 \le \Delta_{sih}^{L} \le (E_{i}^{\max} - M_{sih}^{L-1}) j_{sih}^{L-1} \quad \forall s, \forall i, \forall h \quad (11)$$

In (5), the slope of each segment is $T_i^l$, the segment power is $\Delta_{sih}^l$. In (6), the binary variable $b_{sih}$ guarantee that the power production is equal to 0 if the unit is in the state offline. In (7), if the binary variable $j_{sih}^l$ has a null value, then the segment power $\Delta_{sih}^l$ can be lower than the segment $l$ maximum power (MaxPow); otherwise and in conjunction with (8), if the unit is in the state on, then $\Delta_{sih}^l$ is equal to the segment $l$ MaxPow. In (9), if the binary variable $j_{sih}^l$ has a null value, then the segment power $\Delta_{sih}^l$ can be lower than the segment $l$ MaxPow; otherwise and in conjunction with (10), if the unit is in the state on, then $\Delta_{sih}^l$ is equal to the segment $l$ MaxPow. The exponential nature of the ST-UP costs functions, $u_{sih}$ is approached by a linear formulation [21] is:

$$u_{sih} \ge K_i^\alpha \left( b_{sih} - \sum_{r=1}^{\alpha} b_{sih-r} \right) \quad \forall s, \forall i, \forall h \quad (12)$$

The constraints to limit the power produced by the unit are:

$$E_i^{\min} b_{sih} \le E_{sih} \le E_{sih}^{\max} \quad \forall s, \forall i, \forall h \quad (13)$$

$$E_{sih}^{\max} \le E_i^{\max} (b_{sih} - z_{sih+1}) + SD z_{sih+1} \quad \forall s, \forall i, \forall h \quad (14)$$
$$E_{si h}^{\max} \leq E_{si h-1}^{\max} + RU b_{si h-1} + SU y_{si h} \quad \forall s, \forall i, \forall h \quad (15)$$

$$E_{si h-1} - E_{si h} \leq RD b_{si h} + SD z_{si h} \quad \forall s, \forall i, \forall h \quad (16)$$

In (13) and (14), the upper bound of $E_{si h}^{\max}$ is established, which is the maximum available power of the unit. The minimum down time (MDT) constraint is imposed by a formulation:

$$\sum_{h=1}^{F_i} b_{si h} = 0 \quad \forall s, \forall i \quad (17)$$

$$\sum_{h=k}^{k+DH_i-1} (1-b_{si h}) \geq DH_i z_{si h} \quad \forall s, \forall i, \forall k = F_i + 1 \dots H - DH_i + 1 \quad (18)$$

$$\sum_{h=k}^{H} (1-b_{si h} - z_{si h}) \geq 0 \quad \forall s, \forall i, \forall k = H - DH_i + 2 \dots H \quad (19)$$

$$F_i = \min\{H, (DH_i - t_{si 0})(1 - b_{si 0})\}$$

The MUT constraint is also imposed:

$$\sum_{h=1}^{N_i} (1 - b_{si h}) = 0 \quad \forall s, \forall i \quad (20)$$

$$\sum_{h=k}^{k+UH_i-1} b_{si h} \geq UH_i y_{si h} \quad \forall s, \forall i, \forall k = N_i + 1 \dots H - UH_i + 1 \quad (21)$$

$$\sum_{h=k}^{H} (b_{si h} - z_{si h}) \geq 0 \quad \forall s, \forall i, \forall k = H - UH_i + 2 \dots H \quad (22)$$

$$N_i = \min\{H, (UH_i - U_{si 0})b_{si 0}\}$$

The relation between the binary variables to identify start-up, shutdown and forbidden operating zones is:

$$y_{si h} - z_{si h} = t_{si h} - b_{si h-1} \quad \forall s, \forall i, \forall h \quad (23)$$

$$y_{si h} + z_{si h} \leq 1 \quad \forall s, \forall i, \forall h \quad (24)$$

The total power produced by the TU is:

$$E_{st}^{g} = \sum_{i=1}^{I} E_{si h} \quad \forall s, \forall h \quad (25)$$

Objective function: The total offer is:

$$E_{sh}^{\text{offer}} = E_{sh}^{th} + E_{sh}^{D} \quad \forall s, \forall h \quad (26)$$

The physical delivering is:

$$E_{sh}^{\text{act}} = E_{sh}^{g} + E_{sh}^{\omega d} \quad \forall s, \forall h \quad (27)$$

In (27), $E_{sh}^{g}$ is the physical delivering by the TU and $E_{sh}^{\omega d}$ is the physical delivering by the wind farm for scenario s. The expected revenue of the GNCO is:

$$\sum_{s=1}^{N_c} \sum_{h=1}^{N_i} p_s \left[ \left( \pi_{sh}^{D} E_{sh}^{\text{offer}} + \pi_{sh}^{D} p r_{sh}^{+} D_{sh}^{+} - \pi_{sh}^{D} p r_{sh}^{-} D_{sh}^{-} \right) - \sum_{i=1}^{I} T_{si h} \right] \quad \forall s, \forall h \quad (28)$$
Subject to:

$$0 \le E_{sh}^{offer} \le E_{sh}^{M} \quad \forall s, \forall h \quad (29)$$

$$D_{hs} = (E_{sh}^{act} - E_{sh}^{offer}) \quad \forall s, \forall h \quad (30)$$

$$D_{hs} = D_{hs}^{+} - D_{hs}^{-} \quad \forall s, \forall h \quad (31)$$

$$0 \le D_{hs}^{+} \le E_{hs} x_s \quad \forall \omega, \forall h \quad (32)$$

The maximum Ther generation is:

$$E_{sh}^{M} = \sum_{i=1}^{I} E_{si h}^{\max} + E^{W \max} \quad \forall s, \forall h \quad (33)$$

An additional constraint for (28) appears:

$$(E_{s h}^{offer} - E_{s' h}^{offer})(\pi_{s h}^{D} - \pi_{s' h}^{D}) \ge 0 \quad \forall s, s', \forall h \quad (34)$$

## 5 Case Study

The case study is from a GNCO with a WTPP, with 1440 MW of installed capacity. The used data is available in [6]. The energy prices are from the Iberic Market of electricity and available in [23], considering 10 days of June. The EPr and the energy generated from wind are displayed in Fig. 1.

<figure><img src="image_2.png" /><figcaption>Fig 1. Market Iberic: June 2014; left: EPr, energy from wind: right.</figcaption></figure>

The energy generated is obtained using the total energy generated from the wind farm having 360 MW of rated power. The expected revenue for CoConf and unCoConf are displayed in Table 1.

<table><caption>Table 1. Expected revenue for CoConf and uncoordinated configurations</caption><thead><tr><td>Case</td><td>Expected revenue</td></tr></thead><tbody><tr><td>Wind uncoordinated (€)</td><td>119 200</td></tr><tr><td>Ther uncoordinated (€)</td><td>516 848</td></tr><tr><td>Coordinated Wind and Ther (€)</td><td>642 326</td></tr><tr><td>Gain (%)</td><td>0,99</td></tr></tbody></table>
The non-decreasing energy bid for the unCoConf approach is displayed in Fig. 2.

<figure><img src="image_3.png" /><figcaption>Fig 2. Bids of energy.</figcaption></figure>

In Fig. 2 the CoConf permits for a Min value of offered power upper than the one offered in the unCoConf and permits for a lesser price of the offering, which is a possible operation benefit.

**6 Conclusion**

Smart Systems can play an important role for a Ther and WP producer since the operation till the bidding in day-ahead EMars. The ability to provide real-time data from the wind production may result in foremost decisions for the decision-maker and therefore higher revenues. As result of the proposed approach for uncoordinated and coordinated operations optimal schedule of the TU and the short-term bidding strategies are obtained. The presented approach is appropriate for the GNCO involvement with TU and a wind farm. The offer coordinated of Ther with WP power permits providing foremost outcomes than the sum of the lonely offers. The Uncer are modelled using established scenarios for the prices of the energy and power production. In the literature of all trading problems and management involving production by wind prove to be optimization problems under Uncer.

**Acknowledgments.** Portuguese Funds partially support this work through the FCT-Foundation for Science and Technology, LAETA 2015/2020, UID-EMS-50022-2013.

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