Testing the Procedure for Optimization of Cascaded Hydropower Plants and Wind Power Plants Using the IEEE 14 Bus System

Abstract

In real-life power system, optimization methods are used on a daily basis to obtain optimal dispatch for various types of power plants. In this paper, the existing optimization method is used to maximize profit of three cascaded Hydro Power Plants (HPP) and to co-ordinate HPPs dispatch according to Wind Power Plants (WPP) production forecast. A two-stage optimization method was used with first stage design to obtain optimal HPPs dispatch plan and to maximize profit by selling electricity on a day-ahead market. First optimization stage is defined as mixed-integer liner programming (MILP) problem. Obtained dispatch plan together with the WPPs production forecast is used to check power flows in the transmission network and if line congestion occurs, second optimization stage is needed, and it is defined as quadratic programming (QP) problem. The objective of second optimization stage is to find re-dispatch plan that would be minimal deviation from an initial dispatch but it would eliminate line congestions. Proposed optimization strategy is tested using IEEE 14 Bus Test System and the results are presented in this paper.

1 Introduction

The problem of finding optimal short-term operation schedule (dispatch) of cascaded hydropower plants (HPPs) is well studied in the research literature. For example, good optimization models can be found in [1]. Due to electricity market reform, researchers developed optimization models that take into consideration variable electricity market prices [2, 3]. The main challenge regarding cascading HPPs is how to properly model nonlinear discharge characteristics of the HPP as well as take into consideration hydrological couplings between the HPPs in the cascade [4]. The situation becomes more complex in the modern power system with connected wind power plants (WPPs). Uncertainty of WPP production can be captured with stochastic programming and good examples are [5, 6]. Paper [7] presents the two-stage optimization model for optimization of cascaded HPPs in the presence of the WPPs. In the first stage, an optimal day-ahead dispatch plan of the HPPs is obtained and it maximizes profit from selling energy to the day-ahead electricity market [7]. The second optimization stage is started only when the optimal dispatch plan of the HPPs together with the forecasted WPPs production causes congestion in the transmission network. As it is explained in [7]: “The objective of the second optimization stage is the minimization of the re-dispatching of cascaded hydropower plants in order to avoid possible congestion.”

In this paper, IEEE 14-bus Test system is used to evaluate the optimization procedure developed in [7]. The original test system [8] consists of 5 generators, 11 loads, 16 lines, and 3 transformers. It is used as a reference system for many case studies in its original or modified configuration. Transient stability after three-phase faults with different locations and fault clearing time is analyzed in paper [9] using IEEE 14 Bus System. In [10] IEEE 14 bus system was used to test Static Var Compensator (SVC) integrated into the system at the weakest node. The load was changed in the range of 5–45% in order to test functionality and justification of SVC installation. Due to its complexity 14 bus system is used often in transient stability studies, for example [11, 12]. Beside above-mentioned application, 14 bus system can also be used for system optimization and minimizing production costs while respecting system constraints and managing congestions [13, 14], minimizing power losses [15] and other possibilities regarding electricity market (for example calculating locational marginal price (LMP) of electricity market [16]).

Paper has a following structure: in Sect. 2 optimization procedure is described and presented with block diagram. Model and modifications made on a IEEE 14-bus Test System along with data regarding WPPs and HPPs are given in Sect. 3. Optimization results, given graphicly and in tables are given in Sect. 4. At the end of the paper short conclusion and comments are given.

2 Used Optimization Procedure

The used optimization procedure is initially introduced in [7] and it is developed for minimization of hydropower plants re-dispatching due to wind power plants output uncertainty. The stochastic nature of wind power can lead to re-dispatching of cascaded HPP’s, in case of a surplus in power production, in order to alleviate transmission network congestion. The object of the proposed optimization is to minimize the given effect with two optimization stages. The optimization is conducted day-ahead (d − 1) on a daily basis and split into hourly segments. The necessary input parameters, given in the block diagram in Fig. 1, are the HPP’s technical data, data from the previous optimization period, hydrological data, and local water inflow forecast to the river and reservoirs supplying the HPP’s as well as the forecast for the electricity market, both the day-ahead and future market prices. The data from the previous optimization period provides the starting reservoir levels as well as the last known water discharge which can be necessary due to the water delay. Hydrological data provides the water inflow from upstream to downstream reservoirs as well as the reservoir levels when the optimization period ends.

The first optimization stage uses mixed-integer linear programming (MILP) and calculates the day-ahead HPP dispatch plans (hourly discharge or production plan) and the day-ahead reservoir management plans (hourly reservoir water levels). Then, the transmission system operator (TSO) uses the calculated HPP dispatch plans and the hourly WPP forecast in order to determine the day-ahead hourly transmission network power flow. If the results do not show any congestions in the transmission network, the optimization ends as there is no need for further optimization in the second stage. However, if the hourly power flow shows congestions, the second optimization stage is needed.

The second stage inputs are all outputs from the first stage, hourly wind forecast with the given forecast error, congested transmission line capacities, and power transfer distribution factors (PTDF) for the congested lines. The second optimization stage uses quadratic programming (QP), in combination with chance-constrained programming (CC), and gives the changed hourly dispatch planes of the HPPs and the changed hourly reservoir management plans so that no congestions occur in the transmission network. The TSO once again calculates the hourly power flow with the newly optimized data and if there still are congestions in the network, the second stage optimization is once again started. Once there are no congestions, the optimization ends. The second stage optimization ensures that re-dispatching is minimized so that HPPs water management plan changes minimally.

Fig. 1.

Block diagram of the used optimization procedure

The objective function is the first optimization stage is defined as maximization of the expression (1):

$$ \begin{gathered} \mathop \sum \limits_{t = 1}^{24} \left[ {\rho \left( t \right) \cdot \mathop \sum \limits_{i = 1}^{{n_{i} }} \mathop \sum \limits_{a = 1}^{{n_{ai} }} \left[ {z_{a} \left( {i,a,t} \right) \cdot \mu_{amin} \left( {i,a} \right) \cdot Q_{amin} \left( {i,a} \right) + \mathop \sum \limits_{j = 1}^{{n_{jai} }} \mu_{as} \left( {i,a,j} \right) \cdot Q_{as} \left( {i,a,j,t} \right)} \right]} \right] \hfill \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; + \;\rho_{f} \cdot \mathop \sum \limits_{i}^{{n_{i} }} \left[ {V\left( {i,t = 24} \right) \cdot \mathop \sum \limits_{{i \in {\Gamma }_{i} }} \mu_{f} \left( i \right)} \right] - C \cdot S\left( {i,t} \right) \hfill \\ \end{gathered} $$

(1)

Part I of the objective function represents the income from selling energy to the day-ahead electricity market where:

t :

- Time interval in the optimization period (discrete variable) [hour]

i :

- HPP index where i = 1, …, ni; ni – total number of HPPs

a :

- Production unit index where a = i, …, n_ai; n_ai – total number of production units in the HPP i

j :

- Index of the linear segment in the discharge characteristic of the production unit, j = 1, …, n_jai; n_jai - total number of linear segments in the discharge characteristic of the production unit a in the HPP i

$\rho \left( t \right)$:

- Forecasted day-ahead electricity market price [€/MWh]

$z_{a} \left( {i,a,t} \right)$:

- An integer variable that is used to model forbidden zones of the discharge characteristic of production unit a in HPP i

$\mu_{amin}$:

- Production equivalent that describes a minimum discharge segment of production unit a in HPP i [MWh/Heh].

$Q_{amin}$:

- Minimal discharge of production unit a in HPP i [HE]

$\mu_{as} \left( {i,a,j} \right)$:

- Production equivalent of the linear segment j in the discharge characteristic of production unit a in HPP i [MWh/Heh]

$Q_{as} \left( {i,a,j,t} \right)$:

- Discharge of the linear segment j in the discharge characteristic of production unit a in HPP i during the hour t [HE]

Part II is the expected income from the usage of water that remained in reservoirs at the end of the optimization period, with following parameters:

$\rho_{f}$:

- Forecasted average future electricity price (the average price for the next optimization period - next day) [€/MWh]

$V\left( {i,t} \right)$:

- Volume of water in the reservoir of HPP i at the end of hour t [HE]

$\mu_{f} \left( i \right)$:

- Production equivalent of HPP i that is used for calculation of HPP production in the future optimization period [MWh/HEh]

Part III represents the penalty cost for spillage of water where:

C:

- Artificial penalty cost associated with water spillage [€/MWh]

$S\left( {i,t} \right)$:

- The total spillage of water by HPP i during hour t [HE]

Optimization constraints are defined as:

$$ 0\,\, \le \,\,Q_{as} \left( {i,a,j,t} \right)\,\, \le \,\,Q_{{as,{\text{max}}}} \left( {i,a,j} \right) $$

(2)

$$ z_{a} \left( {i,a,t} \right) \in \left\{ {0,1} \right\} $$

(3)

Where

$Q_{{as,{\varvec{max}}}}$:

- Maximal discharge of the linear segment j in the discharge characteristic of production unit a in HPP i. The unit is HE

Discharge constraints can be divided on constraints regarding discharge of the generation units and constraints which take in regard output power. Total discharge of the production units and the discharge through the individual discharge are described with Eqs. (4) and (5). Output electrical power and discharge are described with Eqs. (6) and (7):

$$ Q_{a} \left( {i,a,t} \right)\, = \,z_{a} \left( {i,a,t} \right) \cdot \,Q_{{a{\text{min}}}} \left( {i,a} \right)\, + \,\,\,\mathop \sum \limits_{j\, = \,1}^{{n_{jai} }} Q_{as} \left( j \right) $$

(4)

$$ Q\left( {i,t} \right)\, = \,\mathop \sum \limits_{a\, = \,1}^{{n_{ai} }} Q_{a} \left( {i,a,t} \right) $$

(5)

$$ P_{a} \left( {i,a,t} \right) = z_{a} \left( {i,a,t} \right) \cdot \mu_{amin} \left( {i,a} \right) \cdot Q_{amin} \left( {i,a} \right) + \mathop \sum \limits_{j = 1}^{{n_{jai} }} \mu_{as} \left( {i,a,j} \right) \cdot Q_{as} \left( {i,a,j,t} \right) $$

(6)

$$ P\,\left( {i,t} \right) = \,\mathop \sum \limits_{a = 1}^{{n_{a} }} P_{a} \left( {i,a,t} \right) $$

(7)

Where parameters:

$Q_{a} \left( {i,a,t} \right)$:

- Total discharge through production unit a in HPP i during hour t [HE]

$Q_{{a{\text{min}}}} \left( {i,a} \right)$:

- Minimal discharge of production unit a in HPP i [HE]

$\overline{Q}_{as} \left( j \right)$:

- Maximal discharge through linear segment j of production unit a [HE]

$Q\left( {i,t} \right)$:

- Total discharge of HPP i during hour t [HE]

$P_{a} \left( {i,a,t} \right)$:

- Output electric power of the unit a in the HPP i during hour t [MW]

$Q_{as} \left( {i,a,j,t} \right)$:

- Discharge of the linear segment j in the discharge characteristic of production unit a in HPP i during the hour t [HE]

$P\left( {i,t} \right)$:

- Total output power of HPP i during hour t [MW]

Hydrological constraints of the HPP reservoirs are defined as follows:

$$ V_{{{\text{min}}}} \left( i \right)\,\, \le \,\,V\left( {i,t} \right)\,\, \le \,\,V_{{{\text{max}}}} \left( i \right); $$

(8)

$$ Q\left( {i,t} \right) = \mathop \sum \limits_{a = 1}^{{n_{ai} }} Q_{a} \left( {i,a,t} \right); $$

(9)

$$ 0 \le S\left( {i,t} \right). $$

(10)

Where:

$V\left( {i,t} \right)$:

- Volume of water in the reservoir of HPP i at the end of hour t [HE]

$V_{{{\text{min}}}} \left( i \right)$:

- Minimal volume of the reservoir HPP i [HE]

$V_{{{\text{max}}}} \left( i \right)$:

- Maximal volume of the reservoir HPP i [HE]

As stated, water reservoirs located at the same river are coupled and coupling is expressed with equation:

$$ V\left( {i,t} \right)\, = \,V\left( {i,t - 1} \right)\, - \,Q\left( {i,t} \right) - S\left( {i,t} \right) + Q\left( {i - 1,t - \tau } \right) + S\left( {i - 1,t - \tau } \right) + w\left( {i,t} \right) $$

(11)

$\tau$:

- Time delay

$w\left( {i,t} \right)$:

- Local inflow in the reservoir i during hour t [HE]

Constraint expressed with Eq. (12) needs to be fullfiled if the owner of the HPPs contracted a long-term bilateral contract:

$$ P\left( {i,t} \right) \ge P_{f} \left( {i,t} \right). $$

(12)

$P_{f} \left( {i,t} \right)$:

- The amount of electric energy in every hour t for every HPP i that is contracted by bilateral contract, expressed as constant power per hour [MW]

After first optimization stage is carried out, if there are congestions, second optimization stage needs to be carried out. Objective function of the second optimization stage is the minimization of the equation:

$$ \mathop \sum \limits_{t = 1}^{24} \mathop \sum \limits_{i = 1}^{{n_{i} }} \left[ {D\left( {i,t} \right) - \mathop \sum \limits_{a = 1}^{{n_{ai} }} \mu_{a} \cdot Q_{a} \left( {i,a,t} \right)} \right]^{2} + C \cdot S\left( {i,t} \right) $$

(13)

where first part of the equation is minimization of the square deviation of old dispatch plan and the new dispatch plan and second part of the equation is to minimize the possible spillage. In this equation following parameters are used:

$D\left( {i,t} \right)$:

- Output active power of HPP i during hour t, obtained in the first optimization stage [MWh/h]

$\mu_{a} \left( {i,a} \right)$:

- Production equivalent of the discharge characteristic around the operating point for production unit a in the HPP i, obtained in the first optimization stage [MWh/HEh]

Second optimization stage uses constraints (2) and (8–12). It is also assumed that result regarding water volume at the end of the last hour is optimal but if re-dispatch is needed, this value can be altered as long as these alterations remain minimal. Minimal and maximum value (Vsetmin and Vsetmax) are defined by user:

$$ V_{{set\,{\text{min}}}} \left( i \right)\,\, \le \,\,V\left( {i,t = 24} \right)\,\, \le \,\,V_{{set\,{\text{max}}}} \left( i \right) $$

(14)

In second optimization stage, it is necessary to obtain power flows. If there is congestion in transmission line, re-dispatch is necessary. Available transfer capacity (ATC) of the lines need to be calculated and included in optimization process because active power flow beyond ATC results in HPP output power reduction. The connection between ATC and HPP output power is defined as

$$ \mathop \sum \limits_{v = 1}^{{n_{v\,} }} P_{v} \left( {v,t} \right) \cdot {\text{PTDF}}_{WPP\,v,k} \,\, \ge \,\,ATC_{k} \left( t \right) $$

(15)

$P_{v} \left( {v,t} \right)$:

- Forecasted production of WPP v in the hour t [MW]

v:

- WPP index, v = 1, …, n_v, where n_v is the total number of WPPs

PTDF_WPPv,k:

- PTDF which determines the change of active power flow of transmission line k due to active power change of the WPP i

ATCk(t):

- The available transfer capacity of the transmission line k in the hour t, [MWh/h]

Additional constraint regarding the HPP re-dispatch is defined as

$$ \mathop \sum \limits_{v = 1}^{{n_{v\,} }} P_{v} \left( {v,t} \right) \cdot {\text{PTDF}}_{WPP\,v,k} \,\, - \,\mathop \sum \limits_{i = 1}^{{n_{i} }} \left[ {\left( {D\left( {i,t} \right) - P\left( {i,t} \right)} \right) \cdot {\text{PTDF}}_{HPP\,i,k} } \right]\,\, \le \,\,ATC_{k} \left( t \right) $$

(16)

where

PTDF_HPPv,k:

- PTDF which determines the change of active power flow of transmission line k due to active power change of the HPP i

Re-dispatch and spillage need to be minimized and this is obtained by objective function (13) in case when WPP production causes congestion. WPP production value is deterministic which means that also constraint (16) is deterministic. On the other hand, wind speed is stochastic so Eq. (16) needs to be modified as

$$ \, - ATC_{k} \left( t \right)\, - \,\mathop \sum \limits_{i = 1}^{{n_{i} }} \left[ {\left( {D\left( {i,t} \right) - P\left( {i,t} \right)} \right) \cdot {\text{PTDF}}_{HPP\,i,k} } \right]\,\, \le \,\,\mathop \sum \limits_{v = 1}^{{n_{v\,} }} P_{v,cr} \left( {v,t} \right) \cdot {\text{PTDF}}_{WPP\,v,k} $$

(17)

where

P_v,cr(v, t):

- Critical value of WPP power plant [MW].

3 Modifications of IEEE 14 Bus Test System

For the purpose of the optimization procedure testing, the original IEEE 14 Bus Test System [8] is modified and modeled in Power World Simulator [17]. The aforementioned model was modified for the sake of this paper and the following changes were made:

• Synchronous compensators connected to buses 3, 6, and 8 were replaced with generators;

• Generators connected to buses 2, 3, and 6 represent HPP generators;

• A generator connected to bus 8 represents a wind farm;

• A generator connected to bus 1 replaces the neighboring power system in which energy is exported through interconnected lines 1–2 and 1–5;

• Assumed peak load in a 24-h period are shown in Table 1 and assumed 24-h wind speed forecast together with forecasting error and forecasted WPP output power are shown in Table 2;

• Loads of the remaining hours in the observed day were recalculated according to the daily load diagram of the Croatian power system taken on 27.11.2012. [18] and they are shown in Fig. 2;

The day-ahead electricity price was taken from the EEX for 27.11.2012. and is shown in Fig. 2.

Fig. 2.

Electricity prices on the day-ahead market and load diagram of the test system

The assumed electricity price, at which it will be possible to sell electricity in the future, is 50 €/MWh. The hydrological system is modeled containing 3 cascaded accumulation HPPs with its generators connected to buses 2, 3, and 6. The first one in line (HPP 1) is located at bus 6 and it has a large accumulation of seasonal character. The next in line (HPP 2) is located at bus 3 and has a smaller capacity accumulation (compensational pool for daily accumulation). The last one (HPP 3) is located at bus 2 and is the largest one of the 3 with its accumulation being also daily. Each HPP consists of 2 generators. The technical data of the HPPS are given in Table 3. The unit HE (hour equivalent) is used for water discharge as well as water volume. It corresponds to the water flow of 1 m³/s during one hour.

The assumed wind park consists of 42 wind turbines. The rated power of each turbine is 1.5 MW so the WPPs total rated power is 63 MW. The IEEE.14 Bus Test System with the aforementioned modifications is shown in Fig. 3.

Table 1. Active and reactive bus load

Table 2. Wind speed forecast and WPP output power for observed planning period

Fig. 3.

Modified IEEE 14 bus test system

Table 3. HPPs technical data

4 Results of the Optimization Procedure

4.1 First Optimization Stage

The first optimization stage gives optimal dispatch for HPPs in order to achieve maximal profit by selling electricity on the day-ahead market and from future selling of hydro energy saved in reservoirs at the end of the planning period. Key parameters which define the HPP dispatch plan are forecasted electricity price on the day-ahead market and forecasted future electricity price at the end of the planning period.

Obtained HPPs production plan is correlated with electricity prices and is given in Fig. 4. During nighttime planned production is equal to planned bilateral contracts and HPPs do not participate in the day-ahead market; instead, they operate with their minimum capacity. During the day, when electricity prices are higher, planned production is equal to rated power.

Fig. 4.

Calculated day-ahead dispatch plan of the HPPs during the planning period

Water volume in reservoirs also changes as is shown in Fig. 5. HPP1 reservoir dictates discharge for all downstream reservoirs, since the incoming inflow is low, the volume of HPP1 reservoir constantly decreases at a rate of 3500 HE (12600000 m³). In order to maximize profit, all of the water needs to be used. HPP2 and HPP3 volumes are in margins between the minimum and maximum volume with HPP2’s tendency to increase volume in the last 4 h of the planning period. This increase is due to electricity prices which are lower than the presumed future price (50 €/MWh). Water stored in HPP2 reservoir can be used in both HPP2 and HPP3 which leads to bigger profit from saving water than selling it on the day-ahead market during the last 4 h.

Fig. 5.

Calculated water volume in reservoirs at the end of each hour

4.2 Second Optimization Stage

In the first optimization stage optimal HPPs dispatch plan is obtained and these data, along with forecasted production from WPPs, are used as input data for second optimization stage. Using the aforementioned data, power flows are calculated in order to check if any congested (critical) lines exist. Power flow results show that line 1–2 could be congested and it is considered to be critical.

Fig. 6.

Power flows on the critical line

Power flows on a critical line during the planning period are shown in Fig. 6. Transfer capacity of line 1–2 is 200 MWh/h with the red line representing this transfer capacity. All values which are above the transfer capacity are considered to be congestions.

Since congestions are possible, the second optimization stage is needed to re-dispatch HPPs output power and remove congestion. Figure 7 shows the re-dispatching of HPPs power during the planning period. The first hour of congestion is hour 11 but due to the time delay of water, HPP1 is the first one to reduce its output power one hour before (hour 10). The biggest change is in the output power of HPP 3. This HPP has the biggest contribution in power flow changes of the critical line since it is nearest to it.

Fig. 7.

HPPs output power re-dispatching

As in the previous optimization stage, the reservoir volumes are also calculated, and they are shown in Fig. 8. In this case, volumes are similar to the first optimization stage. The goal of the second optimization stage is to find optimal dispatch to ensure minimal offset from the reservoir volume values obtained during the first optimization stage.

Fig. 8.

Water volume in reservoirs at the end of each hour obtained in the second optimization stage

Results obtained in the second optimization stage and re-dispatched HPPs are used for new power flow calculations in order to eliminate line congestions. Figure 9 shows a comparison of line active power flow before and after re-dispatching was made. Congestions are eliminated and critical line 1–2 is uncongested during the whole planning period and optimization process can end.

Fig. 9.

Active power flow comparison after the first and second optimization stage

5 Conclusion

Optimization of cascaded HPPs dispatch plan is quite challenging problem especially in modern power systems where WPPs are connected. Two-stage optimization procedure proposed in [7] is tested using IEEE 14 Bus Test Case. In first optimization stage optimal dispatch and accumulation plan was obtained and since congestions occurred it was necessary to perform second optimization stage. As the results showed, if there are congestions in the system, optimal plan needs to be modified but these modifications need to be minimal. Parallel to HPPs dispatch plan, accumulation management plan needs to be made. Accumulated water can be used when electricity prices are higher, so it is necessary to make a decision when the right time is to use it. Since WPP production is stochastic, HPPs production needs to be dispatched in a way to eliminate congestions. Results shows that two-stage optimization model is useful tool for alleviating congestion in transmission network that are caused by WPPs uncertainty and second optimization stage ensures that deviations from original HPPs dispatch plan are minimal.